# Ratio sentence example

ratio
• The ratio of criminal proceedings to population is, as a rule, much higher in the south than in the north.
• This equation does not give us the value of the unknown factor but gives us a ratio between two unknowns.
• In the election of 1896 all the parties in the state declared in favour of the free and unlimited coinage of silver at the ratio of 16 to 1.
• The ratio "weight to volume" is the absolute density.
• The working expenses were reduced in a progressively larger ratio, e.g.
• Now this ratio is the same as that which gives the relative chemical equivalents of hydrogen and copper, for r gramme of hydrogen and 31.8 grammes of copper unite chemically with the same weight of any acid radicle such as chlorine or the sulphuric group, SO 4.
• If we assume that no other cause is at work, it is easy to prove that, with non-dissolvable electrodes, the ratio of salt lost at the anode to the salt lost at the cathode must be equal to the ratio of the velocity of the cation to the velocity of the anion.
• Thus, the ratio of the losses at the two ends is two to one - the same as the ratio of the assumed ionic velocities.
• Now Hittorf's transport number, in the case of simple salts in moderately dilute solution, gives us the ratio between the two ionic velocities.
• Its history does not run parallel with the scientific side, but rather varies in inverse ratio with scientific activity.
• In such cases the speed with which the wings are driven is increased in the direct ratio of the mutilation.
• When none of the radiations which fall on a body penetrates through its substance, then the ratio of the amount of radiation of a given wave-length which is absorbed to the total amount received is called the "absorptive power" of the body for that wave-length.
• In the case of solutions, if the absorption of the solvent is negligible, the effect of increasing the concentration of the absorbing solute is the same as that of increasing the thickness in the same ratio.
• In a similar way the absorption of light in the coloured gas chlorine is found to be unaltered if the thickness is reduced by compression, because the density is increased in the same ratio that the thickness is reduced.
• This distinction between Logos as ratio and Logos as oratio, so much used subsequently by Philo and the Christian fathers, had been so far anticipated by Aristotle's distinction between the g w Xo'yos and the Xoyos iv rff Ox?j.
• In that year the Swiss government reduced the rate for inland telegrams by one-half, and the traffic immediately doubled, but the cost of carrying on the service increased in a larger ratio.
• He applied himself more particularly to the oxygen compounds, and determined with a fair degree of accuracy the ratio of carbon to oxygen in carbon dioxide, but his values for the ratio of hydrogen to oxygen in water, and of phosphorus to oxygen in phosphoric acid, are only approximate; he introduced no new methods either for the estimation or separation of the metals.
• If, however, an amount of energy a is taken up in separating atoms, the ratio is expressible as C p /C„= (5+a)/(3-Fa), which is obviously smaller than 5/3, and decreases with increasing values of a.
• The following table gives a comparative view of the specific heats and the ratio for molecules of variable atomic content.
• For a further discussion of the ratio of the specific heats see Molecule.
• 7V k / R, and since Vk is proportional to the volume at absolute zero, the ratio T k /P k should exhibit additive relations.
• It is there shown that every substance, transparent to light, has a definite refractive index, which is the ratio of the velocity of light in vacuo to its velocity in the medium to which the refractive index refers.
• A crystal may be regarded as built up of primitive parallelepipeda, the edges of which are in the ratio of the crystallographic axes, and the angles the axial angles of the crystals.
• The average attendance of enrolled black and white pupils is practically identical, but the enrolment of whites (about 52% in 1902) is somewhat higher and that of the blacks about a third lower than their ratio in the population.
• The great abstract ideas (considered directly and not merely in tacit use) which have dominated the science were due to them - namely, ratio, irrationality, continuity, the point, the straight line, the plane.
• A better basis of comparison would be the ratio of the actual to the limiting conductivity, but since the conductivity of acids is chiefly due to the mobility of the hydrogen ions, its limiting value is nearly the same for all, and the general result of the comparison would be unchanged.
• When the solutions may be taken as effectively dilute, so that the gas laws apply to the osmotic pressure, this relation reduces to E _ nrRT to c1 ey gE c2 where n is the number of ions given by one molecule of the salt, r the transport ratio of the anion, R the gas constant, T the absolute temperature, y the total valency of the anions obtained from one molecule, and c i and c 2 the concentrations of the two solutions.
• = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.
• If the form, sometimes termed a quantic, be equated to zero the n+I coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.
• A volume entitled Opera posthuma (Leiden, 1703) contained his "Dioptrica," in which the ratio between the respective focal lengths of object-glass and eye-glass is given as the measure of magnifying power, together with the shorter essays De vitris figurandis, De corona et parheliis, &c. An early tract De ratiociniis tin ludo aleae, printed in 16J7 with Schooten's Exercitationes mathematicae, is notable as one of the first formal treatises on the theory of probabilities; nor should his investigations of the properties of the cissoid, logarithmic and catenary curves be left unnoticed.
• In the latter he advocated the unlimited coinage of silver, irrespective of international agreement, at a ratio of 16 to 1, a policy with which his name was afterwards most prominently associated.
• The theoretical element is the ratio of the parallactic inequality to the solar parallax.
• The determination of this ratio is one of the most difficult problems in the lunar theory.
• If the magnet is not uni - form, the magnetization at any point is the ratio of the moment of an element of volume at that point to the volume itself, or I = m.ds/dv.
• The internal force F is opposite to the direction of the magnetization, and equal to NI, where N is a coefficient depending only on the ratio of the axes.
• 3 In general, the greater the ratio of length to section, the more nearly will the poles approach the end of the bar, and the more nearly uniform will be the magnetization.
• - The ratio B/H is called the permeability of the medium in which the induction is taking place, and is denoted byµ.
• The ratio I/H is called the susceptibility of the magnetized substance, and is denoted by «.
• The ratio of the residual magnetization to its previous maximum value measures the retentiveness, or retentivity, of the metal.'
• (35) Du Bois has shown that _when the dimensional ratio in (= length/ diameter) exceeds t00, Nm 2 =constant=45, and hence for long thin rods N = 45/ m2.
• The ratio M/H is then found by one of the magnetometric methods which in their simplest forms are described below.
• It is often sufficient to find the ratio of the moment of one magnet to that of another.
• If the cardboard scale upon which the beam of light is reflected by the magnetometer mirror is a flat one, the deflections as indicated by the movement of the spot of light are related to the actual deflections of the needle in the ratio of tan 20 to 0.
• The specimen upon which an experiment is to be made generally consists of a wire having a " dimensional ratio " of at least 300 or goo; its length should be rather less than that of the magnetizing coil, in order that the field Ho, to which it is subjected, may be approximately uniform from end to end.
• They consider, however, that Kirchhoff's theory, which assumes change of magnetization to be simply proportional to strain, is still in its infancy, the present stage of its evolution being perhaps comparable with that reached by the theory of magnetization at the time when the ratio I/H was supposed to be constant.
• (ii.) The general theory of ratio and proportion requires the use of general symbols.
• Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of xn (ï¿½ 45)ï¿½ This conception of a limit enables us to deal with algebraical expressions which assume such forms as -° o for particular values of the variable (ï¿½ 39 (iii.)).
• We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.
