Product sentence examples

product
  • Either way, the words were the product of a kind heart.

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  • Sainte-Claire Deville obtained a grey product, from which, on dissolving out the aluminium with sodium hydroxide, they obtained a crystalline product, which they thought to be a modification of boron, but which was shown later to be a mixture of aluminium borides with more or less carbon.

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  • Good information on a product can mitigate this problem.

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  • Yelp, online product reviews, and trust.

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  • A product of being an only child of aging parents, no doubt.

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  • The total value of the factory product in 1905 was $4,951,964, an increase of 82.3% since 1900.

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  • It is usually made by distilling tartaric acid with potassium bisulphate at about zoo-250° C., the crude product being afterwards fractionated.

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  • Now, we know that the number of electrochemical equivalents electrolysed is proportional to the whole amount of electricity which passed through the circuit, and the product of this by the electromotive force of the battery is the work done by the latter, so that in this case also Joule showed that the heat generated was proportional to the work done.

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  • And every day, their product gets better because it is being fed more data.

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  • the product of the total capacity and the total resistance, both the capacity and the resistance having a retarding effect on the signals.

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  • Flax covers about 160,000 acres, with a product, in fibre, amounting to about 20,000 tons.

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  • Vegetables.Potatoes are not a special product of any region, though grown in great quantities in the Bresse and the Vosges.

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  • The principal product is quinine, the manufacture of which has acq~iired great importance, owing to its use as a specific against malaria.

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  • Donations, bequests and the product of collections in churches.

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  • In 1905 the value of Troy's factory product was $31,860,829.

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  • She continued to the small women.s boutique that stocked every kind of facial and body care product she.d ever heard of—and many she hadn.t.

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  • In their exposition, an historic character is first the product of his time, and his power only the resultant of various forces, and then his power is itself a force producing events.

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  • One could hardly expect that a colourless deity of this description, so completely the product of priestly speculation, could ever have found a place in the hearts of the people generally.

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  • Of this $11,271,708 was the value of collars and cuffs (89.5% of the value of the total American product), an industry which gave employment to 49.3% of the wage-earners in Troy, and paid 42.1% of the wages.

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  • Boron chloride BC1 3 results when amorphous boron is heated in chlorine gas, or more readily, on passing a stream of chlorine over a heated mixture of boron trioxide and charcoal, the volatile product being condensed in a tube surrounded by a freezing mixture.

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  • You're the product of a single mother, probably raised in near-poverty, based on your simple tastes.

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  • The only sound Sarah had made throughout the ordeal was an occasional sniff, as she attempted to wipe away the product of her distress with the back of her hand.

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  • The dark product obtained is washed with water, hydrochloric acid and hydrofluoric acid, and finally calcined again with the oxide or with borax, being protected from air during the operation by a layer of charcoal.

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  • To the extent that I get accurate information from other consumers of the product, I will tend to make better choices.

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  • The chief product is copra.

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  • The chief product of the islands are bananas; the chief export sandal-wood.

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  • The value of the factory product in 1905 was $99,040,676.

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  • This way, as soon as one car makes its drop, the other car is signaled to drop the product at another location and it comes out even.

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  • The product is principally used for industrial purposes, and partly in the preparation of food, but the amount is decreasing.

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  • The product of certain fines.

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  • Innumerable little streams overlap and interlace one with another, exhibiting a sort of hybrid product, which obeys half way the law of currents, and half way that of vegetation.

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  • When you look at a product on one of its web pages, Amazon suggests other products you might like as well.

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  • An ongoing debate is whether a high amount of energy raises a nation or region's gross national product (GNP) or whether rising GNP increases the consumption of energy.

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  • The overall economic output of the planet, GWP (gross world product), will rise dramatically in the years to come, but its distribution will be quite skewed.

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  • There are several old pictures of merit, and the shrine of St Eleuthere, the first bishop of Tournai in the 6th century, is a remarkable product of the silversmith's art.

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  • The value of the product has, however, proportionately increased, so that the sum realized was little less, while less than half the number of men was employed.

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  • The city is a product of the industrial transformation in the southern states since the Civil War.

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  • A larger product of the vine was in the form of raisins and currants, of which quantities were exported to Russia.

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  • Adding to this 1,240,000 of communal and provincial subsidies, the product of the labor of inmates, temporary subscriptions, &c., the net revenue available for charity was, during i88o, 3,860,000.

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  • Closely connected with the manufacture of lumber is the making of paper and wood pulp, centralized at Bellows Falls, with waterpower on the Connecticut river and with the raw materials near; the product was valued in 1905 at $3,831,448.

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  • It is obtained by the oxidation of orthophenylene diamine with ferric chloride; when a mixture of para-aminodimethylaniline and meta-toluylenediamine is oxidized in the cold, toluylene blue, an indamine, being formed as an intermediate product and passing into the red when boiled; and also by the oxidation of dimethylparaphenylene diamine with metatoluylene diamine.

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  • As national income increases in a given country, the size of government as a portion of gross national product (GNP) rises and the range of services people expect the government to offer rises.

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  • Momentum (quantity of motion) is the product of mass and velocity.

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  • Often, a buying decision hinges on a piece of arcane information about a product that is difficult to locate.

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  • In 1900 the Birmingham district produced six-sevenths of the total pig iron exported from the United States, and in 1902 nine-tenths of Alabama's coal, coke and pig iron; in 1905 Jefferson county produced 67.5% of the total iron and steel product of the state, and 62.5% of the pig iron produced by the state.

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  • A salt basin underlies the city, and, next to the lumber industry, the salt industry was the first to be developed, but its importance has dwindled; the product value in 1905 being $20,098 out of $5,620,866 for all factory products.

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  • The silver product from 1879 to 1901 was about $4,154,000.

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  • The value of the factory product for 1905, however, was 3.3% less than that for 1900, though it represented 36.6% of the product of the state as a whole.

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  • According to this view the power of historical personages, represented as the product of many forces, can no longer, it would seem, be regarded as a force that itself produces events.

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  • An important product of oak woods is the bark that from a remote period has been the chief tanning material of Europe.

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  • The product of stone manufactures in 1905 was $9,570,436.

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  • 1 They are particularly important in that they counteracted the popular and interestingly written books of Max Muller: for instance, Muller, like Renan and Wilhelm von Humboldt, regarded language as an innate faculty and Whitney considered it the product of experience and outward circumstance.

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  • Product Product Years.

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  • The conductivity, which varies as the product of n into the mobility, will thus vary inversely as the pressure, and so at 36 kilometres will be one hundred times as large as close to the ground.

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  • The man who breaks the law is himself a product of social evolution and cannot be regarded as solely responsible for his disposition to transgress.

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  • In military affairs the strength of an army is the product of its mass and some unknown x.

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  • The rules then are sine of the middle part = product of tangents of adjacent parts = product of cosines of opposite parts.

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  • The individual is the product of sexual reproduction; a person is an individual of lower rank, which may be produced asexually.

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  • Another objection is advanced from the standpoint of naturalism, which, whether it issues in materialism or not, seeks to explain man as but a product of the process of nature.

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  • The value of Joplin's factory product in 1905 was $3,006,203, an increase of 2 9.3% since 1900.

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  • If you made a product the military could use, government contracts came a-flowin'.

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  • Deidre made a deal with the Dark One, one good enough to bring her soul back from the dead, combine the two Deidres, cure the tumor of one and release the final product from Hell.

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  • Sheep's milk cheese (pecorino) is largely made, but sold as the Roman product.

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  • Salt, &c.Rock-salt is worked chiefly in the department of Meurthe-et-Moselle,which produces more than half the average annual product of salt.

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  • Polyporus igniarius and other species are also used, but yield an inferior product.

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  • The association of this metal with silver in the Broken Hill mines of New South Wales adds very greatly to the value of the product.

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  • In the modern doctrine of evolution the cosmic system appears as a natural product of elementary matter and its laws.

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  • The doctrine of evolution in its finished and definite form is a modern product.

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  • By this writer the world is explained as a product of three principles - dead matter, and two active forces, heat and cold.

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  • Thus man is said to be the highest product of nature, and as such to be dependent on all lower products.

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  • In his Naturgeschichte des Himmels, in which he anticipated the nebular theory afterwards more fully developed by Laplace, Kant sought to explain the genesis of the cosmos as a product of physical forces and laws.

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  • The chief mineral product is the asphalt of the mines of Seyssel on the eastern frontier, besides which potter's clay, building stone, hydraulic lime and cement are produced in the department.

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  • The only product is cocoa-nut oil, of which about 106,000 gallons are annually exported.

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  • The phellogen derives its name from the fact that its external product is the characteristic tissue known as cork.

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  • Substances contained in the Protoplastn.Starch may be found in the chlorophyll bodies in the form of minute granules as the first visible product of the assimilation of carbon dioxide, and it occurs in large quantities as a reserve food material in the cells of various parts of plants.

