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multiply

Machines multiply our labor and increase our ability to do work.

7237But the old abuses continuing to multiply, the Prussian towns and gentry at last took their affairs into their own hands, and formed a so-called Prussian League, which demanded an equal share in the government of the country.

3825The outcome has been to raise the dignity of the calling, to induce persons of a superior class to adopt it in increasing numbers, to enlarge the demand for their services, and to multiply the means of educating them.

2017To obtain the real form we multiply out, and, in the result, substitute for the products of symbols the real coefficients which they denote.

1210We can multiply out this expression so as to obtain a series of monomials of the form 9(sl is2 2 s3 3 ...).

1110(ii.) In the same way, if we multiply the table by c, the sum of the first n numbers in any column is equal to the nth number in the next following column.

106(ii.) In the same way, if we multiply the table by c, the sum of the first n numbers in any column is equal to the nth number in the next following column.

106Others have a special faculty of consuming dry, powdery vegetable and animal refuse, and are liable to multiply in manufactured products of this nature, such as mouldy cheese.

108without a hydroid phase; the medusa develops directly from the actinula larva, which may, however, multiply by budding.

85without a hydroid phase; the medusa develops directly from the actinula larva, which may, however, multiply by budding.

85(f, 4)) k +(f, 4)) k +ï¿½(f, 4/) k +aï¿½(1, 4)')k; and, moreover, if we require to find the kth transvectant of one linear system of forms over another we have merely to multiply the two systems, and take the k th transvectant of the separate products.

811(v.) When we have to multiply two multinomials arranged according to powers of x, the method of detached coefficients enables us to omit the powers of x during the multiplication.

65In the first, a cleavage follows each nuclear division; in the second, the nuclei multiply by division a number of times, and then the ovum divides into as many blastomeres as there are nuclei present.

66To perpetuate and multiply the choicer varieties, peaches and nectarines are budded upon plum or almond stocks.

67In the scale of C the intervals from the key-note, the frequency ratios with the key-note, the successive frequency ratios and the successive intervals are as follows: - If we pass through two intervals in succession, as, for instance, if we ascend through a fourth from C to F and then through a third from F to A, the frequency ratio of A to C is a, which is the product of the ratios for a fourth 4, and a third I That is, if we add intervals we must multiply frequency ratios to obtain the frequency ratio for the interval which is the sum of the two.

54each number in the table) by a, and for the second series we multiply it by b, and the terms and their successive sums are given for the first series by the first and the second columns, and for the second series by the second and the third columns.

56Nothing could be more false than the common opinion that as a financier his sole expedient was to multiply the emissions of assignats.

44In a multiply connected space, like a ring, with a multiply valued velocity function ¢, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example, 4) =mB=m tan - l y/x (5) will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.

32His Hamburg interests continued from that date onwards to multiply in something like geometrical progression.

32mesentericus do not develop below 15° C.; but if it be introduced into the alimentary canal of a child the spores will rapidly multiply, and in such cases large quantities of gas, giving rise to flatulency, will be formed, and possibly also poisonous decomposition products of albuminoid matter.

32The use of the slips for the purpose of multiplication is now evident; thus to multiply 2085 by 736 we take out in this manner the multiples corresponding to 6, 3, 7, and set down the digits as they are obtained, from right to left, shifting them back one place and adding up the columns as in ordinary multiplication, viz.

33The actinula, when free, may multiply by larval budding, but in all cases both the original actinula and all its descendants become converted into medusae, so that there is no alter nation of generations.

33To multiply A 1 by a scalar, we apply the rule A = A1E = E (Eat) ea, and similarly for division by a scalar.

34i was 40 fathoms in depth, we multiply the depth by the rate of inclination, 40 X 8 = 320 fathoms, which gives the point at which the coal seam A should reach the surface.

34The set of ten rods is thus equivalent to four sets of slips as described above, and by their means we may multiply every number less than II,irr, and also any number (consisting of course FIG.

35The set of ten rods is thus equivalent to four sets of slips as described above, and by their means we may multiply every number less than II,irr, and also any number (consisting of course FIG.

35But if the little bit I've got's not stopping me from trying to multiply it to something better, how can you be worried about this kid of yours?

37Thus the Society for Promoting Christian Knowledge (founded 1698), besides its other activities, has done much to cheapen and multiply copies of the Scriptures, not only in English and Welsh, but in many foreign languages.

21The principle of the method is to multiply the given prime (supposed to consist of 4, 5 or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by I or 2, the prime factors may all be within the range of the logarithmic tables.

21appertaining to the function (li'l32l3...), each separation having a specification m" ` ' 8 m ï¿½2s m ï¿½38 multiply b P (is 2s 3s .ï¿½ï¿½), P Y by ls!

23Suppose that the dielectric has a constant K, then we must multiply both sides by K and the expression for the energy per unit of volume of the field is equivalent to z DE where D is the displacement or polarization in the dielectric.