• Archimedes' problem of dividing a sphere by a plane into two segments having a prescribed ratio,was first expressed as a cubic equation by Al Mahani, and the first solution was given by Abu Gafar al Hazin.
• His notation is based on that of Vieta, but he introduced the sign X for multiplication, - for continued proportion, :: for proportion, ' and denoted ratio by one dot.
• This last character has since been entirely restricted to multiplication, and ratio is now denoted by two dots (:).
• Accordingly, the amplitude of the resultant will be less than if all its components had the same phase, in the ratio +17r -17r or 2: 7.
• If, however, the primary wave be spherical, and of radius a at the wave-front of resolution, then we kno* that at a distance r further on the amplitude of the primary wave will be diminished in the ratio a:(r+a).
• At the central point there is still complete agreement of phase; but the amplitude is diminished in the ratio of a: a+d.
• The only effect of the ruling is to diminish the amplitude in the ratio a: a+d; and, except for the difference in illumination, the appearance of a line of light is the same as if the aperture were perfectly free.
• The effect of each of the elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a: d, the resultant amplitude becomes comparatively very great.
• Rabelais not only lectured on Galen and Hippocrates, but edited some works of the latter; and Michael Servetus (1511-1553), in a little tract Syruporum universa ratio, defended the practice of Galen as compared with that of the Arabians.
• Libitina was the goddess of funerals; her officers were the Libitinarii our undertakers; her temple in which all business connected with the last rites was transacted, in which the account of deaths - ratio Libitinae - was kept, served the purpose of a register office."
• To prevent excessive bending stresses the diameter of drum and sheave must bear a proper ratio to that of the rope.
• Thus, at every complete stroke of the piston, the air in the vessel or receiver was diminished by that fraction of itself which is expressed by the ratio of the volume of the available cylindrical space above the outward opening valve to the whole volume of receiver, nozzle and cylinder.
• He had discovered a contraction in the vein of fluid (vena contracta) which issued from the orifice, and found that, at the distance of about a diameter of the aperture, the section of the vein was contracted in the subduplicate ratio of two to one.
• 2, it will be balanced by a thrust W lb applied to the other piston of area B ft.', where p = P/A=W/B, (I) the pressure p of the liquid being supposed uniform; and, by making the ratio B/A sufficiently large, the mechanical advantage can be increased to any desired amount, and in the simplest manner possible, without the intervention of levers and machinery.
• It is used to determine the density of a body experimentally; for if W is the weight of a body weighed in a balance in air (strictly in vacuo), and if W' is the weight required to balance when the body is suspended in water, then the upward thrust of the liquid (I) (2) "F r an Minim ' 'i n or weight of liquid displaced is W-W, so that the specific gravity (S.G.), defined as the ratio of the weight of a body to the weight of an equal volume of water, is W/(W-W').
• He weighed out a lump of gold and of silver of the same weight as the crown; and, immersing the three in succession in water, he found they spilt over measures of water in the ratio:: A or 33: 24: 44; thence it follows that the gold: silver alloy of the crown was as I I: 9 by weight.
• The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2 a b1c1 a -b -b12 abc (Bo-Ao)+(B1-A1) a 2 + b 2 a1 2 + b1 2 alblcl Make c= CO for confocal elliptic cylinders; and then _, 2 A? ?
• C - a 2 - 2 b2 +A, = and then as above in § 31, with a= c ch a, b=c sh a, a =-1 (a 2 +X) =c ch al, b1= c sh a (13) the ratio in (II) agrees with § 31 (6).
• I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/W is so small that the square may be neglected, and formula (II) can be replaced for practical purpose in artillery by tan26= n2 = W i (0 - a) (k ð)7()4, (12) if then we can calculate /3, a, or (3-a for the external shape of the shot, this equation will give the value of 6 and n required for stability of flight in the air.
• The ratio of cases to population living in Dublin on loose porous gravel soil for the ten years1881-1891was I in 94, while that of those living on stiff clay soil was but 1 in 145.
• Although the double standard was in force, gold was practically demonetized by the monetary reform of 1872 because of the failure to fix a legal ratio between the two metals.
• If M 1, M2, and P 1, P 2 be the molecular weights and vapour pressures of the components A and B, then the ratio of A to B in the distillate is M 1 P 1 /M 2 P 2.
• P 1 greater than P2, if the molecular weight of A be much less than that of B, then it is obvious that the ratio M 1 P 1 /M 2 P 2 need not be very great, and hence the less volatile liquid B would come over in fair amount.
• These con-, ditions pertain in cases where distillation with steam is successfully practised, the relatively high volatility of water being counterbalanced by the relatively high molecular weight of the other component; for example, in the case of nitrobenzene and water the ratio is I to 5.
• It will be observed that the coast-line is very long in proportion to the area, the ratio being 1 m.
• Males exceed females in the ratio of 2% approximately.
• In the first place, the ratio of the heighi of his head to the length of his body is greater than it is in Euro peans.
• The Englishmans head is often one-eighth of the lengtl of his body or even less, and in continental Europeans, as a rule the ratio does not amount to one-seventh; but in the Japanese it exceeds the latter figure.
• In northern Europeans the leg is usually much more than onehalf of the bodys length, but in Japanese the ratio is one-half or even less; so that whereas the Japanese, when seated, looks almost as tall as a European, there may be a great difference between their statures when both are standing.
• The ratio of marriages is approximately 8.46 per thousand units of the population, and the ratio of divorces is 1.36 per thousand.
• The actual purchase price of the 17 lines was calculated at 43 millions sterling (about double their cost price), on the following basis: (a) An amount equal to 20 times the sum obtained by multiplying the cost of construction at the date of purchase by the average ratio of the profit to the cost of construction during the six business terms of the company from the second half-year of 1902 to the first half-year of 1905.
• Since of a the capacity C is the ratio of charge to potential, the sphere.
• In the absolute determination of capacity we have to measure the ratio of the charge of a condenser to its plate potential difference.
• If one condenser is charged, and then joined in parallel with another uncharged condenser, the charge is divided between them in the ratio of their capacities.
• He constructed two equal condensers, each consisting of a metal ball enclosed in a hollow metal sphere, and he provided also certain hemispherical shells of shellac, sulphur, glass, resin, &c., which he could so place in one condenser between the ball and enclosing sphere that it formed a condenser with solid dielectric. He then determined the ratio of the capacities of the two condensers, one with air and the other with the solid dielectric. This gave the dielectric constant K of the material.
• Blaine, on the other hand, contended that representation should be based on population instead of voters, as being fairer to the North, where the ratio of voters varied widely, and he insisted that it should be safeguarded by security for impartial suffrage.
• Then by relations (2) the heat, H, absorbed in the isothermal change BC, is to the work, W, done in the cycle ABCD in the ratio of o to (o' - o").
• Since the amounts of heat supplied at constant pressure from E to F and from E to C are in the limit proportional to the expansions EF and EC which they produce, the ratio S/s is equal to the ratio ECÃ†F.
• It is often impossible to observe the pressure-coefficient dp/de directly, but it may be deduced from the isothermal compressibility by means of the geometrically obvious relation, BE = (BEÃ†C) XEC. The ratio BEÃ†C of the diminution of pressure to the increase of volume at constant temperature, or - dp/dv, is readily observed.