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  • Even in those cases where the cilia band, which is the product of the centrosome-like body or blepharoplast, enters the ovum, as in Zamia (c in fig.

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  • The total value of the factory product in 1905 was $12,362,600, the city ranking third in product value among the cities of the state.

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  • Columbus is near the Ohio coal and iron-fields, and has an extensive trade in coal, but its largest industrial interests are in manufactures, among which the more important are foundry and machine-shop products (1905 value, $6,259,579); boots and shoes (1905 value, $5,425,087, being more than one-sixtieth of the total product value of the boot and shoe industry in the United States, and being an increase from $359,000 in 1890); patent medicines and compounds (1905 value, $3,214,096); carriages and wagons (1905 value, $2,197,960); malt liquors (1905 value, $2,133,955); iron and steel; regalia and society emblems; steam-railway cars, construction and repairing; and oleo-margarine.

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  • In 1905 Portland was the first manufacturing city of the state, with a factory product valued at $9,132,801 (as against $8,527,649 for Lewiston, which outranked Portland in 1900); here are foundries and machine-shops, planing-mills, car and railway repair shops, packing and canning establishments - probably the first Indian corn canned in the United States was canned near Portland in 1840 - potteries, and factories for making boots, shoes, clothing, matches, screens, sleighs, carriages, cosmetics, &c. Shipbuilding and fishing are important industries.

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  • There is a well-marked processus lateralis anterior (the right and left together equivalent to the mammalian manubrium), which is the product of two or three ribs, the dorsal parts of which reduced ribs remain as cervico-dorsal ribs.

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  • The total value of the city's factory product in 1905 was $1,747,338, being 31.6% more than in 1900.

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  • By conducting the distillation slowly, so that the temperature within the chamber remains at a sufficiently low degree, it is possible to obtain the whole of the product in the form of "flowers."

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  • The colloidal sulphur, Ss, described by Debus as a product of the interaction of sulphuretted hydrogen and sulphur dioxide in aqueous solution, is regarded by Spring (Rec. tra y.

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  • The solid melts to a pale yellow liquid which on continued heating gradually darkens and becomes more viscous, the maximum viscosity occurring at 180°, the product being dark red in colour.

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  • By the action of dilute hydrochloric acid on metallic polysulphides, an oily product is obtained which C. L.

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  • Hofmann, who obtained it by saturating an alcoholic solution of ammonium sulphide with sulphur and mixing the product with an alcoholic solution of strychnine, considered the resulting product to be H2S3; while P. Sabatier by fractionating the crude product in vacuo obtained an oi l which boiled between 60° and 85° C. and possessed the composition H4S5.

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  • rend., 1900, 130, p. 865) by fractionally distilling the product formed in the direct action of fluorine on sulphur.

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  • The dioxide has been known since the earliest times and is found as a naturally occurring product in the gaseous exhalations of volcanoes and in solution in some volcanic springs.

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  • The increase in the value of the "factory" product between 1900 and 1905 was from $59,322,234 to $82,227,950, or 38.6%.

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  • Slaughtering and meat-packing is the most important industry, the value of the product amounting to $ 2 4,45 8, 810 in 1905; this industry dates from about 1835.

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  • In this work, the product, according to Lange, of a fanatical enthusiasm for humanity, he sought to demonstrate the indestructibility of matter and force, and the finality of physical force.

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  • It is the duty of all believers to strive to become one or other of these by subduing the flesh, which is the product of Evil, and all motions of the will.

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  • The rail-failures mentioned above also drew renewed attention to the importance of the thermal treatment of the steel from the time of melting to the last passage through the rolling mill and to the necessity of the finishing temperature being sufficiently low if the product is to be fine grained, homogeneous and tough; and to permit of this requirement being met there was a tendency to increase the thickness of the metal in the web and flanges of the rails.

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  • Though a busy colliery may send off its product by the train-load to an important town, the wagons will usually be addressed to a number of different consignees at different depots in different parts of the town, and therefore the train will have to be broken up somewhere short of its destination and its trucks rearranged, together with those of other trains similarly constituted, into fresh trains for conveyance to the various depots.

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  • If the total resistance against which the train is maintained in motion with an instantaneous velocity of V feet per second is R, the rate at which energy is expended in moving the train is represented by the product RV, and this must be the rate at which energy is supplied to the train after deducting all losses due to transmission from the source of power.

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  • The rate at which work is done on a particular axle is measured by the product where T is the torque or turning moment exerted on the axle by the motor or mechanism applied to it for this purpose, and is the angular velocity of the axle in radians per second.

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  • If there are several driving-axles in a train, the product Tw must be estimated for each separately; then the sum of the products will be equal to RV.

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  • The average value of the product of these percentages, namely o 65 Xo 09 =0.06 say, may be used to investigate generally the working of a locomotive; the actual value could only be determined by experiment in any particular case.

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  • The relation between the b.h.p. and the torque on the driving-axle is 55 o B.H.P. =Tu., (9) It is usual with steam locomotives to regard the resistance R as including the frictional resistances between the cylinders and the driving-axle, so that the rate at which energy is expended in moving the train is expressed either by the product RV, or by the value of the indicated horse-power, the relation between them being 55 0 I.H.P. =RV (Io) or in terms of the torque 55 0 I.H.P.X€=RVe=TW (II) The individual factors of the product RV may have any value consistent with equation (to) and with certain practical conditions, so that for a given value of the I.H.P. R must decrease if V increases.

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  • The crude solid product from the tar distillate is digested with carbon bisulphide to dissolve the pyrene, the solution filtered and the solvent evaporated.

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  • - The book of Deuteronomy was the product of prophetic teaching operating on traditional custom, which was represented in its essential features by the two codes of legislation contained in Ex.

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  • xvii.-xxvi.) is the most obvious product of that influence.

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  • (d) The doctrine of pre-existence is another product of the speculative tendency of the Jewish mind.

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  • In itself a product of the medieval conception of the fool who figured so largely in the Shrovetide and other pageants, it differs entirely from the general allegorical satires of the preceding centuries.

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  • The chief product is amber.

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  • With the growing of grasses as the chief agricultural product, farming in Nevada is necessarily extensive rather than intensive.

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  • It thus appears that the live stock industry is one of the most important in the state; the value of its product in 1899 exceeded its output of gold and silver, which had then reached its lowest point, by over one million dollars.

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  • For the three years1875-1877the production of gold and silver in Nevada was more than the combined product of all the other American states and Territories.

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  • Manufactures based on the products of mines and quarries (chemicals, glass, clay, stone and metal works) constituted about one-fifth of the whole product.

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  • The river furnishes considerable water-power and the total factory product in 1905 was valued at $8,357,993, an increase of 47.2% over that in 1900.

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  • Modern Plymouth has varied and important manufactures comprising cordage, woollens, rubber goods, &c. In 1905 the total value of the factory products was $11,115,713, the worsted goods and cordage constituting about nine-tenths of the whole product.

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  • From 1900 to 1905 the capital invested in manufactures increased 83% and the value of the product Sor %.

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  • These districts are pastoral, and the lower fertile lands are cultivated for sugar, cotton, maize, tobacco, rice, beans, and mandioca - sugar being the principal product.

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  • In 1905 it ranked third among the cities of the state in value of its factory product ($7,3 8 9, 8 44).

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  • The principal mining product is mercury, extracted at Idria, while iron and copper ore, zinc and coal are also found.

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  • They were a unique product of rabbinism; and the authors of the system were also the compilers of its literary expression, the Talmud.

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  • The most important product of the district was the wood from the forests of the Sila, and the pitch produced from it.

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  • There are miscellaneous manufactures, and in 1905 the factory product was valued at $3,090,312.

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  • The lumber and timber product increased in value from $1,920,335 in 1880 to $ 2 4, 0 35,539 in 1905.

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  • The value of the total factory product was $57,45 1, 445 in 1905, when a little more than three-fourths was represented by lumber and timber products, cotton-seed oil and cake, and cotton goods.

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  • In 1902 the value of the gold and silver product combined was $71,287, and in 1908, when the Iola mine 6 m.

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  • In 1908 the product amounted to 48,522 long tons (all magnetite), and was valued at $76,877; almost the entire product is from the Cranberry mines, near Cranberry, Mitchell county.

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  • The most valuable immediate product of the state's mines and quarries for nearly every year from 1890 to 1908 was building stones of granite and gneiss, which are found in all parts of the state west of the " Fall Line "; the best grades of granite are quarried chiefly in Gaston, Iredell, Rowan, Surry and Wilkes counties.

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  • The product increased in value from $4,783,484 in 1890 to $25,488,721 in 1905.

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  • The value of the state's factory product for 1900 was $85,274,083, and that for 1905, $142,520,776, an advance of 67.1%.

