328 Prof. L. Boltzmann on the Assumptions necessary 



Equation (15) is therefore proved by proving the equation 



df drf d% d^' d Vl ' d£ dO' = d% dn dt^i d Vl d^ dO, . (32) 



and vice versa. 



is here the angle between the planes RUR' and R£h? of 

 fig. 1. If on the right-hand side of the equation (32) we 

 introduce, instead of 0, the angle ty, which the former plane 

 makes with the plane ROX (compare fig. 3), £, n, f, £ b rjx, Ji, 

 and therefore also the angle between the planes ROX and 

 ROv remain constant ; and since this is equal to the difference 

 between <fc and ty, it follows that dyjr = dO. If in the same 

 way we introduce upon the left-hand side of the equation (32) 

 yfr' instead of 0', it follows that 



dO' = d$', 



and equation (32) becomes 



rff dif d? d${ d Vl ' d$ } ' city' = d% d v dSd^ d Vl d& df, 



which is exactly the form which H. Stankewitsch gives to the 

 equation. 



We will, however, further multiply each side by cr^S, by 

 which at the same time we indicate that S is to be chosen as 

 the eighth independent variable. The equation thus assumes 

 the form 



d% H dl? d& drji d?i' dip ad$ = d% dn dSd& d^ d& dty <rdS. (33) 



"We now again draw all the 

 lines from the centre O of a Fi £- 3 - 



sphere of unit radius, and de- 

 note in fig. 3 the points of 

 intersection of the two relative 

 velocities before and after im- 

 pact with the surface of the 

 sphere by R and R' ; the ends 

 of the two relative velocities 

 by R x and R/. Let H be the 

 middle point of the arc R R' of 

 a great circle, X the point in 

 which the axis of abscissa in- 

 tersects the surface of the 

 sphere. We now for constants 

 & V, £ f u Vi> Si introduce the 



angles <N = XH and E = ZXH instead of <8=RH and 

 ty = XKR'. Since, again, for a fixed position of the points 

 X, Z, and R, both S and ty as well as N and E are spherical 