23Taking as an example the calculation of the Briggian logarithm of the number 43,867, whose hyperbolic logarithm has been calculated above, we multiply it by 3, giving 131,601, and find by Gray's process that the factors of 1.31601 are (I) 1.316 (5) I.

23These young individuals can themselves multiply by equal binary fission, giving Anterior flagellum; Posterior flagellum; Longitudinal striations nemes); Cytoplasmic vacuole.

23These young individuals can themselves multiply by equal binary fission, giving Anterior flagellum; Posterior flagellum; Longitudinal striations nemes); Cytoplasmic vacuole.

23These varieties tend to multiply, and it is difficult to reduce them all to a few types.

24- As is the case with the Oligochaeta, the Polychaeta furnish examples of species which multiply asexually by budding.

28Of course two sets of rods may be used, and by their means we may multiply every number less than 111,111,111 and so on.

11Of course two sets of rods may be used, and by their means we may multiply every number less than 111,111,111 and so on.

11To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically.

12Thus, to multiply x 3 -2x+1 by 2x 2 +4, we write the process +I +0-2+I +2+0+4 +2+0-4+2 +0+0 - 0+0 +4+0-8+4 +2+0+0+2-8+4 giving 2x 5 + 2x 2 -8x+4 as the result.

12These micro-organisms having found in the tissues everything favourable for their needs, rapidly multiply and very soon produce serious results.

12These being freed from the normal inhibiting power of the neighbouring elements, multiply and go on to the formation of a new growth.

12In ordinary sound waves the displacement is very minute, perhaps of the order 105 cm., so that we multiply it perhaps by ioo,000 in forming the displacement curve.

12Not to multiply examples further, it may suffice to refer to (a) the apparent belief that the serpent tempted Eve to unchastity (2 Cor.

12To destroy the first 6 multiply by 1- 0006 giving 1000063361744, and multiplying successively by i- 00006 and 1-.

12In his history of the Arietidae Hyatt points out that toward the close of the Cretaceous this entire group of ammonites appears to have been affected with some malady; the unrolled forms multiply, the septa are simplified, the ornamentation becomes heavy, thick, and finally disappears in the adult; the entire group ends by dying out and leaving no descendants.

12The indifferent parasites exhibit an alternation of resting, attached phases with active periods, during which they multiply actively and become very abundant in the insect.

12Not to multiply examples further, it may suffice to refer to (a) the apparent belief that the serpent tempted Eve to unchastity (2 Cor.

12In his history of the Arietidae Hyatt points out that toward the close of the Cretaceous this entire group of ammonites appears to have been affected with some malady; the unrolled forms multiply, the septa are simplified, the ornamentation becomes heavy, thick, and finally disappears in the adult; the entire group ends by dying out and leaving no descendants.

12+aï¿½xn multiply each side by I +px, thus introducing a new quantity A; we obtain (1 +a1x) (1+a2x)...(1 -Fanx)(1+,ux) = 1+(a1 +1a)x + (a2+1aa1)x2+...

13Concordia discors pointed out the absurdity of the constant tendency to multiply oaths, while "remonstrances," "narratives," "queries," "prescriptions," "vindications," "declarations" and "statements" were scattered broadcast.

13For a time after entry they multiply, obtaining the nitrogen necessary for their nutrition and growth from the free nitrogen of the air, the carbohydrate required being supplied by the pea or clover plant in whose tissues they make a home.

13As any bacteria present in the milk tend to multiply rapidly on the way to the consumer, it is mainly a question of the time which elapses before consumption.

13- If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -ï¿½-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational integral function of the coefficients an, say f (al °, aol, ...) =f exp(ï¿½dlo+vdol)f d a P-1,4, dot = dapg The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem.

00The parasites multiply in two ways - (a) by binary fission, and (b) by multiple division or segmentation.

00Multiply 970224 By The Year Of The Hegira, Cut Off Six Decimals From The Product, And Add 621.5774.

00If this exhalation is stopped or lessened the digestion in its turn is also stopped, the leaf remains longer than usual in the intestines, the microbes multiply, invading the whole body, and this brings about the sudden death which surprises the rearers.

00They only travel by night; and, staying in congenial places for considerable periods, with unaccustomed abundance of provender, notwithstanding the destructive influences to which they are exposed, they multiply excessively during their journey, having families more numerous and frequent than in their usual homes.

00To place itself in a better posture for combating the simoniacal and concubinary prelates, the court of Rome had had to multiply exemptions and accelerate the movement which impelled the monks to make themselves independent of the bishops.

00The most successful gardening is that which turns to the best account the plastic organization of the plant, and enables it to develop and multiply as perfectly as possible.

00to feed the rest of the race; and that with every decade our food forms a smaller proportion of our needs, so rapidly do these multiply and diversify.

00From the 2nd century onwards references multiply, though the earlier references do not prove the existence of a fixed lectionary or order of lessons, but rather point the other way.