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  • The soul is in turn the image or product of the vas, and the soul by its motion begets corporeal matter..

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  • The soul thus faces two ways - towards the vas, from which it springs, and towards the material life, which is its own product.

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  • When we have mentioned vanilla, which consists of the fleshy pods of an orchid, we have mentioned about the only economic product that now comes into market.

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  • If one excepts the Old Testament, which is a product of the extreme west of Asia, it is remarkable how small has been the influence of Asiatic literature on Europe.

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  • Some authorities hold that Peruvian civilization had no connexion with the north and was an entirely indigenous product, but Kechua is in structure not unlike the agglutinative languages of central and northern Asia.

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  • In 1905 its total factory product was valued at $4,210,265.

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  • Glycerin is also a product of certain kinds of fermentation, especially of the alcoholic fermentation of sugar; consequently it is a constituent of many wines and other fermented liquors.

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  • In 1905 the value of the city's factory product was $2,983,160, being 28.4% more than in 1900.

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  • A consort Antum (or as some scholars prefer to read, Anatum) is assigned to him, on the theory that every deity must have a female associate, but Antum is a purely artificial product - a lifeless symbol playing even less of a part in what may be called the active pantheon than Anu.

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  • In 1905 the value of, the factory product was $11,264,123, an increase of 59 per cent.

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  • The principal product is cigars; most of the tobacco used is imported from Cuba, and the manufacturing is done chiefly by Cubans who live in a district known as Ybor City.

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  • For he proved that the various changes occurring in the several processes of fermentation - as, for example, in the vinous, where alcohol is the chief product; in the acetous, where vinegar appears; and in the lactic, where milk turns sour - are invariably due to the presence and' growth of minute organisms called ferments.

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  • To find three numbers such that the product of any two added to the sum of those two gives a square (III.

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  • 22) To find two numbers such that their product their sum gives a cube (IV.

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  • 29); To find three squares such that their continued product added to any one of them gives a square (V.

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  • The import of the manufactured product from 1875 to 1900 increased at a much greater ratio than that of the raw grain, for whilst in 1875 the former represented less than one-ninth of the total, by 1900 the proportion had risen to nearly one-fourth.

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  • Of phosphoric acid, the cereal crops take up as much as, or more than, any other crops of the rotation, excepting clover; and the greater portion thus taken up is lost to the farm in the saleable product - the grain.

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  • Moreover, in 1885 the imported product was only about one-seventh [[Table Xvi I]].

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  • - Home Product and Imports of Sheep and Mutton into the United Kingdom - Thousands of Tons.

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  • Butchers have palmed off upon their customers imported fresh meat as homegrown, and secured a dishonest profit by charging for it the prices of the latter, which are considerably in excess of those of the imported product.

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  • The value of the township's factory product in 1 9 05 was $3,171,3t8, an increase of 80 6% since 1900, this ratio of increase being greater than that shown by any other "municipality" in the state having a population in 1 9 00 of 8000 or more.

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  • From the product so obtained most of the better sort of "Burgundy pitch" of the druggists is prepared.

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  • It is important here to note that Clausilia furnishes us with an exceptional instance of the continuity of the shell or secreted product of the primitive shell - sac with the adult shell.

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  • Its obligations to other contemporary arts are many and obvious, especially in its later stages; but every borrowed form and motive undergoes an essential modification at the hands of the Aegean craftsman, and the product is stamped with a new character.

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  • The cultivation of tobacco was of little importance in the state until about 1840; but the product increased from 10,454,449 lb in 1850 to 34,735,235 lb in 1880, and to 65,957,100 lb in 1899, when the crop was grown on 71,422 acres; in 1909 the crop was 83,250,000 lb, grown on 90,000 acres.

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  • The value of the factory product was $748,670,855 in 1900 and $960,811,857 in 1905.1 The most important manufacturing industry is that of iron and steel.

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  • It exports filberts (for which product it is the centre), walnuts, hides and timber.

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  • The lemniscate of Bernoulli may be defined as the locus of a point which moves so that the product of its distances from two fixed points is constant and is equal to the square of half the distance between these points.

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  • As a commercial product spider-silk has been found to be equal, if not superior, to the best silk spun by lepidopterous larvae; but the cannibalistic propensities of spiders, making it impossible to keep more than one in a single receptacle, coupled with the difficulty of getting them to spin freely in a confined space, have hitherto prevented the silk being used on any extensive scale for textile fabrics.

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  • The waters of the lake swarm with fish (sturgeons and salmonidae), and its herring (Salmo omul) is the chief product of the fisheries, though notably fewer have been taken within the last forty or fifty years.

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  • In 1900 New Haven was the most important manufacturing centre in Connecticut, and in 1905 it was second only to Bridgeport in the value of its factory product.

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  • In 1905 the total capital inyested in manufacturing was $31,412,715 and the total product $39,666,118 (a gain of 13.7% since 1900).

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  • By careful selection (the methods of which are described below) in the United States, the quality of the product was much improved, and on the recent revival of the cotton industry in the West Indies American " Sea Island " seed was introduced back again to the original home of the species.

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  • The history of no agricultural product contains more of interest and instruction for the student of economics than does that of cotton seed in the United States.

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  • The main product is the refined oil, which is used for a great number of purposes, such as a substitute for olive oil, mixed with beef products for preparation of compound lard, which is estimated to consume one-third of cotton seed oil produced in the States.

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  • These facts indicate that we have here an agricultural product the market price of which is still far below its value as compared, on the basis of its chemical composition, either with other feeding stuffs or with other fertilizers.

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  • For five centuries before the Christian era cotton was largely used in the domestic manufactures of India; and the clothing of the inhabitants then consisted, as now, chiefly of garments made from this vegetable product.

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  • The acetone dicarboxylic acid, CO(CH 2 CO 2 H) 2, so obtained combines with hydrocyanic acid, and this product yields citric acid on hydrolysis.

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  • In 1900 the manufactured product of the city and its immediate suburbs was valued at $31,690,736, of which $19,009,332 were credited to slaughtering and packing.

    0
    0
  • In 1905 the census reports did not include manufactures outside the actual city limits; the total value of the factory product of the city proper in 1905 was $11,573,720; besides slaughtering and packing the other manufactures in 1905 included men's factory-made clothing (valued at $1,556,655) flour and grist-mill products (valued at $683,464), saddlery and harness (valued at $524,918), confectionery ($437,096), malt liquors ($407,054), boots and shoes ($350,384) and farm implements.

    0
    0
  • The poppy is cultivated wherever it will grow, the crop being far more profitable than that of any other product.

    0
    0
  • The earlier supporters of the organic theory held that it was a product of the natural distillation of coal or carbonaceous matter; but though in a few instances volcanic intrusions appear to have converted coal or allied substances into oil, it seems that terrestrial vegetation does not generally give rise to petroleum.

    0
    0
  • This view is supported by the fact that petroleum is found on the Sardinian and Swedish coasts as a product of the decomposition of seaweed, heated only by the sun, and under atmospheric pressure.

    0
    0
  • In most petroleum-producing countries, however, and particularly where the product is abundant, the crude oil is fractionally distilled, so as to separate it into petroleum spirit of various grades, burning oils, gas oils, lubricating oils, and (if the crude oil yields that product) paraffin.

    0
    0
  • The product actually obtained is a mixture of several paraffins and several olefines.

    0
    0
  • Paraffin wax is tested for melting-point (or setting-point), and the semi-refined product is further examined to ascertain the percentage of oil, water and dirt present.

    0
    0
  • For pharmaceutical purposes crude petroleum is no longer generally used by civilized races, though the product vaseline is largely employed in this way, and emulsions of petroleum have been administered internally in various pectoral complaints; while the volatile product termed rhigolene has been largely used as a local anaesthetic.

    0
    0
  • For illuminating purposes, the most extensively-used product is kerosene, but both the more and the less volatile portions Of petroleum are employed in suitable lamps.

    0
    0
  • A wide missionary activity had begun in the 13th century - an activity which was the product of the Crusades and the contact with the Moslem which they brought, but which yet helped to check the Crusades, substituting as it did peaceful and spiritual conquests of souls for the violence and materialism of even a Holy War.

    0
    0
  • In these circumstances the torsional angle becomes a measure of the torque and therefore of the product of the strengths of the currents in the two coils, that is to say, of the square of the strength of the current passing through the two coils if they are joined up in series.

    0
    0
  • Since the current passing through the balance when equilibrium is obtained with a given weight is proportional to the square root of the couple due to this weight, it follows that the current strength when equilibrium is obtained is proportional to the product of the square root of the weight used and the square root of the displacement distance of this weight from its zero position.

    0
    0
  • Each instrument is accompanied by a pair of weights and by a square root table, so that the product of the square root of the number corresponding to the position of the sliding weight and the ascertained constant for each weight, gives at once the value of the current in amperes.

