326 Prof. L. Boltzmann on the Assumptions necessary 



directions of v and Y; if, further, f, 97, £ be the components 

 of v; f 1? rji, & those of V in the directions of the axes of 

 coordinates: then 



d%dndZ = v*advdAdB, .... (28) 



d&dy^&^Yh-dV dftdK (29) 



If we denote the magnitudes with reference to the velocities 

 v f and V after impact by a dash, we have 



dgd v 'd v '=v'** f dv'dA'dB', .... (30) 



d?i H\ dp x = Y'V dV' dT dW. . . . . (31) 



Let us in fig. 2 denote the 

 points of intersection of all Fig. 2. 



these lines drawn from the 

 centre O of a sphere of radius 1 

 with the surface of the sphere 

 in the same way as the lines 

 themselves. The variables v, 

 V, T, S, determine simply 

 the magnitude and relative 

 position of the lines determi- 

 ning the impact ; they deter- 

 mine what I have called the 

 form of the impact; 1/, Y', 

 T', £l f are therefore simple 

 functions of the first-named 

 variables. We will leave these variables constant, so that the 

 whole form of the impact remains unaltered. Only its position 

 in space, and so the variables A, B, and K are to alter ; and 

 the product of the corresponding changes in the variables 

 A', B', K', viz. 



dA! dB' dK' = dAdBdK.$+j^-~- ~, 



— dA dB dK 



is to be determined. It is geometrically evident that dA dB dK 

 must be equal to dA' dB' dK.' ; for both sets of differentials 

 may be supposed to be obtained by supposing that, for fixed 

 position and magnitude of v, v', Y, Y', the axis of abscissae 

 describes the whole interior of a cone of infinitely small aper- 

 ture ; and the system of coordinates revolves about the axis 

 of abscissae at a very small angle. This follows analytically 

 in the following way. We see from fig. 2 that B' = B + <fc vULv'. 

 <fc vKv is simply a function of A, K, and the now constant 

 angles. If, therefore, we now introduce A', K7, B' instead of the 

 variables A, K, B, we have dB'=dB. Therefore 



5-l.^a; bb; bk' ^ba' bk; 



-2>A' BB'dK ~^±dA *^k' 



a 