00With integers, besides adding and subtracting, it was easy to double and to multiply by 10:

00Both plants multiply solely by means of zoospores.

00The field rats (Mus mettada) occasionally multiply so exceedingly as to diminish the out-turn of the local harvest, and to require special measures for their destruction.

00Multiply once more by the number-couple (o, r), and we have (o, 1)(a, 1)(a,(3)=(o, 1)(- 13,a)=(- a,- 13)=(-1,o)(a, 13)= - (a,[3).

00Not to multiply examples, it must suffice to refer to the old popular idea that medical skill could be gained by eating some part of a serpent: the idea that its valuable qualities would thus be assimilated belongs to one of the fundamental dogmas of primitive mankind (cf.

00j If we multiply these equations by mf, mf,.

00If we multiply these by p, q, r and add, we get ~ ~ (3)

00If we multiply theta by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap2 + Bqf + Cr1 and A1p2 + Bfqi + Ciri are both consta~it.

00a,, respectively, multiply the resulting equations by a1t, af,.

00The terms affected with the coefficients ~3,-., on the other hand are such as occur in cyclic systems with latent motion (DYNAMICS, Analytical); they are called the gyrostatic terms. If we multiply (33) by 4~.

00Herbert Spencer, again, before the decline in question set in, put forward the hypothesis that "the ability to maintain individual life and the ability to multiply vary inversely"; in other words, the strain upon the nervous system involved in the struggle for life under the conditions of modern civilization, by reacting on the reproductive powers, tends towards comparative sterility.

00" There had always existed in France, as elsewhere, those who loved traditional stories of a marvellous nature, and tended to multiply the number which were presented as facts rather than legends.

00Among animals, the European rat, goat and pig are naturalized in New Zealand, where they multiply to such an extent as to injure and probably exterminate many native productions.

00It is unnecessary to multiply instances to prove that idealism was never without a protest that there is a heart of existence, life, will, action, which is presupposed by all knowledge and is not itself amenable to explanation.

00These enter the body through various channels, and once they have effected a lodgment they grow, multiply and give rise to various poisons which attack and injure or destroy different tissues or organs in the body.

00Either because running water was not always at hand, or as part of the growing tendency of the church to multiply ceremonies, rituals arose late in the 3rd century for consecrating water.

00Under favourable conditions the elements in the zoogloea again become active, and move out of the matrix, distribute themselves in the surrounding medium, to grow and multiply as before.

00These forms are termed by Fischer Metatrophic, because they require various kinds of organic materials obtained from the dead remains of other organisms or from the surfaces of their bodies, and can utilize and decompose them in various ways (Polytrophic) or, if monotrophic, are at least unable to work them up. The true parasites - obligate parasites of de Bary - are placed by Fischer in a third biological group, Paratrophic bacteria, to mark the importance of their mode of life in the interior of living organisms where they live and multiply in the blood, juices or tissues.

00In the cells of the nodule the bacteria multiply and develop, drawing material from their host.

00In such diseases the bacteria, when introduced into the subcutaneous tissue, rapidly gain entrance to the blood stream and multiply freely in it, and by means of their toxins cause symptoms of general poisoning.

00It would not be difficult to multiply these antitheses in the character and the opinions of this singular man.

00About 1870, camps, summer cottages, summer hotels and boarding houses began to multiply throughout the state.

00His own fortune had all been spent and "troubles did still multiply upon him."

00Like pashas they rarely held their office more than three years, it being the natural policy of the Porte to multiply such lucrative nominations.

00Sand filtration, even when working in the best possible manner, falls short of the perfection necessary to prevent the passage of bacteria which may multiply after the filter is passed.

00- If we multiply n P by n q, we multiply the product of p n's by the product of q n's, and the result is therefore n P + q.

00We therefore use "a of A" to mean that we find a quantity X such that a X X =A, and then multiply X by n.

00To multiply a by is to take 4 times 3.

00Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e.

00Hence, to find a percentage of a percentage, we multiply the two numbers, put o's in front if necessary to make up four figures (not counting fractions), and prefix the point.

00To multiply two decimals exactly, we multiply them as if the point were absent, and then insert it so that the number of figures after the point in the product shall be equal to the sum of the numbers of figures after the points in the original decimals.

00To add or subtract fractional numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index.

00as shillings we multiply 153 by 20 and add 7.

00£153 denotes 15 3 units, each of which is £i or 20s.; and therefore we must multiply 20S.

00multiply 20 by 153 (the unit being now is.) and add 7.

00If we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5.

00We cannot multiply 4 apples by 5 boys, for then we should get 20 " boy-apples," an expression which has no meaning.

00Similarly, to convert £153 into shillings we must multiply it by a factor 2 £I ' ' so that we get 2£I.

00To Multiply By 3 We Use The Row.

00The Following Are Some Special Rules: (I) To Multiply By 5, Multiply By Io And Divide By 2.