    0
    0
  • The lumber industry is important: in 1905 the total factory product of lumber and timber was valued at $10,901,650, and lumber and planing mill products were valued at $1,690,455.

    0
    0
  • The total product of the fisheries in 1902 was valued at about $2,000,000.

    0
    0
  • The principal occupation is agriculture, in which 44% of the labouring population was engaged in 1900, but only 12.6% of the total land surface was enclosed in farms, of which only 34.6% was improved, and the total agricultural product for 1899 was valued at $18,309,104.

    0
    0
  • The cultivation of pineapples, in sub-tropical Florida, is proving successful, the product far surpassing that of California, the only other state in the Union in which pineapples are grown.

    0
    0
  • In 1903, according to the statistics of the United States Department of Agriculture, Indian corn ranked next to fruits .(as given in the state reports), but its product as compared with that of various other states is unimportant - in 1907 it amounted to 7,017,000 bushels only; rice is the only other cereal whose yield in 1899 was greater than that of 1889, but the Florida product was surpassed (in 1899) by that of the Carolinas, Georgia, Louisiana and Texas; in 1907 the product of rice in Florida (69,000 bushels) was less than that of Texas, Louisiana, South Carolina, Arkansas and Georgia severally.

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  • Tobacco culture, which declined after 1860 on account of the competition of Cuba and Sumatra, has revived since 1885 through the introduction of Cuban and Sumatran seed; the product of 1907 (6,937,500 lb) was more than six times that of 1899, the product in 1899 (1,125,600 lb) being more than twice that of 1889 (470,443 lb), which in turn was more than twenty times that for 1880 (21,182 lb)-the smallest production recorded for many decades.

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  • In 1899, 84% of the product was raised in Gadsden county.

    0
    0
  • Their product in 1900 was more than twice the product in 1890, and the product in 1905 (from establishments under the factory system only) was $50,298,290, i.e.

    0
    0
  • The most important industries were those that depended upon the forests, their product amounting to nearly 45% of the entire manufactured product of the state.

    0
    0
  • 2% over the product of 1900.

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    0
  • The manufacture of turpentine and rosin, material for which is obtained from the pine forests, had increased greatly in importance between 1890 and 1900, the product in 1890 being valued at only $191,859, that of 1900 at $6,469,605, and from the latter sum it increased in 1905 to $9,901,905, an increase of more than one-half.

    0
    0
  • In 1900 the state ranked second and in 1905 first of all the states of the country in the value of this product; in 1905 the state's product amounted to 41.4% of that of the entire country.

    0
    0
  • The manufacture of cigars and cigarettes (almost entirely of cigars, few cigarettes being manufactured), carried on chiefly by Cubans at Key West and Tampa, also increased in importance between 1890 and 1900, the products in the latter year being valued at $10,735,826, or more than one-quarter more than in 1890, and in 1905 there was a further increase of 56.2%, the gross value being $16,764,276, or nearly one-third of the total factory product of the state.

    0
    0
  • In the manufacture of fertilizers, the raw material for which is derived from the phosphate beds, Florida's aggregate product in 1900 was valued at $500,239, and in 1905 at $1,590,371, an increase of 217.9% in five years.

    0
    0
  • The Florida East Coast Railway is also the product of one man's faith in the country, that of Henry M.

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    0
  • Soap may be framed and finished in this state, but almost invariably it receives a further treatment called " refining " or " fitting," in which by remelting with water, with or without the subsequent addition of other agents to harden the finished product, the soap may be made to contain from 60 to 70% of water and kept present a firm hard texture.

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    0
  • The oil mixture used differs in the several manufacturing countries, and the commercial name of the product is correspondingly varied.

    0
    0
  • In Germany tallow is the principal fat; in France olive oil occupies the chief place and the product is known as Marseilles or Castile soap; and in England tallow and palm oil are largely used.

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  • Suppose that a pure soap without resin is to be made - a product little seen in the market - the spent lye is run off, steam is again turned on, pure water or very weak lye run in, and the contents boiled up till the whole is thin, close and clear.

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  • On settling the product forms three layers: the uppermost is a thin crust of soap which is worked up again in the pan; the second is the desired soap; next there is a dark-coloured weak soap termed nigre, which, because it contains some soap and alkali is saved for future use; underneath these is a solution of alkaline salts with a little free alkali.

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  • There is no separation of underlyes in potash soap, consequently the product contains the whole constituents of the oils used, as the operation of salting out is quite impracticable owing to the double decomposition which results from the action of salt, producing thereby a hard principally soda soap with formation of potassium chloride.

    0
    0
  • If it fails to form a hard cake on cooling, a known weight of wax may be added and the product re-heated.

    0
    0
  • The city's manufactures include glass, brick, tile, foundry and machine-shop products, &c. In 1905 the factory product was valued at $1,888,894, being 51.4% greater than in 1900.

    0
    0
  • CARBON BISULPHIDE, CS 2, a chemical product first discovered in 1796 by W.

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    0
  • The crude product is very impure and possesses an offensive smell; it may be purified by forcing a fine spray of lime water through the liquid until the escaping water is quite clear, the washed bisulphide being then mixed with a little colourless oil and distilled at a low temperature.

    0
    0
  • Since the area of the triangle FPP' is one half the product of FP into the perpendicular p from P on FP', it follows that if these perpendiculars were equal all round the orbit, the areas described during the infinitesimal time would be smallest at the pericentre and continually increase during the passage of the body to B.

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    0
  • The product is ground and levigated; and when dry it is ready for use.

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    0
  • The fact that its product is shut out of its natural markets, without gaining that of the United States, is also a great handicap. The civic status of the people is still unsettled, but there has been under American rule a notable advance in the well-being of the island.

    0
    0
  • In 1900 the total product was valued at $16,707,027, and the factory product at $14,418,834; and in 1905 the factory product was valued at $25,745,650, an increase of 78.6% in five years.

    0
    0
  • Among the products are cotton goods (the product value of which in 1905 was 1 4% of the total value of the city's manufactures), foundry and machine-shop products, lumber, patent medicines, confectionery, men's clothing, mattresses, spring-beds and other furniture.

    0
    0
  • Consequently the present city is a product of the post-bellum development of Georgia.

    0
    0
  • The value of the city's manufactured products increased from $37,376,322 in 1890 to $77,225,116 in 1900, or 106.6%; in 1905 the factory product alone was valued at $75,740,934, an increase of only 3.9% over the factory product in 1900, this small rate of increase being due very largely to a decline in the value of the products of the sugar and molasses refining industry.

    0
    0
  • The value of the wholesale slaughtering and meat-packing product decreased from $18,551,783 in 1880 and $11,356,511 in 1890 to $6,243,217 in 190o - of this 85,708,763 represented wholesale slaughtering alone; in 1905 the wholesale slaughtering product was valued at $7,568,739.

    0
    0
  • At the beginning of the decade 1891-1901 wheat was the staple product of the Vindhyan and Nerbudda valley districts, and was also grown extensively in all the Satpura districts except Nimar and in Wardha and Nagpur.

    0
    0
  • He discovered that gases always combined in volumes having simple ratios, and that the volume of the product had a simple ratio to the volumes of the reacting gases.

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  • This law states that: - gases combine with one another in simple proportions by volume, and the volume of the product (if gaseous) has a simple ratio to the volumes of the original mixtures; in other words, the densities of gases are simply related to their combining weights.

    0
    0
  • Peligot, who prepared the pure metal, that Klaproth's product was really an oxide.

    0
    0
  • For example: nitric acid and sulphuric acid readily react with benzene and its homologues with the production of nitro derivatives and sulphonic acids, while in the aliphatic series these acids exert no substituting action (in the case of the olefines, the latter acid forms an addition product); another distinction is that the benzene complex is more stable towards oxidizing agents.

    0
    0
  • Not only does the substituent group modify the readiness with which the derivative is attacked, but also the nature of the product.

    0
    0
  • Long-continued treatment with halogens may, in some cases, result in the formation of aromatic compounds; thus perchlorbenzene, C 6 C1 6, frequently appears as a product of exhaustive chlorination, while hexyl iodide, C 6 H 13 I, yields perchlorand perbrom-benzene quite readily.

    0
    0
  • Somewhat similar condensations are: of geranial or citral, (CH 3) 2 CH CH 2 CH :CH C(CH 3) :CH CHO, to p-isopropylmethylbenzene or cymene; of the condensation product of methylethylacrolein and acetone, CH3 CH2 CH:C(CH3) CH:CH CO CH3, to [1.3.

    0
    0
  • 4]-trimethylbenzene or pseudocumene; and of the condensation product of two molecules of isovaleryl aldehyde with one of acetone, C 3 H 7 CH 2 CH:C(C 3 H 7) CH:CH CO.