• The isothermal elasticity - v(dp/dv) is equal to the pressure p. The adiabatic elasticity is equal to y p, where -y is the ratio S/s of the specific heats.
• In thiscase the ratio of the specific heats is constant as well as the difference, and the adiabatic equation takes the simple form, pv v = constant, which is at once obtained by integrating the equation for the adiabatic elasticity, - v(dp/dv) =yp.
• Experiments by Natanson on CO 2 at 17° C. confirm those of Joule and Thomson, but show a slight increase of the ratio do/dp at higher pressures, which is otherwise rendered probable by the form of the isothermals as determined by Andrews and Amagat.
• They are continually changing partners, the ratio c/V representing approximately the ratio of the time during which any one molecule is paired to the time during which it is free.
• The atomic weight of gold was first determined with accuracy by Berzelius, who deduced the value 195.7 (H= i) from the amount of mercury necessary to precipitate it from the chloride, and 195.2 from the ratio between gold and potassium chloride in potassium aurichloride, KAuC1 4.
• The density gives very important information as to the molecular weight, since by the law of Avogadro it is seen that the relative density is the ratio of the molecular weights of the experimental and standard gases.
• This, the only coin minted by the government, should bear a fixed ratio of l000 cash to one tael of silver, but in practice there is no such fixed value.
• Syruporum universa ratio, &c. (Paris, 1537); four subsequent editions; latest, Venice, 1548 (six lectures on digestion; syrups treated in fifth lecture).
• Such a simple formula is only possible because the salts of sea-water are of such uniform composition throughout the whole ocean that the chlorine bears a constant ratio to the total salinity as newly defined whatever the degree of concentration.
• When, however, the air is present in much smaller ratio the combustion is incomplete, and carbon, carbon monoxide, carbon dioxide, hydrogen and water vapour are produced.
• The ratio p is given by e"` e, where e= 2.718; µ is the coefficient of friction and 0 the angle, measured in radians,, subtended by the arc of contact between the rope and the wheel.
• The ratio W/p increases very rapidly as 0 is increased,, and therefore, by making 0 sufficiently large, p may conveniently be made a small fraction of W, thereby rendering errors of observation of the spring balance negligible.
• (20) By division of the values of C, and C„ we find for -y, the ratio of the specific heats.
• The marriage rates in quinquennial periods up to 1905 were 19.6, 18.6, 21.0, 19.8, 15.6, 18.6, 18.6, 18.6, 17.4 and 17.4; the ratio of marriages to the marriageable population was for males (above 16 years) 61.5, for females (above 14) 46.0; the fecundity of marriages seemed to have increased, being about twice as high for foreigners as for natives.
• For cities of above 8000 inhabitants (for which alone comparative statistics are annually available), in 1902-1903 the ratio of average attendance to school enrolment, the average number of days' attendance of each pupil enrolled, and the value of school property per capita of pupils in average attendance were higher than in any other state; the average length of the school term was slightly exceeded in eight states; and the total cost of the schools per capita of pupils in average attendance (\$39.05) was exceeded in six other states.
• (The same year the ratio of wealth productivity was as 66 to 37.) Massachusetts stands " foremost in the Union in the universality of its provision for secondary education."
• Thus, the gases are not present in simple multiples of their combining weights; atmospheric air results when oxygen and nitrogen are mixed in the prescribed ratio, the mixing being unattended by any manifestation of energy, such as is invariably associated with a chemical action; the gases may be mechanically separated by atmolysis, i.e.
• The ratio 47r would thus first appear as the ratio of the average breadth of a circle to the greatest breadth; the interpretation of 7 as the ratio of the circumference to the diameter being a secondary one.
• It is necessary, in applying formulae to specific cases, not only, on the one hand, to remember that the measurements are only approximate, but also, on the other hand, to give to any ratio such as 7r a value which is at least more accurate than the measurements.
• Denoting the constant ratio by fir, the area of a circle is ira 2, where a is the radius, and ir=3.14159 approximately.
• In the case of a wedge with parallel ends the ratio x2/h2 is replaced by x/h.
• The tangent of the angle of deflection 0 of this needle measured from its position, when the shunt coil is disconnected, is equal to the ratio of the voltage of the dynamo to the current through the insulator.
• The exact position of the core, and, therefore, of an index needle connected with it, is dependent on the ratio of the voltage applied to the terminals of the high resistance or insulator and the current passing through it, This, however, is a measure of the insulation-resistance.
• Hence the resistance of the insulator can be ascertained, since it is expressed in ohms by the ratio of the voltage of the battery in volts to the current through the C C galvanometer in amperes.
• Male factory hands greatly outnumbered female, standing in the ratio of four to one.
• Let E be the bulk modulus of elasticity, defined as increase of pressure = decrease of volume per unit volume where the pressure increase is so small that this ratio is constant, w the small increase of pressure, and - (dy/dx) the volume decrease, then E=e/(- dy/dx) or w Ã†= - dy/dx (I) This gives the relation between pressure excess and displacement.
• Experiments, which will be described most conveniently when we discuss methods of determining the frequencies of sources, prove conclusively that for a given note the frequency is the same whatever the source of that note, and that the ratio of the frequencies of two notes forming a given musical interval is the same in whatever part of the musical range the two notes are situated.
• As with light the ratio involved in the second law is always equal to the ratio of the velocity of the wave in the first medium to the velocity in the second; in other words, the sines of the angles in question are directly proportional to the velocities.
• Or, if the same plate be moved in contact with two tuning-forks, we shall, by comparing the number of sinuosities in the one trace with that in the other, be enabled to assign the ratio of the corresponding numbers of vibrations per second.
• A mode of exhibiting the ratio of the frequencies of two forks was devised by Jules Antoine Lissajous (1822-1880).
• If two such flames are placed one under the other they may be excited by different sources, and the ratio of the frequencies may be approximately determined by counting the number of teeth in each in the same space.
• In 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.
• Kundt's dust-tube may also be employed for the determination of the ratio of the specific heats of a gas or vapour.
• If U is the velocity of sound in a gas at pressure P with density p, and if waves of length X and frequency N are propagated through it, then the distanc?e l between the dust-heaps is 2 = N - zN Vyp' where y is the ratio of the two specific heats.
• The rainfall is sufficient for good grazing, but except in the Flathead valley cultivation was long considered to be dependent on irrigation; and consequently farming was only incidental to stock raising and mining until after 1870, and as late as 1900 the ratio of improved farm land to the total land area was less than in any other state or territory except New Mexico, Wyoming, Arizona and Hawaii.
• But with the sole exception of proving that the volumes of spheres are in the triplicate ratio of their diameters, a theorem probably due to Eudoxus, no mention is made of its mensuration.
• It would be economical, therefore, to make the girder very deep. This, however, involves a much heavier web, and therefore for any type of girder there must be a ratio of depth to span which is most economical.
• The actual bridge must have the section of all members greater than those in the provisional design in the ratio k/(i -k).
• It was pointed out as early as 1869 (Unwin, Wrought Iron Bridges and Roofs) that a rational method of fixing the working stress, so far as knowledge went at that time, would be to make it depend on the ratio of live to dead load, and in such a way that the factor of safety for the live load stresses was double that for the dead load stresses.