00(And Conversely, To Divide By 5, We Multiply By 2 And Divide By Io.) (Ii) In Multiplying By 2, From The Left, Add I If The Next Figure Of The Multiplicand Is 5, 6, 7, 8 Or 9.

00(Iv) To Multiply By 7, 8, 9, Ii Or 12, Treat The Multiplier As 10 3, 10 2, Io I, 10 1 Or 10 2; And Similarly For 13, 17, 18, 19, &C.

00(V) To Multiply By 4 Or 6, We Can Either Multiply From The Left By 2 And Then By 2 Or 3, Or Multiply From The Right By 4 Or 6; Or We Can Treat The Multiplier As 5 I Or 5 1.

00To Multiply 4273 By 200, We Use The Commutative Law, Which Gives 200.4273 = 2 X100 X4273 2X4273X100=8546X100=854600; And Similarly For 30.4273.

00The principle is that 162.4 27 =100.4 2 7+ 60.4 2 7+ 2.4 2 7 = 1.42700+6.4270+2.427; but, instead of 427 writing down the separate products, we 6 427 (in effect) write 42700, 4270 and 427 in 2 4?7 separate rows, with the multipliers 1, 6, 2 69 174 in the margin, and then multiply each number in each column by the corresponding multiplier in the margin, making allowance for any figures to be " carried."

00a lb, we multiply is.

00Thus, to multiply L7, 13s.

00In incomplete partition the quotient is 3, and the remainders 11 and 17 are in effect disregarded; if, after finding the quotient 3, we want to know what remainder would be produced by'a direct division, the simplest method is to multiply 3 by 2 4 0 and subtract the result from 935.

00When we have to multiply two numbers, and the product is only required, or can only be approximately correct, to a certain number of significant figures, we need only work to two or three more figures (§ 83), and then correct the final figure in the result by means of the superfluous figures.

00A better method is to ignore the positions of the decimal points, and multiply the numbers as if they were decimals between I and i o.

00Thus, 0859 to multiply 2 7.343 by 3.14159 2 7 to one decimal place, we require 2+ I+ I =4 figures in the product.

00For example, 3.14 16= 3Xi g t o = 3x l g = 3X g fxz 9= 3(I + 1 1) (I - 2-50-6) Hence, to multiply by 3.1416, we can multiply by 34, and subtract 2 0 o (_ .0004) of the result; or, to divide by 3 .

00The number of persons holding immediately of the crown began at once to multiply by leaps and bounds.

00Yet they continued to multiply, and exercised at times considerable influence; though they had few supporters among the baronage, yet among the lesser gentry and still more among the burgher class and in the universities they were strong.

00External troubles continued to multiply during Henrys earlier years.

00Nearly all the Baltic goods, and most of those from Denmark and Norway, had been reaching London or Hull in foreign bottoms. Henry allied himself with John of Denmark, who was chafing under the monopoly of the Hansa, and obtained the most ample grants of free trade in his realms. The Germans murmured, but the English shipping in eastern and northern waters continued to multiply.

00Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.

00By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.

00They became petty local tyrants, all the more despotic because they had nothing to fear save the distant authority of the kings missi, and the more rapacious because they had no salary save the fines they inflicted and the fees that they contrived to multiply.

00If the figure be entirely to one side of any face the polyhedron is said to be " convex, " and it is obvious that the faces enwrap the centre once; if, on the other hand, the figure is to both sides of every face it is said to be concave, " and the centre is multiply enwrapped by the faces.

00the Platonic solids, while the four polyhedra in which the centre is multiply enclosed are referred to as the Kepler-Poinsot solids, Kepler having discovered three, while Poinsot discovered the fourth.

00They bear a relation to the Platonic solids similar to the relation of " star polygons " to ordinary regular polygons, inasmuch as the centre is multiply enclosed in the former and singly in the latter.

00Some of the apterous young that are hatched from these form fresh galls and continue to multiply in the leaves, others descend to the root of the plant, becoming what are known as root-forms. These, like the parent form of spring, reproduce parthenogenetically, giving rise to generation after generation of egg-laying individuals.

00"Imagine all the baggage a human has after one lifetime and multiply that by a few thousand," Fate replied.

00But if the little bit I've got's not stopping me from trying to multiply it to something better, how can you be worried about this kid of yours?

00To get a rough approximation of how much sums in the 1890s would be in today's terms we multiply by 60.

00Technologies, Synergies and Force Multipliers Using technology to multiply the impact of military forces seems almost axiomatic.

00Delbrueck chose to use bacteriophage, viruses that infect bacteria and then can multiply very rapidly.

00In order to cause the disease, the single-celled parasite must multiply in the human bloodstream.

00The protein subunits in a virus capsid are multiply redundant, i.e. present in many copies per particle.

00coax these stem cells to multiply faster.

00The first sentence is an algorithm rule often employed by students to explain how to multiply decimals.

00Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and explain the effect.