    0
    0
  • CH :CH (OH), formed by acting on formic ester with acetone in the presence of sodium ethylate, readily yields [1.3.51-triacetylbenzene, C 6 H 3 (CO CH 3) 3; oxymethylene acetic ester or formyl acetic ester or ß-oxyacrylic ester, (HO)CH :CH CO 2 C 2 H 51 formed by condensing acetic ester with formic ester, and also its dimolecular condensation product, coumalic acid, readily yields esters of [1.3.

    0
    0
  • Industrial chemistry makes many claims upon the chemist, for it is necessary to determine the purity of a product before it can be valued.

    0
    0
  • Sulphur is detected by heating the substance with sodium, dissolving the product in water, and adding sodium nitroprusside; a bluish-violet coloration indicates sulphur.

    0
    0
  • Phosphorus is obtained as a soluble phosphate (which can be examined in the usual way) by lixiviating the product obtained when the substance is ignited with potassium nitrate and carbonate.

    0
    0
  • The product is dissolved in water, and the calcium haloid estimated in the usual way.

    0
    0
  • This states that " the atomic heat (the product of the atomic weight and specific heat) of all elements is a constant quantity."

    0
    0
  • We now proceed to discuss molecular heats of compounds, that is, the product of the molecular weight into the specific heat.

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  • Lehmann it melts at 168° (or at a slightly lower temperature in its water of crystallization) and on cooling forms optically isotropic crystals; at 125.6° the mass becomes doubly refracting, and from a solution rhombohedral (optically uniaxial) crystals are deposited; by further cooling acicular rhombic crystals are produced at 82.8°, and at 32.4° other rhombic forms are obtained, identical with the product obtained by crystallizing at ordinary temperatures.

    0
    0
  • Those who hold to the genuineness of Colossians find it easier to explain the resemblances as the product of the free working of the same mind, than as due to a deliberate imitator.

    0
    0
  • Equally certain is a second observation of a general character that the epic originating as the greater portion of the literature in Assur-bani-pal's collection in Babylonia is a composite product, that is to say, it consists of a number of independent stories or myths originating at different times, and united to form a continuous narrative with Gilgamesh as the central figure.

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  • It was discovered in 1838 by Piria as a decomposition product of salicin.

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    0
  • Potassium persulphate oxidizes it in alkaline solution, the product on boiling with acids giving hydroquiirone carboxylic acid (German Patent 81,297).

    0
    0
  • Whatever drug of this group be taken, the product absorbed by the blood is almost entirely sodium salicylate.

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    0
  • The vegetable product - which is extremely expensive - must be prescribed or the synthetic product guaranteed "physiologically pure," i.e.

    0
    0
  • The product is at first white, but soon turns green ("green ultramarine") when it is mixed with sulphur and heated.

    0
    0
  • A blue product is obtained at once, but a red tinge often results.

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  • For several centuries it was wholly lost sight of, and it was not till the 13th century that it was rediscovered through the agency of Robert Grosseteste, bishop of Lincoln, who translated it into Latin, under the misconception that it was a genuine work of the twelve sons of Jacob, and that the Christian interpolations were a genuine product of Jewish prophecy.

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  • The total value of tle city's factory product in 1905 was $4,159,410, being an increase of 118.3% over that of 190o.

    0
    0
  • The total value of the factory product in 1905 was $5,976,637 (67.2% more than in 1900).

    0
    0
  • No single man appears as creator of the tendency of thought they represent; they are the product of a period extending over several centuries, but they form an intellectual unity, and presuppose a great body of thinkers.

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  • They were to be redeemed by the product of the sales, and from certain other sources, at the rate of 120,000,000 francs in 1791, ioo,000,000 francs in 1792, 80,000,000 francs in 1793 and 1794, and the surplus in 1795.

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    0
  • The timber product of 1900 ($17,294,444) was almost ten times that of 1880 ($1,764,640); and in 1905 the product value ($35,192,374) was more than twice that of 1900.

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    0
  • The average product of the ten seasons1894-1904was 299,745 tons.

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    0
  • The total value of the tobacco crop of 35,000 ib in 1907 was only $to,000, an amount exceeded by each of the other 24 tobacco-growing states, and the crop was about one-twentieth of% of the product of the whole United States.

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    0
  • From 1879 to 1899 the product increased twenty-five fold.

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    0
  • Among nuts the native pecan is exceptionally abundant, the product (637,470 lb in 1899) being much greater than that of any other state save Texas.

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    0
  • Slightly above one-half of the product of 1900 was from New Orleans, and in 1905 about 45.4%.

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    0
  • Other manufactures with a product value in 1905 of between $4,000,000 and $1,000,000 were: bags (not paper); foundry and machine-shop products; planing-mill products; railway cars, construction and repairs; malt liquors; men's clothing; cooperage; food preparations; roasted and ground coffee and spice; fertilizers; cigars and cigarettes; cotton goods; and manufactured ice.

    0
    0
  • Four or five annual crops grow from one plant, but not more than three can be marketed, unless locally, as the product deteriorates.

    0
    0
  • In recent years the growth of the leaf under cloth tents has greatly increased, as it has been abundantly proved that the product thus secured is much more valuable - lighter in colour and weight, finer in texture, with an increased proportion of wrapper leaves, and more uniform qualities, and with lesser amounts of cellulose, nicotine, gums and resins.

    0
    0
  • In these respects the finest Cuban tobacco crops, produced in the sun, hardly rival the finest Sumatra product; but produced under cheese-cloth they do.

    0
    0
  • Iron is now the most important mineral product.

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    0
  • Nearly the entire product goes to the United States.

    0
    0
  • The commercial product of stones, brick and cement is of rapidly increasing importance.

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    0
  • The product is used to bleach sugar, as well as for construction and disinfection purposes.

    0
    0
  • It behaves as a powerful reducing agent, and on hydrolysis with dilute mineral acids is decomposed into formaldehyde and hydroxylamine, together with some formic acid and ammonia, the amount of each product formed varying with temperature, time of reaction, amount of water present, &c. This latter reaction is probably due to some of the oxime existing in the form of the isomeric formamide HCO NH 2.

    0
    0
  • The first product of the reaction is nitric oxide, which on cooling with the residual gases produces nitrogen peroxide.

    0
    0
  • Glauber in 1648 as a product of the reaction between nitric acid and arsenious oxide.

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    0
  • Sodium nitrite, the most commonly used salt of the acid, is generally obtained by heating the nitrate with metallic lead; by heating sodium nitrate with sulphur and sodium hydroxide, the product then being fractionally crystallized;(Read, Holliday & Sons): 3NaNO 3 +S+2NaOH = Na2S04+3NaN02+H20; by oxidizing atmospheric nitrogen in an electric arc, keeping the gases above 300° C., until absorption in alkaline hydroxide solution is effected (German Pat.

    0
    0
  • The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and 13, satisfying the conditions (1) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in common, and then by defining by reference to a and (3 two other suitable classes whose cardinal numbers are defined to be respectively the required sum and product of the cardinal numbers in question.

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  • terms in common; (2) their relation-numbers are the two relationnumbers in question, and then by defining by reference to R and S two other suitable relations whose relation-numbers are defined to be respectively the sum and product of the relation-numbers in question.

    0
    0
  • But an indefinite number of definitions of the product of two complex numbers yield interesting results.

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    0
  • The product of two complex numbers of the second order - namely, l e l +x 2 e 2 and y i e l +y 2 e 2, is in this case defined to mean the complex (x i y i - x 2 y 2)e i +(x i y 2 +x 2 y 1)e 2.

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  • Four-fifths of the net product of the revenues, after deduction of the first charge of £T590,000, was to be applied.to the service of the interest on the new reduced debt, and provided that the four-fifths were sufficient to allow the distribution of 1% interest, one-fifth was to be devoted to sinking fund; but this latter fifth was to be reduced, if necessary, by an amount sufficient to maintain the rate of interest at i %.

    0
    0
  • When the net product of the ceded revenues amounts to £T2,157,375, the surplus is divisible as to 75% to the Turkish government and 25% to the public debt administration.

    0
    0
  • Table A gives the produce of the revenues in 1881-1882, the last year of the administration of the " Galata Bankers," the average product of the first, second, third, fourth and fifth quinquennial periods since the public council was established, and of the year 1907-1908.

    0
    0
  • Roanoke's factory product in 1905 was valued at $5,544,907 (2.7% more than in 1900).

    0
    0
  • Cleveland is the headquarters of the largest shoddy mills in the country (value of product, 1905, $ 1, 0 84,594), makes much clothing (1905, $ 10, 4 26, 535), manu factures a large portion of the chewing gum made in the United States, and is the site of one of the largest refineries of the Standard Oil Company.

    0
    0
  • The product of Cleveland breweries in 1905 was valued at $3,986,059, and of slaughtering and meatpacking houses in the same year at $10,426,535.

    0
    0
  • Yttria is an exceedingly complex mixture, which has been decomposed, yielding as an intermediate product terbia.

