5° 



SCIENCE 



[Vol. XX. No, 494. 



other. Let 1 contain the smaller atoms. Suppose one face made 

 fast to the plank p, and both sheered slightly till they have the posi- 

 tion shown by the dotted lines. It is evident that the ratio of work 

 done in bringing the atom at G over to H to that done in bring- 

 ing £ to X), or C to A. will be the mean ratio of the force of attrac- 

 tion between K and G to that between E and F. This latter varies 

 inversely as the square of the distance, according to the well- 

 known electrical law, and, consequently, as the distance G K i's 

 twice that of E F, the work done in moving E to D will be four 

 times that done in moving G to H. Again, in Fig. 1 there will 

 be 2'' as many atoms to be displaced as in Fig. 3. so that, on the 

 whole, there will be 2^^ 4- 2* as rmioh work done in displacing 

 the cube in Fig. 1 as in Pig. 2. In other words, the rigidity will 

 vary inversely as the fifth power of the distance between the cen- 

 tres of the atoms, or as (atomic volume) ^. Col. IV. gives the 

 results calculated on this theory. As will be seen, they agree fairly 

 well, as svell as could be expected, considering the fact that we 

 have left out one factor. This is the variation of rigidity with 

 temperature, and as it would be obviously unfair to compare lead 

 and silver at 600° C. it is obvious that our calculated results should 

 only be applied when the metals are at some one point, say, at, a 

 temperature which is J the temperature of their melting-point 

 As those metals having the greatest atomic volume, as a rule, 

 melt at lowest temperature (though there are many exceptions to 

 this) we may make a rough sort of formula, which shall give the 

 rigidity at ordinary temperatures by multiplying again b}' the 

 atomic radius, so we get (atomic volume) ^ as the rate at which 



Fig. 3. 



rigidity varies with size of atoms, 

 way from the rough formula : — 



Rigidity. 28x10- 



Col. V. is calculated in this 



Equation I. 



(atomic volume) = 



The formula for Col. IV., and the more correct one, if we 



neglect variation of rigidity with temperature, is 



13560 X 109 



i Equation II. 



(atomic volume) " 



The other moduli are related to that of rigidity. For if we 



represent Young's modulus by 

 1 



-, then the modulus of rigidity 



is represented by 



and the bulk modulus by 



1 



2{a + b) "8 (a-2b)' 



where b represents the lateral shortening accompanying the longi- 

 tudinal lengthening a. So if 6 bears to a any constant ratio, then 

 Young's modulus and the bulk modulus will each be some fraction 

 of the modulus of rigidity. The continental writers, at least a 

 b 1 



good many of them, hold that — = 



Kelvin, Tait, and 



b. 



Stokes say there is no relation. On the one hand, it is certain that 



is not constantly equal to J. On the other hand, it does not 



follow that there is no relation between the two, and the evidence 

 which has been brought to prove this has no value, for we have 



no right to argue from the facts that in india-rubber _ = _, 



while in cork 



say. 



100 



that _ does not have any con- 



stant ratio in metals. The laws which govern the moduli of com- 

 pounds and non-homogeneous substances like india-ruhber and 

 cork are not the same as those which govern homogeneous sub- 

 stances like gold and silver. 



The following is a table of the metals and their Young's moduli. 

 Col. I. contains the observed moduli taken from Sutherland's 

 paper, and Col. II. contains the calculated values from the 

 equation. 



78 X 10^^ 



Equation III. Young's modulus = l (corres- 



(atomic volume)^ 

 ponding to Equation I.). 



Metals. I. II. 



Iron 2,000X10" 1,560x10=" 



Copper 1,220 1,560 



Zinc 930 920 



Silver 740 750 



Gold 760 750 



Aluminium 680 690 



Cadmium 480 465 



Magnesium 390 395 



Tin 420 295 



Lead 190 235 



There is only one metal which does not agree with theory. antJ 

 that is tin (iron, of course, on account of its impurities does not^ 

 but we know that, as we obtain iron more pure, we find its rigidity 

 less, so there is very little doubt but that if it were absolutely 

 pure the agreement would be closer). But it is easy to sbovsr 

 that the observed results of tin are wrong. For the rigidity is 

 given as 136 x 10' and the Young's modulus as 420 x 10". There- 

 fore, if we represent Young's modulus by — , then = 



a 2{a + b) 



I. Therefore the bulk modulus 



Solving this vve get b = 



130 



1 



about 



Poisson's ratio foir 



is negative, and the more tin is compressed the larger 



3 (a - 2 b) "" ' ^ 



it swells, a result which is absurd. This will emphasize the- 

 fact that the agreement between theory and experiment is as close 

 as that between the experiments themselves. 



It will be noticed that the ratio-rigidity, Young's modulus, is 



— Therefore, as = — , 



78 2 (a + b) 2.7' 



these metals is, on the average, 0.35. Therefore the bulk modu- 

 lus = 1.1 times Young's modulus, which agrees with the only 

 datum I find in Everett, i.e., Wertheims's figures for brass, which 

 gives the ratio 9 48 : 10,3 = 1.08, very closely. All these moduli 

 must contain the atomic volume to the same power, but this is 

 not the case with the tensile strength ; for, according to this elec- 

 trostatic theory of cohesion, we may look at a wire as made up of 

 thin discs, each disc consisting of a layer of atoms. The attrac- 

 tive force between any two such layers would vary inversely as- 

 the square of the distance between them and directly as the num- 

 ber of atoms in a layer. Combining these we find that it would^ 

 vary as the fourth power of the atomic radius, or as (atomic vol- 

 ume)', making no allowance for the effect of temperature on the 

 tensile strength. The following table gives in Col. I. the ob- 

 served tensile strengths, taken from Wertheim for wires 1 milli- 

 meter in diameter; in Col. II. the atomic volumes of the ele- 

 ments, raised to the |-power; and in Col. III. the calculatedJ 

 tensile strengths, as found by the formula. 



638 



Equation IV. Tensile strength = 4 in kilo- 



(atomic volume) » 



grams for wires 1 millimeter in diameter. 



Metal. 



Iron 



Copper 



Platinum 



Zinc 



Silver 



Gold 



Aluminium 



Tin 



Lead 



I. 



65 



41 



35 



15.77 



29.6 



28.46 



18 

 3.40 

 2.36 



IL 



13.7 



13.7 



17.8 



19.3 



22,2 



22.3 



23.3 



41 



47.8 



36 

 33 

 39 

 29 



27 

 15 

 13 



IV. 

 2,000 (?> 

 1,327 

 1,800 (?> 



690 

 1,223 

 1,313 



898 



504 



60O 



Col. IV. contains the melting-points in degrees Centigrade 

 from absolute zero. Here we have to deal with a much more^ 