• Let A be the dead load and B the live load, producing stress in a bar; p =B / A the ratio of live to dead load; f i the safe working limit of stress for a bar subjected to a dead load only and f the safe working stress in any other case.
• To prevent temperature from affecting the shunt ratio, Edison joined in series with the electrolytic cell a copper coil the resistance of which increased with a rise of temperature by the same amount that the electrolyte decreased.
• On the ground that the aim of every prosperous community should be to have a large proportion of hardy country yeomen, and that horticulture and agriculture demand such a high ratio of labour, as compared with feeding and breeding cattle, that the country population would be greatly increased by the substitution of a fruit and vegetable for an animal dietary.
• Lord Kelvin assumed as a superior limit of k, the ratio of amplitude to wave-length, the value io 2, which is a very safe limit.
• But it is found not to vary at all, even up to the second order of the ratio of the earth's velocity to that of light.
• This theory secures that the times of passage of the rays shall be independent of the motion of the system, only up to the first order of the ratio of its velocity to that of radiation.
• Aristotelian dialectics had always been taught in the schools; and reason as well as authority had been appealed to as the foundation of theology; but for the theologians of the 9th and 10th centuries, whose method had been merely that of restatement, ratio and auctoritas were in perfect accord.
• But the maximum pressure may exceed the mean in the ratio of 2 or 3 to I, as shown in fig.
• In ocean water the ratio of soda (Na 2 O) to potash (K 2 0) is loo: 3.23 (Dittmar); in kelp it is, on the average, ioo: 5.26 (Richardson).
• The well-known ratio of 25:24 between the 12.16 foot and this we see to have arisen through one being (1/6)th of 100 and the other 16 digits--16+2/3: 16 being as 25: 24, the legal ratio.
• The other ratio of Revillout and Hultsch, 320 hons = cubit cubed, is certainly approximate.
• The log and kab are not found till the later writings; but the ratio of hin to issaron is practically fixed in early times by the proportions in Num.
• This gives the same approximate ratio 96: 100 to the libra as the usual drachma reckoning.
• Hultsch reckons on a ratio of 24:25 between them, and this is very near the true values; the full Attic being 67.3, the Assyrian should be 129.2, and this is just the full gold coinage weight.
• We may perhaps see the sense of this ratio through another system.
• Suppose that the ratio of 10, or any other particular number, to i is compounded of a very great number of equal ratios, as, for example, 1,000,000, then it can be shown that the ratio of 2 to i is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculae, that the ratio of 3 to I is very nearly equal to a ratio compounded of 477,121 of them, and so on.
• The small ratio, or ratiuncula, is in fact that of the millionth root of to to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to I will be nearly the same as that of a301'°30 to i, and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a 301,030, 3 will be nearly equal to a477,1u, and so on.
• The Ratio Studiorum took its shape during this time.
• If the refracting angleof the prism is small, then the ratio of the dispersion to the mean deviation of the two rays is the dispersive power of the material of the prism.
• The form of the limacon depends on the ratio of the two constants; if a be greater than b, the curve lies entirely outside the circle; if a equals b, it is known as a cardioid; if a is less than b, the curve has a node within the circle; the particular case when b= 2a is known as the trisectrix.
• This extraordinary production is memorable as having announced the discovery of the "third law" - that of the sesquiplicate ratio between the planetary periods and distances.
• An ingenious, though ineffective, proposal for the reform of the calendar was put forward in his Elenchus Calendarii Gregoriani (Frankfort, 1612); and he published a book on music, Melodiae condendae ratio (Erfurt, 1592), still worth reading.
• Not a state of the Union as it existed in 1850 showed an increase, during the half-century following, in the ratio of white children under 16 to fooo white females over f6 years: the ratio declined for the whole country from 1600 to iioo; and it has fallen for the census area of 1790 from 1900 in that year to 1400 in 1850 and 1000 in 1900.
• In the North there was little difference in 1900 in the ratios shown by city and country districts, but in the South the ratio in the latter was almost twice that reported for the former.
• In Alabama (70.8% in 1880), North and South Carolina, amid Arkansas the ratio exceeded 5o% in 1900.
• Thus, in the period 1791 to 1811 their ratio to total government expenditure ranged from 41.6 to 189.6%; during the years 1812-1817, from 17.2 in 1814, when war finances reached their weakest point, to 131 ~4% in 1817, showing how rapid was their response under the return of peace; in the period1817-1859from 29-9% in the crisis year of 1837 to 158.9%; in the period1860-1869from 6.5% in 1865, when the governments bonds fell in price to \$50.93 per hundred and the war policy of loans was most desperate, tO 84.1%; in the years 1870I 893 from 5f.4 to 85%; and, finally, in the years 1893f 909, from 36.9% (in 1898) to 52.7%.
• It may be readily shown that the external and internal centres are the points where the line joining the centres of the two circles is divided externally and internally in the ratio of their radii.
• It may be shown to be the locus of the vertex of the triangle which has for its base the distance between the centres of the circles and the ratio of the remaining sides equal to the ratio of the radii of the two circles.
• All exact relations pertaining to the mensuration of the circle involve the ratio of the circumference to the diameter.
• This ratio, invariably denoted by 7r, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number.
• Probably the earliest value for the ratio was 3.
• Accordingly, we find in Vieta a formula for the ratio of diameter to circumference, viz.
• 1 V 2+2A/ From this point onwards, therefore, no knowledge whatever of geometry was necessary in any one who aspired to determine the ratio to any required degree of accuracy; the problem being reduced to an arithmetical computation.
• His book, Van den Circkel (Delft, 1596), gave the ratio correct to 20 places, but he continued his calculations as long as he lived, and his best result was published on his tombstone in St Peter's church, Leiden.
• This gives the ratio correct to 35 places.
• In Germany the "Ludolphische Zahl " (Ludolph's number) is still a common name for the ratio.
• Grienberger, using Snell's method, calculated the ratio correct to 39 fractional places. ?
• In a very curious manner, by viewing the circle y= (1 - x2): as a member of the series of curves y= (I -x 2 )', y = (I -x 2) 2, &c., he was led to the proposition that four times the reciprocal of the ratio of the circumference to the diameter, i.e.
• About the same time John Machin calculated it correct to Ioo places, and, what was of more importance, gave for the ratio the rapidly converging expression 4 _ I I 5 I 3152+ 5 1 54 - 7 1 5b?.
• 8 With him, apparently, began the usage of denoting by 71 the ratio of the circumference to the diameter.'
• This achievement was anticipated or outdone by an unknown calculator, whose manuscript was seen in the Radcliffe library, Oxford, by Baron von Zach towards the end of the century, and contained the ratio correct to 152 places.
• De Rationis Sectione had for its subject the resolution of the following problem: Given two straight lines and a point in each, to draw through a third given point a straight line cutting the two fixed lines, so that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio.
• De Sectione Determinata resolved the problem: Given two, three or four points on a straight line, to find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has to the square on the remaining one or the rectangle contained by the remaining two, or to the rectangle contained by the remaining one and another given straight line, a given ratio.