    0
    0
  • At the same time it was decided that the deputies to that convention should be elected by all Frenchmen 25 years old, domiciled for a year and living by the product of their labour.

    0
    0
  • Eight years later (1892) the much richer Mesabi Range, the most productive iron range in the world, was opened up; it soon surpassed the Vermilion in its output, and by 1902 the product was nearly ten times greater.

    0
    0
  • It was difficult to be sure as to the variations in the actual number of fish caught, but it was easy to show that there was a real variability in the yield of cod-liver oil (an important product of the fishery).

    0
    0
  • Gelatin forms a white amorphous powder; the commercial product, however, generally forms glassy plates.

    0
    0
  • " Amyloid " occurs as a pathological product, and also in the healthy aorta and in old cartilage.

    0
    0
  • In 1905 the total value of the factory product was $3,926,632, an increase of 28.2% since 1900.

    0
    0
  • Burauen is situated in a rich hemp-growing region, and hemp is its only important product.

    0
    0
  • Date sugar is a valuable commercial product of the East Indies, obtained from the sap or toddy of Phoenix sylvestris, the toddy palm, a tree so closely allied to the date palm that it has been supposed to be the parent stock of all the cultivated varieties.

    0
    0
  • On the view of the process of conduction described above, the amount of electricity conveyed per second is measured by the product of the number of ions, known from the concentration of the solution, the charge carried by each of them, and the velocity with which, on the average, they move through the liquid.

    0
    0
  • In 1905 Dover ranked fourth among the manufacturing cities of the state, and first in manufactures of woollens; the value of the city's total factory product in that year was $6,042,901.

    0
    0
  • It appears to be synthesized in the plant tissues from carbon dioxide and water, formaldehyde being an intermediate product; or it may be a hydrolytic product of a glucoside or of a polysaccharose, such as cane sugar, starch, cellulose, &c. In the plant it is freely converted into more complex sugars, poly-saccharoses and also proteids.

    0
    0
  • This product melts at 86° C., and becomes anhydrous when heated to 110° C. The anhydrous compound can also be prepared, as hard crusts melting at 146°, by crystallizing concentrated aqueous solutions at 30 to 35°.

    0
    0
  • The acid employed may be hydrochloric, which gives the best results, or sulphuric, which is used in Germany; sulphuric acid is more readily separated from the product than hydrochloric, since the addition of powdered chalk precipitates it as calcium sulphate, which may be removed by a filter press.

    0
    0
  • "Syrup glucose" is the commercial name of the product; by continuing the concentration further solid glucose or grape sugar is obtained.

    0
    0
  • The rubber is of good quality, though, owing to the method of preparation adopted, the product is often impure and discoloured, and consequently usually brings a lower price than the best rubbers of commerce.

    0
    0
  • There are many substances produced by plants which can be synthetically prepared by chemical means, but, as with quinine, the process involved is too costly to enable the synthetic product to compete with the natural product.

    0
    0
  • The washed product contains in its pores a notable proportion of water, which is removed by hanging the rubber for some days in a warm room.

    0
    0
  • The total factory product in 1905 was valued at $13,420,863; of this $2,890,301 was the value of agricultural implements, in the manufacture of which Auburn ranked fifth among the cities of the United States.

    0
    0
  • Let there be given n 2 quantities all a,2 a13 ��� aln a21 a22 a23 ��� a2n a3, a32 a33 ��� a3n and an, an 3 ��� ann and form from them a product of n quantities ala a2 0 a37 ...

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    0
  • Now a11A11= alla22a33...ann, wherein all is not to be changed, but the second suffixes in the product a 22 a 33 ...a nn assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.

    0
    0
  • a ll a33 ��� a32 a33 ��� a3n an2 an3 ��� ann Similarly A ik, the cofactor of aik, is shown to be the product of (-) i+k and the determinant obtained by erasing from A the ith row and k th column.

    0
    0
  • From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, ��� ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.

    0
    0
  • We thus obtain for the product a determinant of order n.

    0
    0
  • If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +��� +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i.

    0
    0
  • Hence the product determinant has the principal diagonal elements each equal to A and the remaining elements zero.

    0
    0
  • The resultant being a product of mn root differences, is of degree mn in the roots, and hence is of weight mn in the coefficients of the forms; i.e.

    0
    0
  • Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third.

    0
    0
  • Hence this product is the required resultant of the three equations.

    0
    0
  • (x y m - x m y), ar _ y1(x y 2 - and substituting in the latter any root of f and forming the product, we find the resultant of f and d, viz.

    0
    0
  • and, dividing by y1y2...ym, the discriminant of f is seen to be equal to the product of the squares of all the differences of any two roots of the equation.

    0
    0
  • The discriminant of the product of two forms is equal to the product of their discriminants multiplied by the square of their resultant.

    0
    0
  • A separation is the symbolic representation of a product of monomial symmetric functions.

    0
    0
  • = 11(1 +�alxl+� g al x2+�3a1x3+...), the product on the right involving a factor for each of the quantities a l, a2, a3..., and u being arbitrary.

    0
    0
  • in the product Xm1Xm2Xm3...

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  • and therefore of the product of those expressions.

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  • 1 2 3 We have found above that the coefficient of (x 1 1 x 12 x 13...) i n the product XmiXm2X m3 ...

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  • a Product of Symmetric Functions.

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  • - Suppose f to be a product of symmetric functions f i f 2 ...f m .

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    0
  • Suppose we wish to find the coefficient of (52413) in the product (20(2' 4)(0).

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    0
  • If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of umbrae arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 .

    0
    0
  • = a k; and if we wish to denote, by umbrae, a product of coefficients of degree s we employ s sets of umbrae.

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  • We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.

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  • (ab)i(ac)j(bc)k..., that the symbolic product (ab)i(ac)j(bc)k..., possesses the invariant property.

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    0
  • Notice, therefore, that the symbolic product (ab)i(ac)j(bc)k...

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  • may be the same or different, it is necessary that every product of umbrae which arises in the expansion of the symbolic product be of degree n, in a l, a 2; in the case of b,, b 2 of degree n 2; in the case of c 1, c 2 of degree n3; and so on.

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  • For these only will the symbolic product be replaceable by a linear function of products of real coefficients.

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    0
  • = t) 1 v ...axbxcx..., and assert that the symbolic product (ab)i(ac)'(bc)k...aibxc2...

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  • +P=n1, j+k+...+T =n3, It will also be a covariant if the symbolic product be factorizable into portions each of which satisfies these conditions.

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  • -2 _ ab 2an-2bn-2Crz z x () x x x, Each term on the right-hand side may be shown by permutation of a, b, c to be the symbolical representation of the same covariant; they are equivalent symbolic products, and we may accordingly write 2(ac) (bc)ai -1 bi -1 cx 2 =(ab)2a:-2b:-2c:, a relation which shows that the form on the left is the product of the two covariants n (ab) ay 2 by 2 and cZ.

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  • (ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant.

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  • Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...

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  • It has been shown by Gordan that every symbolic product is expressible as a sum of transvectants.

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  • n of k; if k = o we have the product of the two forms, and for all values of k>n the transvectants vanish.

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  • We have seen that transvection is equivalent to the performance of partial differential operations upon the two forms, but, practically, we may regard the process as merely substituting (ab) k, (OW for azbx, 4x t ' respectively in the symbolic product subjected to transvection.

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  • It is essentially an operation performed upon the product of �two forms. If, then, we require the transvectants of the two forms f+Xf', 0+14', we take their product fc5+xf'95+,-ifct'+atif'cb', and the kth transvectant is simply obtained by operating upon each term separately, viz.

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  • If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on.

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  • In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.

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  • and that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x.

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  • Calling the discriminate D, the solution of the quadratic as =o is given by the formula a: = o (a0+a12_x2 (a0x+aix2 If the form a 2 be written as the product of its linear factors p.a., the discriminant takes the form -2(pq) 2.

    0
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  • x x x To form an invariant or covariant we have merely to form a product of factors of two kinds, viz.

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  • Such a symbolic product, if its does not vanish identically, denotes an invariant or a covariant, according as factors az, bz, cz,...

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  • When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.

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    0
  • The general term in a solution involves the product ao°ai 1 a2 2 ...an" wherein Tr =0, Zs7r s =w; the number of such products that may appear depends upon the number of partitions of w into B or fewer parts limited not to exceed n in magnitude.

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  • If we expand the symbolic expression by the multinomial theorem, and remember that any symbolic product ai 1 a2 2 a3 3 ...

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  • In general the coefficient, of any product A n A m A 7, 3 ..., will have, as coefficient, a seminvariant which, when expressed by partitions, will have as leading partition (preceding in dictionary order all others) the partition (Tr1lr2lr3�..).

