794 



NATURE 



[February 17, 192 1 



second cos^5 = o, and /i=i, the change will be the 

 same as would be produced by a lengthening of 

 L, in the ratio of i to i +t;2/2c2. As no displace- 

 ment of the fringes has been observed, we are led 

 to the well-known hypothesis of a contraction of 

 moving bodies in the direction of translation, in 

 the ratio of i to x—v^jzc^. 



We could now try to extend the above con- 

 siderations to cases in which v/c, though always 

 below I, is no longer a small fraction. This 

 would require somewhat lengthy calculations, into 

 which, however, we need not enter here, because 

 we know by the theory of relativity that the true 

 value of the coefficient of contraction is \/ 1 — t^^/zc^. 

 I may remark here that there can be no question 

 about the reality of this change of length. Sup- 

 pose that, in studying the phenomena, we use a 

 system of rectangular co-ordinates Xj, x^, X3, and 

 a time f, and that in this system the velocity of 

 light is c in all directions. Further, let there be 

 two rods, I. and II., exactly equal to each other, 

 and both placed in the direction of x^, I. at rest 

 in the system of co-ordinates, and II. moving in 

 the direction of its length with a velocity v. 

 Then, certainly, if the length of a rod is measured 

 by the differences of the values which the co- 

 ordinate x-^ has at the two ends at one and the 

 same instant t, II. will be shorter than I., just as 

 it would be if it were kept at a lower temperature. 

 I need scarcely add that if, by the ordinary trans- 

 formation of the theory of relativity, we pass to 

 new co-ordinates x-^ , x^, xj, t' in such a manner 

 that in this system the rod II. is at rest, and if 

 now we measure the lengths by the difference 

 between the values of x-^' which correspond to a 

 definite value of t', I. will be found to be the 

 shorter of the two. 



The question arises as to how far the dimensions 

 of a solid body will be changed when its parts 

 have unequal velocities, when, for example, it 

 has a rotation about a fixed ajfis. It is clear that 

 in such a case the different parts of the body will, 

 by their interaction, hinder each other in their 

 tende ncy to c ontract to the amount determined 

 by v^i — w'/c'. The problem can be solved by the 

 ordinary theory of elasticity, provided only that 

 this theory be first adapted to the principle of rela- 

 tivity. Indeed, we can still use Hamilton's prin- 

 ciple : — 



«/"'V/f(T-uys=o . 



(3) 



(dS, element of volume ; T, kinetic, and U, poten- 

 tial, energy, both per unit of volume), if, by some 

 slight jnodifications, the integral is made to be 

 independent of the particular choice of co-ordin- 

 ates. That this can be done, even in the general 

 theory of relativity (theory of gravitation), is due 

 to the possibility of expressing the length of a 

 line-element in the four-dimensional space x^, x.,. 

 «3, x^ {Xf = t) in "natural units" — i.e. in such a 

 manner that the number obtained for it is the 

 same whatever be the co-ordinates chosen — and 

 of measuring angles in a similar way. As is well 

 NO. 2677, VOL. 106] 



known, the length ds of a line-element is given 

 by the formula : — 



<is* = 2{a6)gai^x.tixt, .... (4) 



where the ten quantities gab (gab = gba) are the 

 gravitation potentials, and the angle 8 between 

 two elements is determined by 



coitdsds = 1{ab)g^Xad.xi, ... (5) 



In the sums, each of the indices a and b is 

 to be given the values i, 2, 3, 4. When the 

 value 4 is excluded, as will be the case in some 

 of the following formulae, a Greek letter will be 

 used for the index. 



We can also find an invariant value I for the 

 distance between two material particles P and P' 

 infinitely near each other. To this effect we 

 have to consider the word-lines L and L' of these 

 particles in the space Xj, x^, Xj, x^. Let Q be 

 the point of L corresponding to the chosen time 

 X4, and Q' a point of L' such that QQ' is at right 

 angles to L. Then the length of QQ', determined 

 by means of (4), will be the value required. Simi- 

 larly, if P" is a third particle, infinitely near P 

 and P', and Q" the point of its word-line so 

 situated that QQ" is perpendicular to L, the angle 

 P'PP" will be taken to be the angle between the 

 elements QQ' and QQ" determined according to 



As to the co-ordmates Xj, x-j^ «g, X4, it may be 

 recalled that, in a field free from gravitation, they 

 may be chosen in such a manner (xj, x^, x^ being 

 at right angles to each other) that the velocity 

 of light has the constant magnitude c ; the 

 potentials gah will in this case have the values 



g\\ =g-a =.«'33 = - 1 . ^44 ="^'- .?'«i> = ° foi" " * *• 



These may be called the normal values of the 

 potentials, and a system of co-ordinates for which 

 they hold a normal system. 



Let us now consider a solid body M, and let 

 us first conceive it to be placed in a normal system 

 of co-ordinates (Sq), and to be at rest in that 

 system, free from all external forces. The body 

 may then be said to be in its natural state, and 

 its particles may be distinguished from each other 

 by their co-ordinates £, »;, ^ with respect to three 

 rectangular axes fixed in the body. In all that 

 follows, these quantities will be constant, and so 

 will be the mass pd^d.r\d\ of an element, p being 

 the density in the natural state. 



We shall now suppose the body to be placed in 

 a system of co-ordinates Xj, x„, Aj, x^ (S), not 

 necessarily normal, and to have some kind of 

 motion in that system. It is this motion, in which 

 X,, .-xTj, X3 will be definite functions of ^, r\, ^, and 

 X4, which we want to determine by means of 

 Hamilton's principle properly modified. 



In order to get the new U, I shall introduce 

 the dilatations f(, 17,, t. , and shearing strains ^„, i/j, ^ , 

 with respect to the axes ^, »;, {. These quantities 

 are defined as follows : — 



Let P, P' be the particles ^, r;, J;, and ^ + d^, ri,t,, 

 and let I be their distance in the state considered 