• From the city of Quebec westwards there is a constantly increasing ratio of southern forms, and when the mountain (so called) at Montreal is reached the representative Ontario flora begins.
• For the computation of the Christian date, the ratio of a mean year of the Hegira to a solar year is Year of Hegira 354s =0.970224.
• If V = N/A then N expresses the ratio of the volume of the instrument up to the zero of the scale to that of one of the scale-divisions.
• The total trade of the port increased from £3,853,593 in 1897 to £5, 6 75, 28 5 in 1905 and £7,009,758 in 1906 (the large increase being mainly due to a rise of over Li,000,000 in imports - mainly of coal, building materials and machinery), the average ratio of imports to exports being as three to two.
• Thus weighting, which was until recently thought to apply only to black silks, and from which coloured silks were comparatively free, is now cheapening and deteriorating the latter in pretty much the same ratio as the former.
• Bradley recognized the fact that the experimental determination of the aberration constant gave the ratio of the velocities of light and of the earth; hence, if the velocity of the earth be known, the velocity of light is determined.
• Duffield finds that the iron lines divide themselves into three groups with pressure shifts which are approximately in the ratio i: 2: 4.
• Curiously enough this is also approximately the ratio of the displacements found by Humphreys in the trunk series, the side branch and main branch in the order named, in cases where these displacements have been measured.
• If K has a value nearly equal to unity, the pressure shift is z k2 " +3 (K- 1 - 1); and it is significant that for different values of n, the shifts should be in geometric ratio, because as stated above, the ratio occurring in the amounts observed with different lines of the same element are as 1: 2: 4.
• The fact that in certain simple cases where a line when looked at equatorially splits into a triplet, the ratio of the charge to the mass is found by Lorentz's theory to be equal to that observed in the carrier of the kathode ray, shows that in these cases the electron moves as an independent body and is not linked in its motion to other electrons.
• Pappus gives several solutions of this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one.
• In the same preface is included (a) the famous problem known by Pappus's name, often enunciated thus: Having given a number of straight lines, to find the geometric locus of a point such that the lengths of the perpendiculars upon, or (more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to the product of the remaining ones; (Pappus does not express it in this form but by means of composition of ratios, saying that if the ratio is given which is compounded of the ratios of pairs - one of one set and one of another - of the lines so drawn, and of the ratio of the odd one, if any, to a given straight line, the point will lie on a curve given in position), (b) the theorems which were rediscovered by and named after Paul Guldin, but appear to have been discovered by Pappus himself.
• The change in the character of the immigration of Japanese is shown by the fact that in the fiscal year 1906-1907 the ratio of female immigrants to males was as i to 8, in the fiscal year 1907-1908 it was as I to 2, and in the latter year, of 4593 births in the Territory, 2445 were Japanese.
• He generalized Weber's law in the form that sensation generally increases in intensity as the stimulus increases by a constant function of the previous stimulus; or increases in an arithmetical progression as the stimulus increases in a geometrical ratio; or increases by addition of the same amount as the stimulus increases by the same multiple; or increases as the logarithm of the stimulus.
• What is known as the method of sines is used, for since the axes of the two magnets are always at right angles when the mirror magnet is in its zero position, the ratio M/H is proportional to the sine of the angle between the magnetic axis of the mirror magnet and the magnetic - = meridian.
• This angle depends on the ratio of the magnetic moment of the needle b to the total force of the earth's field.
• Hence the above observation gives us a means of obtaining the ratio of the magnetic moment of the needle' b to the value of the earth's total force.
• But if the bodies are of different substances, say one of iron and the other of gold, the ratio of these magnitudes is found to depend upon something else besides bulk.
• When, as in the case of contact, a mutual relation is perceived between the motions of two particles, the changes of velocity are in opposite directions, and the ratio of their magnitudes determines the ratio of the masses of the particles; the motion being reckoned relative to any base which is unaffected by the change.
• It is found that this gives a consistent result; that is to say, if by an experiment with two particles A and B we get the ratio of their masses, and by an experiment with B and a third particle C we get the ratio of the masses of B and C, and thus the ratio of the masses of A and C, we should get the same ratio by a direct experiment with A and C. For the numerical measure of mass that of some standard body is chosen as a unit, and the masses of other bodies are obtained by comparison with this.
• The various expansions and developments have made it difficult to maintain the ratio between accommodation and requirements, and although overcrowding is troublesome only during some three or four hours a week, at "high 'Change" on market days, various complaints and suggestions provoked in 1906 an appeal from the chairman of directors to the Manchester corporation.
• The following may be taken as examples: - When dealing with gases it is usually more convenient to express the solubility as the ratio of the volume of the gas absorbed to the volume of the absorbing liquid.
• The relative lowering of vapour pressure of the solution compared with that of the solvent is measured by the ratio of the extra mass absorbed from the solvent bulbs to the total mass absorbed from both series of bulbs.
• Substituting these values, we find that the relative lowering of vapour pressure in a very dilute solution is equal to the ratio of the numbers of solute and solvent molecules, or (p - p')/p = n/N.
• Now measurements of osmotic properties of these solutions show that their osmotic pressures are abnormally great and that, at extreme dilution, the ratio of their osmotic pressures to that of equivalent solutions of non-electrolytes is equal to the number of ions indicated by the electrolytic properties.
• Let us assume that the ratio p/p' of the vapour pressures of the solvent and solution is equal to the ratio of the number of free molecules of solvent to the whole number of molecules in the solution.
• If there are n molecules of solute to N of solvent originally, and each molecule of solute combines with a molecule of solvent, we get for the ratio of vapour pressures p/p'=(N - an)/(N - an+n), while the relative lowering of vapour pressure is (p - p')/p=n/(N - an).
• The most remarkable physical property of argon relates to the constant known as the ratio of specific heats.
• The ratio of specific heats of the principal gases is I.
• If, as for Boscovitch points, the whole energy is translatory, the ratio of specific heats must be I.
• These gases agree with argon in respect of the ratio of the specific heats and in being non-oxidizable under the electric spark.
• In the Snow telescope the ratio of aperture to focal length is I: 30.
• This operation is varied in detail according to the kind of plant to be propagated, but it is essential in all cases that the affinity between the two plants be near, that the union be neatly effected, and that the ratio as well as the season of growth of stock and scion be similar.
• The heat evolved by this process of solidification retards the fall of temperature; but after this the rate of cooling remains regular until T (750°) on the line Sa (Ar 3) is reached, when a second retardation occurs, due to the heat liberated by the passage within the pasty mass of part of the iron and carbon from a state of mere solution to that of definite combination in the ratio Fe 3 C, forming microscopic particles of cementite, while the remainder of the iron and carbon continue dissolved in each other as austenite.
• This formation of cementite through the rejection of carbon by both the primary and the eutectic austenite continues quite as in the case of 1.00% carbon steel, with impoverishment of the austenite to the hardenite or eutectoid ratio, and the splitting up of that hardenite into pearlite at Ari, so that the mass when cold finally consists of (1) 1 Note the distinction between the " eutectic " or alloy of lowest freezing-point, 1130°, B, with 4.30% of carbon, and the " eutectoid," hardenite and pearlite, or alloy of lowest transformation-point, 690° S, with 0.90% of carbon.