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  • The existence of such a relation, as 0-1+0-2+.,.+cr2=0, necessitates the vanishing of a certain function of the coefficients A2, A 3, ...A 9, and as a consequence one product of these coefficients can be eliminated from the expanding form and no seminvariant, which appears as a coefficient to such a product (which may be the whole or only a part of the complete product, with which the seminvariant is associated), will be capable of reduction.

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  • The generating function is I - z2' 52 For 0 =3, (alai +a2a2+a3a3) 10; the condition is clearly a1a2a3 = A3 = 0, and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing A3, all such are perpetuants.

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    0
  • Hence every product of A 1, A2, A3, A4, which contains the product A 4 A 3 disappears before reduction; this means that every seminvariant, whose partition contains the parts 4, 3, is a perpetuant.

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  • For the case 8=1, 0' =2, the condition is a i r 1 72 = A032=0; and the simplest perpetuant, derived directly from the product A 1 B 21 is (I)a(2)b-(21)b; the remainder of those enumerated by z3 I z.

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  • We will choose from the forms in such manner that the product of letters A is either a power of A i, or does not contain A i; this rule leaves us with A2B 1 B 2 and A 2 B,Bs; of these forms we will choose that one which in letters B is earliest in ascending dictionary order; this is A2B 1 B 21 and our earliest perpetuant is (22)a(21)b - (221)a(2)b, and thence the general form enumerated by the generating function Z7 is (1-z)(1 - z2)2 (2 A2+2) a (2�2 +1 1�1 +1) b - (2 A2+2 1)a (2 M2+1 1, ai)b ...

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  • By the rules we select the product A2B 3 B 2 Bi, giving the simplest perpetuant of weight 15, viz: (24) a (3212) b - (241)a(321)b+(2412)a(32)b; and thence the general form (2A2+4)a(3�3+12�2+11�1+ 2)b -...

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    0
  • (2A2+ 41�i +2)a(3�3+12�2+ 1) b, due to the generating function 2 15 (1 -z)(1 -z 2) 2 (1 -z3) For the case 0=r, 0'=4., the condition is a 1 r 1 T 2 r 3 7 4 (a i + T 1)(a + T 2)(a 1 T (a i + T 4) II (T s T t) 0; the calculation gives Selecting the product A;B 4 B 3 B2, we find the simplest perpetuant A1B4(A 1 B2+A1B3+B4)(-B3-A1B2B3-ATB4) =0.

    0
    0
  • The number of different symbols a, b, c,...denotes the the covariants are homogeneous, but not in general isobaric functions, of the coefficients of the original form or forms. Of the above general form of covariant there are important transformations due to the symbolic identities: - (�b) 2 2)2 = a b - a b; (x� = as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven power may be expressed as the product of its first power and a function of the other symbolic factors.

    0
    0
  • Or, lastly, we may leave the exponents h, k, j,1, untouched and consider the product i i i 2 .

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  • There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

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  • In 1905 the borough ranked fifth among the cities of the United States in the manufacture of glass (plateglass, lamp chimneys and bottles), its product (valued at $1,841,308) being 2.3% of that of the whole country.

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  • The value of the factory product in 1905 was $2,081,019, an increase of 20.6% since 1900.

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    0
  • The slag is a waste product, and the flue-dust, collected by special devices in dust-chambers, is briquetted by machinery, with lime as a bond, and then resmelted with the ore-charge.

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    0
  • The suboxide, Pb 2 0, is the first product of the oxidation of lead, and is also obtained as a black powder by heating lead oxalate to 300° out of contact with air.

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    0
  • The conclusions arrived at by earlier writers are combated by Joseph Bedier in the first volume, "Le Cycle de Guillaume d'Orange" (1908), of his Legendes epiques, in which he constructs a theory that the cycle of Guillaume d'Orange grew up round the various shrines on the pilgrim route to Saint Gilles of Provence and Saint James of Compostella - that the chansons de geste were, in fact, the product of 11th and 12th century trouveres, exploiting local ecclesiastical traditions, and were not developed from earlier poems dating back perhaps to the lifetime of Guillaume of Toulouse, the saint of Gellone.

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  • When we know the mass of the earth in gravitational measure, its product by the denominator of the fraction just mentioned gives the mass of the sun in gravitational measure.

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    0
  • Other conditions remaining unchanged, the force between two poles is proportional to the product of their strengths; it is repulsive or attractive according as the signs of the poles are like or unlike.

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  • If m, and m 2 are the strengths of two poles, d the distance between them expressed in centimetres, and f the force in dynes, f=ml m2/d2 (I) The force is one of attraction or repulsion, according as the sign of the product m l m 2 is negative or positive.

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    0
  • The moment, M, M or V, of a uniformly and longitudinally magnetized bar-magnet is the product of its length into the strength of one of its poles; it is the moment of the couple acting on the magnet when placed in a field of unit intensity with its axis perpendicular to the direction of the field.

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  • Such a filament is called a simple magnetic solenoid, and the product aI is called the strength of the solenoid.

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  • A thin sheet of magnetic matter magnetized normally to its surface in such a manner that the magnetization at any place is inversely proportional to the thickness h of the sheet at that place is called a magnetic shell; the constant product hI is the strength of the shell and is generally denoted by 4, or 4.

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    0
  • The potential at any point due to a magnetic shell is the product of its strength into the solid angle w subtended by its edge at the given point, or V = Fu.

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    0
  • The product MH is first determined by suspending the magnet horizontally, and causing it to vibrate in small arcs.

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  • Du Bois believes this to be an important general law, applicable to the case of every paramagnetic substance, and suggests that the product KB should be known as " Curie's constant " for the substance.

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    0
  • ausserkanonischen Evangelien, 1901, p. 2, which show that it was a product of pantheistic Gnosticism.

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    0
  • Chrome alum, K 2 SO 4 �Cr 2 (SO 4) 3.24H 2 O, appears chiefly as a by-product in the manufacture of alizarin, and as a product of the reaction in bichromate batteries.

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  • Moissan by reducing the oxide with carbon in the electric furnace (the product obtained always contains from 2-3% of combined carbon), and by H.

    0
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  • Another vegetable product of the Amazon region is made from the fruit of the Paullinia sorbilis, Mart., and is known by the name of guarand.

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    0
  • Tobacco is also widely cultivated, and the product of some states, such as Bahia, Minas Geraes and Goyaz, has a high local reputation for its excellence.

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    0
  • The product of the elevated inland regions is good, but the costs of transportation and the small profits afforded have prevented its extensive cultivation, and it is imported from the La Plata republics for consumption along the coast.

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    0
  • Vegetable wax, which is an excellent substitute for beeswax, is a product of the carnahuha palm (Copernicia cerifera), and is an important export from Ceara.

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  • The crude methods of preparing jerked beef were also modified to some extent by better equipped abattoirs and establishments for preparing beef extract, preserved meats, &c. There were also mills for crushing the dried mate leaves, cigar and 1 The " bran " exported is from imported wheat and cannot be considered a national product.

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  • 2 The " old metals " consist of old iron, brass, &c., derived from railway material, machinery, &c., all imported, and should not be considered a Brazilian product.

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  • In 1904 the total value of the city's factory product was $6,177,170, an increase of 31.1% over that of 1900.

    0
    0
  • When methyl iodide is used, nitromethane is the sole product, but the higher homologues give more or less of the isomeric nitrous esters.

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  • The former is the principal product of this province.

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  • Several members of the order are used medicinally for the strong purging properties of the milky juice (latex) which they contain; scammony is the dried latex from the underground stem of Convolvulus Scarnmonia, a native of the Levant, while jalap is the product of the tubercles of Exogonium Purga, a native of Mexico.

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  • We find that the product a m X a m X a m is equal to a im; and, by definition, the product; /a X -la X Ala is equal to a, which is a'.

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    0
  • The numerical factor 6 is called the coefficient of a 5 bc 2 (� 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.

    0
    0
  • Suppose, for instance, that n=5, so that we take five factors (A+a) (B+b) (C+c) (D+d) (E+e) and find their product.

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  • If, however, the 3.4.5 results of making changes like this are written down, it will be seen that any one term in the required product is written down several times.

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  • (iii.) The product 1.2.

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  • the product of any r consecutive integers is divisible by r!

    0
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  • (vi.) The product r!

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  • r we regard the multiplications as taking place from left to right; and similarly in r A product in which multiplications are taken in this order is called a continued product.

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  • (b) Let us assume that the product of every set of p consecutive integers is divisible by p!, and let us try to prove that the product of every set of p+ I consecutive integers is divisible by (p+i)!.

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  • Denote the product n(n +I) ..

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  • - The discussion, in � of the number of terms of a particular kind in a particular product, forms part of the theory of combinatorial analysis, which deals with the grouping and arrangement of individuals taken from a defined stock.

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  • ., the sum of the indices in each product being r.