00faith-based initiatives, all of you are helping to multiply the compassion of our government.

00flagellate stage can multiply.

00His defining moment was when he showed his teacher how to multiply mixed fractions.

00Researchers found that germanium reduces the spread of cancer by slowing down the process that causes cancer cells to multiply.

00How has it been possible to multiply the stock of german loanwords in English?

00The organisms penetrate the gut mucosa via Peyer's patches and multiply within intestinal lymph nodes.

00multiply mixed fractions.

00Bacteria start to multiply within the blockage, which leads to inflammatory lesions or red papules.

00In order to cause the disease, the single-celled parasite must multiply in the human bloodstream.

00BLOCKS Blocks are multiply aligned ungapped segments corresponding to the most highly conserved regions of proteins.

00To find the distance the strands should be unlaid for an eye splice, multiply the diameter of the wire by 36 inches.

00syndicated nationally, they can multiply with surprising ease.

00theorizessay explores the implications of reading a translated text as multiply original by theorizing the practice of rendering ' Matha ' in English.

00Rivalries and tensions multiply and there is no shared external enemy to keep the triumvirate together.

00The use of the slips for the purpose of multiplication is now evident; thus to multiply 2085 by 736 we take out in this manner the multiples corresponding to 6, 3, 7, and set down the digits as they are obtained, from right to left, shifting them back one place and adding up the columns as in ordinary multiplication, viz.

00In the first, a cleavage follows each nuclear division; in the second, the nuclei multiply by division a number of times, and then the ovum divides into as many blastomeres as there are nuclei present.

00The actinula, when free, may multiply by larval budding, but in all cases both the original actinula and all its descendants become converted into medusae, so that there is no alter nation of generations.

00The revolutionary terrorists took advantage of the situation to multiply outrages; popular agitation was fomented by a multitude of new journals preaching every kind of extravagant doctrine, now that the censor no The longer dared to act; in December the trouble "union culminated in a formidable rising in Moscow.

00They occupy and destroy the red corpuscles, converting the haemoglobin into melanin; they multiply in the blood by sporulation, and produce accessions of fever by the liberation of a toxin at the time of sporulation (Ross).

00- As is the case with the Oligochaeta, the Polychaeta furnish examples of species which multiply asexually by budding.

00It is not necessary to multiply authors, as is done, for example, by Siegfried, who supposes four principal writers (a pessimistic philosopher, an Epicurean glossator, a sage who upholds the value of wisdom, and an orthodox editor) besides a number of annotators; it is sufficient to assume that several conservative scribes have made short additions to the original work.

00To find its scale divide the mean diameter of the earth (1,273,500 m.) by the diameter of the globe; to find its circumference multiply the diameter by Map Printing.

00To perpetuate and multiply the choicer varieties, peaches and nectarines are budded upon plum or almond stocks.

00If we multiply the elements of the second row by an arbitrary magnitude X, and add to the corresponding elements of the first row, A becomes Zai,A18+XEa28A13 = Lia13A18 =A, showing that the value of the determinant is unchanged.

00Such a determinant when of uneven degree vanishes, for if we multiply each row by - I we multiply the determinant by (- I) n = -1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered.

00We have to multiply a01; -alas+a2 by ao, -aif32+a2 and we obtain ao (3 - aoal(f31N2 +01133) +aoa2(SI+13) -i-a?31a2 - aIa2(31 + 02) + al, 131+02 = b, 131132 = b t'i +s2 = 2bob2, and clearing of fractions R 1,5 = (a o b 2 - a2 b o) 2 + (a i b o - aobi) (aib2 - a2b1).

00We can multiply out this expression so as to obtain a series of monomials of the form 9(sl is2 2 s3 3 ...).

00appertaining to the function (li'l32l3...), each separation having a specification m" ` ' 8 m Ã¯¿½2s m Ã¯¿½38 multiply b P (is 2s 3s .Ã¯¿½Ã¯¿½), P Y by ls!

00+aÃ¯¿½xn multiply each side by I +px, thus introducing a new quantity A; we obtain (1 +a1x) (1+a2x)...(1 -Fanx)(1+,ux) = 1+(a1 +1a)x + (a2+1aa1)x2+...

00- If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -Ã¯¿½-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational integral function of the coefficients an, say f (al Ã‚°, aol, ...) =f exp(Ã¯¿½dlo+vdol)f d a P-1,4, dot = dapg The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem.

00(V.) As an illustration multiply (IV.) throughout by az 2b x 2cz 2 so that each term may denote a covariant of an niÃ‚°.

00To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically.

00(f, 4)) k +(f, 4)) k +Ã¯¿½(f, 4/) k +aÃ¯¿½(1, 4)')k; and, moreover, if we require to find the kth transvectant of one linear system of forms over another we have merely to multiply the two systems, and take the k th transvectant of the separate products.