• In 1561, at the age of fourteen, he produced a work upon Latin spelling, called Orthographiae ratio.
• It can be easily seen that this ratio, according to Henry's law, must correspond to that of vapour-pressures, and so be independent of the solvent; in fact, in alcohol the figures are o 0066 and o o052.
• The first desideratum here mentioned - the want, namely, of an accurate statement of the relation between the increase of population and food - Malthus doubtless supposed to have been supplied by the celebrated proposition that "population increases in a geometrical, food in an arithmetical ratio."
• The thermal conductivity of the substance is the constant ratio of the rate of transmission to the temperature gradient.
• If Q is expressed in terms of this unit in equation (I), it is necessary to divide by c, or to replace k on the right-hand side by the ratio k/c. This ratio determines the rate of diffusion of temperature, and is called the thermometric conductivity or, more shortly, the diffusivity.
• The diffusivity can be deduced from observations at different depths x' and x", by observing the ratio of the amplitudes, which is (x '- x ") for a simple-harmonic wave.
• One important result, which might be regarded as established by this work, was that the ratio k/k of the thermal to the electrical conductivity, though nearly constant for the good conductors at any one temperature such as 0° C., increased with rise of temperature nearly in proportion to the absolute temperature.
• The value found for this ratio at o° C. approximated to 1500 C.G.S.
• The increase of resistance with temperature was also very small, so that the ratio varied very little with temperature.
• Lorenz, assuming that the ratio k/k'=aG, had previously given 0 2 maz Bo g = E2/4a, (12) which is practically identical with the preceding for small differences of temperature.
• Surface.In respect of physical structure Germany is divided into two entirety distinct portions, which bear to one another a ratio of about 3 to 4.
• In case of injury, involving incapacity for more than, thirteen weeks (for the earlier period the Krankenkassen provide), the weekly sum payable during complete or permanent incapacity is fixed at the ratio of two-thirds of the earnings during the year preceding the accident, and in case of partial disablement, at such a proportion of the earnings as corresponds to the loss through disablement.
• While the proportion of like weights of fine gold and fine silver in 1866-1870 averaged J to 1555, it was I to 17-79 in 1876, I to 17.18 in 1877, and, in 1902, in consequence of the heavy fall in silver, the ratio became as much as I to 39.
• The total number of bullocks in the island is calculated to be less than 200,000; and although the ratio of consumption of meat is low in proportion to the population, some of the cattle for slaughter have to be imported.
• The male sex remains in excess until about the twentieth year, from which age the female sex preponderates in increasing ratio with advancing age.
• Through this space the fresh surface water finds its way, and dissolving the salt below rises in the inner tube as brine, but only to such a level that the two columns bear to one another the relation of ten to twelve, this being the inverse ratio of the respective weights of saturated brine and fresh water.
• We see therefore that while leaves, gathered in equal numbers from each of loo trees, are distributed about their mean with a standard deviation of 1.735 veins, the leaves gathered from a single tree are distributed about their mean with a standard deviation of 1.426 veins, the ratio between variability of the race and variability of the individual tree being 1 - (0-5699)1=0.822.
• The births registered in Singapore during 1898 numbered 3751, namely, 1960 males and 1791 females, being a ratio of 16.55 per mille.
• The deaths registered during the same period numbered 7602, namely, 5894 males and 1708 females, a ratio of 33.54 per mille.
• Realizing that the total weight of all the products of a chemical reaction must be exactly equal to the total weight of the reacting substances, he made the balance the ultima ratio of the laboratory, and he was able to draw correct inferences from his weighings because, unlike many of the phlogistonists, he looked upon heat as imponderable.
• The tendency is for property valuations to decline, the estimated valuation from 1873 to 1893 decreasing 27% in Cook county and 39% in the other counties, while the assessments from 1888 to 1898 were in inverse ratio to the increase of wealth.
• This ratio will become more equal for larger sizes on account of the additional thickness of larger object-glasses and the consequent additional absorption of light in transmission.
• The specific heat of a substance is sometimes defined as the thermal capacity of unit mass, but more often as the ratio of the thermal capacity of unit mass of the substance to that of unit mass of water at some standard temperature.
• The ratio of the times of cooling is equal to the ratio of the thermal capacities of the calorimeter and its contents in the two cases.
• The Value 4.180 Joules At 20° C. Is The Mean Between Rowland'S Corrected Result 4.181 And The Value 4.179, Deduced From The Experiments Of Reynolds And Moorby On The Assumption That The Ratio Of The Mean Specific Heat O° To 100° To That At 20° Is 1.043'6, As Given By The Formulae Representing The Results Of Callendar And Barnes.
• Its Relation To The Calorie At Any Given Temperature, Such As 15° Or 20°, Cannot Be Determined With The Same Degree Of Accuracy As The Ratio Of The Specific Heat At 15° To That At 20°, If The Scale Of Temperature Is Given.
• If The Molecules Are Supposed To Be Like Smooth, Hard, Elastic Spheres, Incapable Of Receiving Any Other Kind Of Energy Except That Of Translation, The Specific Heat At Constant Volume Would Be The Increase Per Degree Of The Kinetic Energy Namely 3Pv/20=3R/2, That At Constant Pressure Would Be 5R/2, And The Ratio Of The Specific Heats Would Be 5/3 Or 1.666.
• In 1879 Maxwell Considered It One Of The Greatest Difficulties Which The Kinetic Theory Had Yet Encountered, That In Spite Of The Many Other Degrees Of Freedom Of Vibration Revealed By The Spectroscope, The Experimental Value Of The Ratio S/S Was 1.40 For So Many Gases, Instead Of Being Less Than 4/3.
• The Theoretical Value Of The Ratio S/S In This Case Would Be The Required 7/5.
• It Is Not At All Clear, However, That Energy Of Vibration Should Bear A Constant Ratio To That Of Translation, Although This Would Probably Be The Case For Rotation.
• For Such Gases, Assuming A Constant Ratio Of Rotation To Translation, The Specific Heat At Low Pressures Would Be Very Nearly Constant.
• This Would Account For An Increase Of S, And A Diminution Of The Ratio S/S, With Rise Of Temperature Which Apparently Occurs In Many Vapours.
• The Direct Methods Of Measuring The Ratio S/S, By The Velocity Of Sound And By Adiabatic Expansion, Are Sufficiently Described In Many Text Books.
• In this system one star is defined to be unit magnitude higher than another if its light is less in the ratio I:2 512.
• This ratio is adopted so that a difference of five magnitudes may correspond to a light-ratio of i: loo.
• (If only the relative orbit is known, the sum of the masses can be determined; but if absolute positions of one component have been observed, both masses can be determined separately.) But even when, as in most cases, the parallax is unknown or uncertain, the ratio of the brightness to the mass can be accurately found.
• But notwithstanding the great variety of intrinsic brightness of the stars, the ratio of the number of stars of one magnitude to the number of the magnitude next lower (the " star-ratio ") is a guide to the uniformity of their distribution.
• Seeliger, who investigated this ratio for the stars of the Bonn Durchmusterung and Southern Durchmusterung, came to the conclusion (as summarized by Simon Newcomb) that for these stars the ratio ranges from 3.85 to 3.28, the former value being found for regions near the Milky Way and the latter for regions near the galactic poles.