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  • These laws can be established either by tracing the individual terms in a sum or a product or by means of the general theorem in � 52 (vi.).

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  • (a+r+ p - i �b), we divide the product of the p+1 factors which occur either in the nth or in the (n+i)th term by p+ 1, and by the common difference of the factors, and add to a constant, whose value is found by putting n= o.

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  • Another kind of sequence is that which is formed by introducing the successive factors of a continued product; e.g.

    0
    0
  • A continued product of this kind can, by taking logarithms, be replaced by an infinite series.

    0
    0
  • The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.

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  • Every unit of the rth species which does not vanish is the product of r different units of the first species; two such units are independent unless they are permutations of the same set of primary units e i, in which case they are equal or opposite according to the usual rule employed in determinants.

    0
    0
  • If r+s>n, a product such as E r E 3, worked out by the previous rules, comes out to be zero.

    0
    0
  • If E is any extensive unit, there is one other unit E', and only one, such that the (progressive) product EE' =- r.

    0
    0
  • Now when r+s>n, the product E r E B is defined to be that unit of which the supplement is the progressive product IE,4E 8.

    0
    0
  • The symbol AB, from this point of view, is called the inner product of A and B, as distinguished from the outer product IAB.

    0
    0
  • An inner product may be either progressive or regressive.

    0
    0
  • In the first two cases the product is said to be pure, in the third case mixed.

    0
    0
  • A pure product is associative; a mixed product, speaking generally, is not.

    0
    0
  • The sum and product of two quantities are defined in the first instance by the formulae zae -IE(3e = E (a +0) e, Za,ei X E ai e j = (a iai) eie9, so that the laws A, C, D of � 3 are satisfied.

    0
    0
  • One of the most important questions in investigating a linear algebra is to decide the necessary relations between a and b in order that this product may be zero.

    0
    0
  • The name l'arte magiore, the greater art, is designed to distinguish it from l'arte minore, the lesser art, a term which he applied to the modern arithmetic. His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms toss or algebra, cossic or algebraic, cossist or algebraist, &c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square.

    0
    0
  • The value of the factory product in 1905 was $10,591,143, being 10 6% more than the product value of 1900.

    0
    0
  • The value of the city's factory product was $48,036,593 in 1905, $4 1, 74 1, 9 8 0 in 1900.

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  • The manufacture of textiles is the most important industry; in 1905 the city produced worsteds valued at $30,926,964 and cotton goods worth $5,745,611, the worsted product being greater than that of any other American city.

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    0
  • By reducing the oxide with charcoal at a high temperature, he obtained a product which he took to be metallic uranium.

    0
    0
  • In any case the titles are manifestly the product of the same uncritical spirit as we have just been speaking of, for not only are many of the titles certainly wrong, but they are wrong in such a way as to prove that they date from an age to which David was merely the abstract psalmist and which had no idea whatever of the historical conditions of his age.

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    0
  • Scholtz (Ber., 1894, 2 7, p. 2 95 8) from piperonyl acrolein (the condensation product of piperonal and acetaldehyde) and acetic acid.

    0
    0
  • Anyhow, it is much to be regretted that no Syriac writing from Harran has survived.3 Syriac literature continued in life from the 3rd to the 14th century A.D., but after the Arab conquest it became an increasingly artificial product, for Arabic gradually killed the vernacular use of Syriac.

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  • za`faran), a product manufactured from the dried stigmas and part of the style of the saffron crocus, a cultivated form of Crocus sativus; some of the wild forms (var.

    0
    0
  • In 1905 its factory product was valued at $6,731,740, an increase of 41 6% since 1900.

    0
    0
  • Crum was probably the first to recognize that some hydrogen atoms of the cellulose had been replaced by an oxide of nitrogen, and this view was supported more or less by other workers, especially Hadow, who appears to have distinctly recognized that at least three compounds were present, the most violently explosive of which constituted the main bulk of the product commonly obtained and known as guncotton.

    0
    0
  • This particular product was insoluble in a mixture of ether and alcohol, and its composition could be expressed by the term tri-nitrocellulose.

    0
    0
  • The nitrated product retains the outward form of the original cellulose.

    0
    0
  • No effort is made to improve the Venezuelan product, a part of which is exported to Cuba for cigar making.

    0
    0
  • Cacau (Theobroma cacao) is an indigenous product and is extensively cultivated on the Caribbean slopes.

    0
    0
  • Its principal product is " papelon," or brown sugar, which is put on the market in the shape of small cylindrical and cubical masses of 14 to 31 lb weight.

    0
    0
  • Among its manufactures are foundry and machineshop products, boilers, carriages and wagons, agricultural implements, pipe and fittings, working-men's gloves, &c. In 1905 the total factory product was valued at $6,729,381, or 61.5% more than in 1900.

    0
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  • In a similar manner, formic acid and dicyanophenylhydrazine yields a phenyl-triazole carboxylic acid, in which the phenyl group may be nitrated, the nitro group reduced to the amino group, and the product oxidized to a triazole carboxylic acid, which, by elimination of carbon dioxide, yields the free triazole: HO 2 C C=N H02C C =N, NPh-> N :CH N :CH They also result when the acidylthiosemicarbazides are strongly heated, the mercapto-triazoles so formed being converted into triazoles on oxidation with hydrogen peroxide (M.

    0
    0
  • The value of the factory product in 1904 was $13,141,767, an increase of 82.9% since 1900.

    0
    0
  • It wastes in the disease known as " myxoedema," and the above product gathers in the tissues, in that disease, to such an extent as to give rise to what has been termed a " solid oedema."

    0
    0
  • Nowak, however, found later that he could generate it where the turpentine failed to induce suppuration; he believes that it may arise quite apart from the influence of the organisms of suppuration, that it is not a biological product of the micro-organisms of disease, and also that it has nothing to do with emaciation.

    0
    0
  • In fact the whole Samaj movement is as distinct a product of the contest of Hinduism with Christianity in the 19th century, as the Panth movement was of its contest with Islam 300 years earlier.

    0
    0
  • The action is very rapid, and the product, which rises to the top of the acids, is separated and washed successively with cold and then tepid water, and finally with water made slightly alkaline with sodium carbonate or hydroxide, to remove all adhering or dissolved acids which would otherwise render the product very unstable.

    0
    0
  • It was the first definite product of Greek medicine on Roman soil, but was destined to be followed by others, which kept up a more or less successful rivalry with it, and with the Hippocratic tradition.

    0
    0
  • officinalis, and that all that is found in modern commerce is the product of the Liquidambar orientalis of Cyprus and Anatolia.

    0
    0
  • In deep mines the pillars may furnish the bulk of the product, and the control of the fall of the roof, so as to permit the successful extraction of the mineral, demands a well-schemed plan of operation.

    0
    0
  • When the mineral is transported by rail or water to concentration or metallurgical works for treatment, or to near or distant markets for sale, provision must be made for the economical loading of railway wagons or vessels, and for the temporary storage of the mineral product.

    0
    0
  • The value of the total factory product in 1905 was $20,208,715, which was 17% greater than that for 1900.

    0
    0
  • In 1905 the value of the factory product was $5,231,084, Newport ranking third among the manufacturing centres of the state.

    0
    0
  • The cornmercial product (which is known in Germany as "Kalkstickstof") contains from 14 to 22% of nitrogen, which is liberated as ammonia when the substance is treated with water; to this decomposition it owes its agricultural value.

    0
    0
  • The examination of small test-pieces of the glass withdrawn from the crucible by means of an iron rod having shown that the molten mass is free from bubbles, the stirring process may be begun, the object of this manipulation being to render the glass as homogeneous as possible and to secure the absence of veins or striae in the product.

    0
    0
  • The raw materials used in this manufacture are chosen with considerable care, since the requirements as to the colour of the product are somewhat stringent.

    0
    0
  • with great care so as to secure a product as free from colour as possible, since the relatively great thickness of the sheets FIG.

    0
    0
  • The product is a crystalline solid of specific gravity 2.34, and melts at about 1430° C. See also German Patent 108817 for the production of crystallized silicon from silica and carborundum.

    0
    0
  • These are passed through a vessel surrounded by a freezing mixture and on fractionating the product the hydride distils over as a colourless liquid which boils at 52° C. It is also obtained by the decomposition of lithium silicide with concentrated hydrochloric acid.

    0
    0
  • Silicon iodoform, SiHI 3, is formed by the action of hydriodic acid on silicon, the product, which contains silicon tetraiodide, being separated by fractionation.

    0
    0
  • Iron, for instance, is converted first into FeC1 21 ultimately into FeCl 3, which practically means a mixture of the two chlorides, or pure FeC1 3 as a final product.

    0
    0
  • The ultimate chlorination product of copper, CuC1 2, when heated to redness, decomposes into the lower chloride, CuCI, and chlorine.

    0
    0
  • I n a straight uniform current of fluid of density p, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current wit