00Put M 1 For M, N I For N, And Multiply Through By (Ab); Then { (F, C6) } = (Ab) A X 2A Y B X 1 M N I 2 (Xy), ?) 2, = (A B)Ax 1B X 2B Y L I Multiply By Cp 1 And For Y L, Y2 Write C 2, C1; Then The Right Hand Side Becomes (Ab)(Bc)Am Lbn 2Cp 1 M I C P (F?) 2 M { N2 X, Of Which The First Term, Writing C P =, ,T, Is Mn 2 A B (Ab)(Bc)Axcx 1 M 2 N 2 P 2 2222 2 2 _2 A X B X C (Bc) A C Bx M N 2 2 2 M2Ã‚°N 2 N 2 M 2 2 A X (Bc) B C P C P (Ab) A B B(Ac) Ax Cp 2 = 2 (04) 2 1 (F,0) 2.4 (F,Y') 2 Ã¯¿½?; And, If (F,4)) 1 = Km " 2, (F??) 1 1 M N S X X X Af A _Af A Ax, Ax Ax Ax1 Observing That And This, On Writing C 2, C 1 For Y 11 Y 21 Becomes (Kc) K X 'T 3C X 1= (F,0 1 ', G 1; Ã¯¿½'Ã¯¿½1(F,O) 1 M 1=1 M 2 0`,4)) 2 0, T (Fm 2.4 (0,0 2 .F ' And Thence It Appears That The First Transvectant Of (F, (P) 1 Over 4) Is Always Expressible By Means Of Forms Of Lower Degree In The Coefficients Wherever Each Of The Forms F, 0, 4, Is Of Higher Degree Than The First In X 1, X2.

00To obtain the real form we multiply out, and, in the result, substitute for the products of symbols the real coefficients which they denote.

00Others have a special faculty of consuming dry, powdery vegetable and animal refuse, and are liable to multiply in manufactured products of this nature, such as mouldy cheese.

00But the present tendency is to regard the early association of arithmetic with linear measurement as important; and it seems to follow that we may properly (at any rate at an early stage of the subject) multiply a length by a length, and the product again by another length, the practice being dropped when it becomes necessary to give a strict definition of multiplication.

00Ã¯¿½ 23) that the algebraic result of performing the operation 3(7+2) is 3.7+3.2; it means that if we convert 7+2 into the single number 9 and then multiply by 3 we get the same result as if we converted 3.7 and 3.2 into 21 and 6 respectively and added the results.

00(v.) When we have to multiply two multinomials arranged according to powers of x, the method of detached coefficients enables us to omit the powers of x during the multiplication.

00Thus, to multiply x 3 -2x+1 by 2x 2 +4, we write the process +I +0-2+I +2+0+4 +2+0-4+2 +0+0 - 0+0 +4+0-8+4 +2+0+0+2-8+4 giving 2x 5 + 2x 2 -8x+4 as the result.

00.; for the first series we multiply the table (i.e.

00each number in the table) by a, and for the second series we multiply it by b, and the terms and their successive sums are given for the first series by the first and the second columns, and for the second series by the second and the third columns.

00To multiply A 1 by a scalar, we apply the rule A = A1E = E (Eat) ea, and similarly for division by a scalar.

00These micro-organisms having found in the tissues everything favourable for their needs, rapidly multiply and very soon produce serious results.

00These being freed from the normal inhibiting power of the neighbouring elements, multiply and go on to the formation of a new growth.

00Nothing could be more false than the common opinion that as a financier his sole expedient was to multiply the emissions of assignats.

00In a multiply connected space, like a ring, with a multiply valued velocity function ¢, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example, 4) =mB=m tan - l y/x (5) will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.

00To find the kinetic energy of such motion in a multiply connected space, the channels must be supposed barred, and the space made acyclic by a membrane, moving with the velocity of the liquid; and then if k denotes the cyclic constant of 0 in any circuit, or the value by which 4) has increased in completing the circuit, the values of 0 on the two sides of the membrane are taken as differing by k, so that the integral over the membrane dS= k f ?

00Concordia discors pointed out the absurdity of the constant tendency to multiply oaths, while "remonstrances," "narratives," "queries," "prescriptions," "vindications," "declarations" and "statements" were scattered broadcast.

00For a time after entry they multiply, obtaining the nitrogen necessary for their nutrition and growth from the free nitrogen of the air, the carbohydrate required being supplied by the pea or clover plant in whose tissues they make a home.

00During this period the bacteria multiply and most of them assume a peculiar thickened or branched form, in which state they are spoken of as bacteroids.

00Suppose that the dielectric has a constant K, then we must multiply both sides by K and the expression for the energy per unit of volume of the field is equivalent to z DE where D is the displacement or polarization in the dielectric.

00Multiply, therefore, the number of Armenian years elapsed by 365; add the number of days from the commencement of the current year to the given date; subtract 176 from the sum, and the remainder will be the number of days from the 1st of January 553 to the given date.

00The enormous rate at which aphids multiply under favourable conditions makes them of the greatest economic importance, since the growth of immense numbers of the same kind of plant in close proximity - as in ordinary farm-crops - is especially advantageous to the insects that feed on them.

00Within this gall the stock-mother lives and surrounds herself with numerous parthenogenetically produced eggs - sometimes as many as two hundred in a single gall; these eggs give birth after six or eight days to a numerous progeny (gallicola), some of which form new galls and multiply in the leaves, whilst others descend to the roots and become the root-dwelling forms already described.

00i was 40 fathoms in depth, we multiply the depth by the rate of inclination, 40 X 8 = 320 fathoms, which gives the point at which the coal seam A should reach the surface.

00His Hamburg interests continued from that date onwards to multiply in something like geometrical progression.

00In ordinary sound waves the displacement is very minute, perhaps of the order 105 cm., so that we multiply it perhaps by ioo,000 in forming the displacement curve.

00In the scale of C the intervals from the key-note, the frequency ratios with the key-note, the successive frequency ratios and the successive intervals are as follows: - If we pass through two intervals in succession, as, for instance, if we ascend through a fourth from C to F and then through a third from F to A, the frequency ratio of A to C is a, which is the product of the ratios for a fourth 4, and a third I That is, if we add intervals we must multiply frequency ratios to obtain the frequency ratio for the interval which is the sum of the two.

00These varieties tend to multiply, and it is difficult to reduce them all to a few types.

00The outcome has been to raise the dignity of the calling, to induce persons of a superior class to adopt it in increasing numbers, to enlarge the demand for their services, and to multiply the means of educating them.

00But the old abuses continuing to multiply, the Prussian towns and gentry at last took their affairs into their own hands, and formed a so-called Prussian League, which demanded an equal share in the government of the country.

00As any bacteria present in the milk tend to multiply rapidly on the way to the consumer, it is mainly a question of the time which elapses before consumption.

00It is, therefore, further recommended (a) that the milk be rapidly cooled or chilled, as the lower the temperature the less do the bacteria multiply, (b) that contamination during railway transit be avoided by dustproof locked milk cans.

00mesentericus do not develop below 15Ã‚° C.; but if it be introduced into the alimentary canal of a child the spores will rapidly multiply, and in such cases large quantities of gas, giving rise to flatulency, will be formed, and possibly also poisonous decomposition products of albuminoid matter.

00Thus the Society for Promoting Christian Knowledge (founded 1698), besides its other activities, has done much to cheapen and multiply copies of the Scriptures, not only in English and Welsh, but in many foreign languages.

00The principle of the method is to multiply the given prime (supposed to consist of 4, 5 or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by I or 2, the prime factors may all be within the range of the logarithmic tables.

00To destroy the first 6 multiply by 1- 0006 giving 1000063361744, and multiplying successively by i- 00006 and 1-.

00Taking as an example the calculation of the Briggian logarithm of the number 43,867, whose hyperbolic logarithm has been calculated above, we multiply it by 3, giving 131,601, and find by Gray's process that the factors of 1.31601 are (I) 1.316 (5) I.

00The indifferent parasites exhibit an alternation of resting, attached phases with active periods, during which they multiply actively and become very abundant in the insect.

00The parasites multiply in two ways - (a) by binary fission, and (b) by multiple division or segmentation.

00They suppressed the chimeras which went with belief in a complete survival after death, chimeras which were homicidal at the time, in so far as they robbed man of the true notion of death and led him to multiply murders."

00(1) Ile /3 c Tou irvpiov, On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Hepi On the Cylindrical Helix (mentioned by Proclus); (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere; (4) `H Ka06Xov lrpa-yµareta, perhaps a work on the general principles of mathematics in which were included Apollonius' criticisms and suggestions for the improvement of Euclid's Elements; (5) ' (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of 7r than the 37 and 3,4-A of Archimedes; (6) an arithmetical work (as to which see Pappus) on a system of expressing large numbers in language closer to that of common life than that of Archimedes' Sand-reckoner, and showing how to multiply such large numbers; (7) a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm.

00Multiply 970224 By The Year Of The Hegira, Cut Off Six Decimals From The Product, And Add 621.5774.

00If this exhalation is stopped or lessened the digestion in its turn is also stopped, the leaf remains longer than usual in the intestines, the microbes multiply, invading the whole body, and this brings about the sudden death which surprises the rearers.

00They only travel by night; and, staying in congenial places for considerable periods, with unaccustomed abundance of provender, notwithstanding the destructive influences to which they are exposed, they multiply excessively during their journey, having families more numerous and frequent than in their usual homes.

00To place itself in a better posture for combating the simoniacal and concubinary prelates, the court of Rome had had to multiply exemptions and accelerate the movement which impelled the monks to make themselves independent of the bishops.

00The most successful gardening is that which turns to the best account the plastic organization of the plant, and enables it to develop and multiply as perfectly as possible.

00to feed the rest of the race; and that with every decade our food forms a smaller proportion of our needs, so rapidly do these multiply and diversify.

00From the 2nd century onwards references multiply, though the earlier references do not prove the existence of a fixed lectionary or order of lessons, but rather point the other way.

00With integers, besides adding and subtracting, it was easy to double and to multiply by 10:

00Both plants multiply solely by means of zoospores.

00The field rats (Mus mettada) occasionally multiply so exceedingly as to diminish the out-turn of the local harvest, and to require special measures for their destruction.

00Multiply once more by the number-couple (o, r), and we have (o, 1)(a, 1)(a,(3)=(o, 1)(- 13,a)=(- a,- 13)=(-1,o)(a, 13)= - (a,[3).

00Not to multiply examples, it must suffice to refer to the old popular idea that medical skill could be gained by eating some part of a serpent: the idea that its valuable qualities would thus be assimilated belongs to one of the fundamental dogmas of primitive mankind (cf.

00j If we multiply these equations by mf, mf,.

00If we multiply by 2d~/dt and integrate, we obtain (dP)22~ cos~+const., (16)

00If we multiply these by p, q, r and add, we get ~ ~ (3)

00If we multiply theta by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap2 + Bqf + Cr1 and A1p2 + Bfqi + Ciri are both consta~it.

00a,, respectively, multiply the resulting equations by a1t, af,.

00The terms affected with the coefficients ~3,-., on the other hand are such as occur in cyclic systems with latent motion (DYNAMICS, Analytical); they are called the gyrostatic terms. If we multiply (33) by 4~.

00Herbert Spencer, again, before the decline in question set in, put forward the hypothesis that "the ability to maintain individual life and the ability to multiply vary inversely"; in other words, the strain upon the nervous system involved in the struggle for life under the conditions of modern civilization, by reacting on the reproductive powers, tends towards comparative sterility.

00" There had always existed in France, as elsewhere, those who loved traditional stories of a marvellous nature, and tended to multiply the number which were presented as facts rather than legends.

00In accordance with the principle of Renovation (see above), the root-idea of the application of religion to economics is not the extorting of boons from an unwilling nature, but rather the stimulation of the sources of life, so that all beings alike may increase and multiply.

00Among animals, the European rat, goat and pig are naturalized in New Zealand, where they multiply to such an extent as to injure and probably exterminate many native productions.

00It is unnecessary to multiply instances to prove that idealism was never without a protest that there is a heart of existence, life, will, action, which is presupposed by all knowledge and is not itself amenable to explanation.

00These enter the body through various channels, and once they have effected a lodgment they grow, multiply and give rise to various poisons which attack and injure or destroy different tissues or organs in the body.

00Either because running water was not always at hand, or as part of the growing tendency of the church to multiply ceremonies, rituals arose late in the 3rd century for consecrating water.

00Under favourable conditions the elements in the zoogloea again become active, and move out of the matrix, distribute themselves in the surrounding medium, to grow and multiply as before.

00These forms are termed by Fischer Metatrophic, because they require various kinds of organic materials obtained from the dead remains of other organisms or from the surfaces of their bodies, and can utilize and decompose them in various ways (Polytrophic) or, if monotrophic, are at least unable to work them up. The true parasites - obligate parasites of de Bary - are placed by Fischer in a third biological group, Paratrophic bacteria, to mark the importance of their mode of life in the interior of living organisms where they live and multiply in the blood, juices or tissues.

00In the cells of the nodule the bacteria multiply and develop, drawing material from their host.

00In such diseases the bacteria, when introduced into the subcutaneous tissue, rapidly gain entrance to the blood stream and multiply freely in it, and by means of their toxins cause symptoms of general poisoning.

00It would not be difficult to multiply these antitheses in the character and the opinions of this singular man.

00About 1870, camps, summer cottages, summer hotels and boarding houses began to multiply throughout the state.

00His own fortune had all been spent and "troubles did still multiply upon him."

00Like pashas they rarely held their office more than three years, it being the natural policy of the Porte to multiply such lucrative nominations.

00Sand filtration, even when working in the best possible manner, falls short of the perfection necessary to prevent the passage of bacteria which may multiply after the filter is passed.

00- If we multiply n P by n q, we multiply the product of p n's by the product of q n's, and the result is therefore n P + q.

00We therefore use "a of A" to mean that we find a quantity X such that a X X =A, and then multiply X by n.

00To multiply a by is to take 4 times 3.

00Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e.

00Hence, to find a percentage of a percentage, we multiply the two numbers, put o's in front if necessary to make up four figures (not counting fractions), and prefix the point.

00To multiply two decimals exactly, we multiply them as if the point were absent, and then insert it so that the number of figures after the point in the product shall be equal to the sum of the numbers of figures after the points in the original decimals.

00To add or subtract fractional numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index.

00as shillings we multiply 153 by 20 and add 7.

00

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