• Thus the figure represents a section the (ideally simplified) uni verse cut perpendicular to C P' D the planes AB and CD between which the stars are contained, 1 This number is the 3/2th power of the ratio of the brightness of stars differing by a unit magnitude.
• In 1807 an account of the magnetic observations made during the tour with Humboldt was published in the first volume of the Memoires d'Arcueil, and the second volume, published in 1809, contained the important memoir on gaseous combination (read to the Societe Philomathique on the last day of 1808), in which he pointed out that gases combining with each other in volume do so in the simplest proportions-1 to 1, 1 to 2, 1 to 3 - and that the volume of the compound formed bears a simple ratio to that of the constituents.
• Comenius also wrote against the Socinians, and published three historical works - Ratio disciplinae ordinisque in unitate fratrum Bohemorum, which was republished with remarks by Buddaeus, Historia persecutionum ecclesiae Bohemicae (1648), and Martyrologium Bohemicum.
• In man, as the noblest of created things, the Trinity is seen most perfectly reflected; intellectus (vous), ratio (X6yos) and sensus (& ivota) make up the threefold thread of his being.
• In 1781 he favoured an amendment 'of the Articles of Confederation giving Congress power to enforce its requisitions, and in 1783, in spite of the open opposition of the Virginia legislature, which considered the Virginian delegates wholly subject to its instructions, he advocated that the states should grant to Congress for twenty-five years authority to levy an import duty, and suggested a scheme to provide for the interest on the debt not raised by the import duty - apportioning it among the states on the basis of population, counting three-fifths of the slaves, a ratio suggested by Madison himself.
• It had also the mathematical meaning of ratio; and in its use for definition it is sometimes transferred to essence as the object of definition, and has a mixed meaning, which may be expressed by " account."
• Babo, 1847, gave the law known by his name, that the " relative lowering" (p - po)lpo of the vapour-pressure of a solution, or the ratio of the diminution of vapour-pressure (p - po) to the vapour-pressure po of the pure solvent at the same temperature, was constant, or independent of the temperature, for any solution of constant strength.
• Raoult (Comptes Rendus, 1886-87) employed other solvents besides water, and showed that the relative lowering for different solvents and different dissolved substances was the same in many cases for solutions in which the ratio of the number of gramme-molecules n of the dissolved substance to the number of molecules N of the solvent was the same, or that it varied generally in proportion to the ratio n/N.
• In this case the ratio of the vapour-pressure of the solution p" to that of the solvent p' should be equal to the ratio of the number of free molecules of solvent N - an to the whole number of molecules N - an+n in the solution.
• Employing Joule's value of the mechanical equivalent of heat, then recently published, in connexion with the value of the ratio of the specific heats of air S/s=I.
• Adopting for steam the same value of the ratio of the specific heats, viz.
• In other words, the increase of pressure per degree (dp/d0) divided by p should be constant and equal to B; but observation shows that this ratio decreases, e.g.
• (20) from which we deduce that the ratio 0'/0" of the temperatures at which the vapour-pressures are the same is a linear function of the temperature 0' of one of the substances.
• Their origin is the fact that where the bars appear nearly to coincide the apparent gaps bear the greatest ratio to the dark spaces; i.e.
• In 1906 the ratio of insane confined to institutions, to the total population, was to every 270.
• The National Democratic Convention declared for the immediate opening of the mints to the free and unlimited coinage of silver at the ratio with gold of 16 to 1; and it nominated for the presidency William Jennings Bryan of Nebraska, who also received the nomination of the People's party and of the National Silver party.
• The state of elementary education is comparatively good, rather more than two-thirds of the population being able to read and write, and the ratio of crime is proportionately low.
• The translation parallel to this axis is lox + mly + nhz (Xf + uv + vi) Ic. (8) The linear magnitude which measures the ratio of translation to rotation in a screw is called the pitch.
• Hence, whatever the ratio f: n, the axis of the resultant screw lies on the conoidal surface z(x1+y1)=cx3~ (13)
• Its type, as distinguished from its absolute magnitude, may be specified by a screw whose axis is the line of action of R, and whose pitch is the ratio G/R.
• In two turns this ratio is squared, and so on.
• Again, the mass-centre of a uniform solid right circular cone divides the axis in the ratio 3: I; that of a uniform solid hemisphere divides the axial radius in the ratio 3: 5.
• Eliminating the ratio A:B we have (ai+aii_~p1) (eI+c1_~q1)_4~a,2=o.
• The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, ~.
• The a/~celeration of G is therefore less than in the case of frictionless sliding in the ratio a2/(K2+a2).
• For a homogeneous sphe:e this ratio is ~, for a uniform circular cylinder or disk ~, for a circular hoop or a thin cylindrical shell 4.
• The ratio of the axes of the ellipse is sec a, the longer axis being in the plane of 0.
• It is obvious that the ratio V (x,y,z) (22)
• The frequency of the gravest mode is to that of a uniform bar in the ratio .98 15.
• That this ratio should be less than unity agrees with the theory of constrained types already given.
• In other cases the safety of the joint against displacement by sliding depends on its power of exerting friction, and that power depends on the law, known by experiment, that the friction between two surfaces bears a constant ratio, depending on the nature 01 the surfaces, to the force by which they are pressed together.
• (3) A parallel projection of a straight line or a plane surface divided in a given ratio is a straight line or a plane surface divided in the same ratio.
• (~) A parallel projection of a pair of solids having a given ratio is a pair of solids having the same ratio.
• For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio.
• By its aid, for example, the whole of the properties a elliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and ellipticconoidal domes from those of hemispherical and circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a certain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.
• The ratio in which the utmost stress before breaking exceeds the safe working stress is called the factor of safety, and is determined empirically.
• It consists of two elements, the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the directional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.
• The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamiltons Quaternions,the tensor expressing the velocity ratio, and the versor the directional relation.
• Velocity Ratio of Components of Motion.As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal.
• Hence also the ratio of the com ponents of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fni and Fn.
• The ratio of the two components of that velocity is = p/2lrr = tan 0.
• In the investigation, therefore, of the comparative motion, of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connection of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of tbe directional relation, so far only as it expresses the connection between forward and backward movements of the driver and follower.
• Williss classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connection of the driver and follower.
• He divides the elementary combinations in mechanism into three classes, of which the characters are as follows: Class A: Directional relation constant; velocity ratio constant.
• Class C: Directional relation changing periodically; velocity ratio constant or varying.
• For otitside gearing that ratio is negative, Cs because the wheels turn contrary ways; for inside gearing it is positive, because they turn the same way.
• If the velocity ratio is to be -constant, as in, fYi Williss Class A, the wheels must be circular; and this is the most common form for wheels.
• If the velocity ratio is to be variable, as in s Williss Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.
• To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity ratio varying from to I ~eccentricitY an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hypereccentricity + I
• When the velocity ratio is variable, the line of contact will shift its position in the plane C1OC2, and the wheels will be cones, with eccentric or irregular bases.
• In every case which occurs in practice, however, the velocity ratio is FIG.
• The directions and positions of the axes being given, and the required angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated: In fig.
• Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.
• They there fore study that the numbers of teeth in each pair of wheels whici work together shall either be prime to each other, or shall hav their greatest common divisor as small as is consistent with velocity ratio suited for the purposes of the machine.
• The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connection of the teeth cuts the Ca line of centres at the pitchpoint.
• Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.
• To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along PiIPi, is less than the pitch in the ratio of cos obliquity: I; and consequently that, if distances equal to the pitch be marked off either way from I towards P~ and Pi respectively, as the extremities of the path of contact, and if, according to Principle IV.
• Pulleys and drums for communi cating a constant velocity ratio are circular.
• The speed-cones are either continuous cones or conoids, as A, B, whose velocity ratio can be varied gradually while they are in motion by shifting the belt, or sets of pulleys whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by shifting the belt from one pair of pulleys to another.
• The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connection n which case the angular velocity ratio at any instant is the recipocal of the ratio of the common perpendiculars let fall from the me of connection upon the respective axes of rotation.
• The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point at the same instant, so that the line of connection is in the plane of the two axes of rotation, in which case the velocity ratio is indeterminate.
• To find the ratio of these velocities, produce C1T~, C2T, till they intersect in K; K is the instantaneous axis of the connecting rod, and the velocity ratio is ci :v2 ::KTi :KT2.
• Should K be inconveniently far off, draw any triangle with its sides respectively parallel to CiT~, CiT2 and TiTi; the ratio of the twiJ sides first mentioned will be the velocity ratio required.
• The mean value of that velocity ratio is that of equality, for each successive quarter-turn is made by both shafts in the same time; but its actual value fluctuates between the limits: a1 I
• Then the resulting velocity ratio is denoted by a,, a~ ~3 & a,,, N, - Ni -.
• It is often a question of importance to determine the number of teeth in a train of wheels best suited for giving a determinate velocity ratio to two axes.
• It was shown by Young that, to do this with the least total number of teeth, the velocity ratio of each elementary combination should approximate as nearly as possible to 3.59., This would in many cases give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6 ought to be the limit of the velocity ratio of an elementary combination in wheelwork.
• Let B/C be the velocity ratio required, reduced to its least terms, and let B be greater than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of teeth (which may be called t) and its double 2t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required.
• Should B or C, or both, be at once inconveniently large, and orime, then, instead of the exact ratio B/C some ratio anproxlmat:ng to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.
• Then, if possible, B and, C themselves are to be resolved each into rnI factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.
• So far as the resultant velocity ratio is concerned, the order of the drivers N and of the followers n is immaterial: but to secure equable wear of the teeth, as explained in 44, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and ii shall be either 1, or as small as possible.
• Double Hookes Coupling.It has been shown in 66 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cos 0 and 1/cos 0.
• Then, from the principles of 60 it is evident that at each instant ai/ai = ai/aa, and consequently that ai; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant.
• Then Ob is the velocity of the point b in magnitude and direction, and cb is the tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the velocities, the instantaneous velocity ratio of the points C and B is given by the ratio Oc/Ob.
• An important property of the diagram is that if points X and x are taken dividing the link CB and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents the acceleration of the point X in magnitude and direction; cb is called the acceleration image of the rod.
• Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds were made, it becomes Pdsfdt = Rds/dt, that is, Pv = Rv, which shows that the force ratio is the inverse of the velocity ratio.
• It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points.
• The virtual centres 0,-,, O,i are at the respective axes of the wheels r and 1, and the centre O,-i divides the line through these two points externally in the ratio of the train of wheels.
• EfficiencyThe efficiency of a machine is the ratio of the useful work to the total workthat is, to the energy exertedand is represented by 1~.
• The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.
• Let Ti be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; 1, the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; 0 the ratio of the length of that arc to the radius of the pulley; do the ratio of an indefinitely small element of that arc to the radius; F=TiT2 the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc do; f the coefficient of friction between the materials of the band and pulley.
• The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the con oecting rod 1 increases.
• The ratio in France was low throughout the r 9th century, and during the last half fell only from 273 to 261, raising the proportion of the old above that resulting in northern Europe and Italy from emigration.
• The tendency of a community towards matrimony, or its "nuptiality," as it is sometimes termed, is usually indicated by the ratio to the total population of the persons married each year.
• The adherents of each of these churches outnumber their communicants in a ratio which is variously estimated.
• Maxwell suggested new methods for the determination of this ratio of the electrostatic to the electromagnetic units, and by experiments of great ingenuity was able to show that this ratio, which is also that of the velocity of the propagation of an electromagnetic impulse through space, is identical with that of light.
• 3 X Io 2 ° of an electromagnetic unit, and the ratio of its charge to its mass is nearly 2 X 10 7 using E.M.
• For the hydrogen atom the ratio of charge to mass as deduced from electrolysis is about Io 1.
• The ratio of the surface speeds of the drawing roller and the feed roller is termed the draft: surface speed of drawing roller _ draft.
• Such pairs of colour may be regarded as infinite in number; but there are three pairs which stand out prominently, and admit of easy expression for the ratio in which each contributes to the total action.
• If the ratio a' la be sufficiently constant, as is often the case, the above relation reduces to the " condition of Airy," i.e.
• This combined condition is exactly fulfilled by holosymmetrical objectives reproducing with the scale 1, and by hemisymmetrical, if the scale of reproduction be equal to the ratio of the sizes of the two components.
• For a single lens of very small thickness and given power, the aberration depends upon the ratio of the radii r: r', and is a minimum (but never zero) for a certain value of this ratio; it varies inversely with the refractive index (the power of the lens remaining constant).
• The rays with an angle of aperture smaller than u* would not have the same distance of intersection and the same sine ratio; these deviations are called "zones," and the constructor endeavours to reduce these to a minimum.
• For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of the Axis Point "); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e.
• Fraunhofer, who defined the colours by means of the dark lines in the solar spectrum; and showed that the ratio of the dispersion of two glasses varied about 20% from the red to the violet (the variation for glass and water is about 50%).
• Thus a continued fraction equivalent to 7r (the ratio of the circumference to the diameter of a circle) is I I I I II 3+ 7+15+7+292+i-1-i+ ..
• For Example, In The Case Of Water Delivered From A Glass Tube, Which Is Cut Off Square And Held Vertically, A Will Be The External Radius; And It Will Be Necessary To Suppose That The Ratio Of The Internal Radius To A Is Constant, The Cases Of A Ratio Infinitely Small, Or Infinitely Near Unity, Being Included.
• The coefficient (q) of the time in the exponential term (e at) may be considered to measure the degree of dynamical instability; its reciprocal 1 /q is the time in which the disturbance is multiplied in the ratio I: e.
• From (3) it appears that the time in which a disturbance is multiplied in a given ratio varies, not as d, but as d z.
• Reasoning from some observations of Savart, Plateau finds in this way 4.38 as the ratio of the length of a division to the diameter of the jet.
• Viewed as a whole, `Ali Riza's forces, scattered as they inevitably were through the need of holding territory, were reasonably well distributed, in that, though the Turks were in the ensemble inferior in the ratio of I to 21, their handicap on the decisive battlefield reduced itself to the ratio of t to about II-.
• Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.
• In the particular case when the radii are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference.