322 Prof. L. Boltzmann on the Assumptions necessary 



If, lastly, instead of v, v' we introduce the energies x 

 and of, 



dZ= <H + * *™d*Udydrdtif(v)V<y). 

 m M vyco 



Now 4*m?flv)dv= s/x^{x } t)dx ; therefore 



then, since for constants #and «' evidently dy=dX, 

 m+M_, ( . $(X ^ rF. diVdMX drdS. 



Hence the quantity formerly denoted by v^X^, X,#') 



is equal to 



m + M ^ffrcWS 



if 



16 \/mM J J *>7"> 



where the integration is to be extended over all possible values 

 with the given x, X, a?. If we interchange the values before 

 and after impact, we have 



Since according to equation (16) v'y'co'^vyco, and also the 

 limits of the two double integrals are the same, it follows at 

 once that 



VxX X (x, X > «0 = ^x'{x-X-x') X {x', ^ + X-^', x). (27) 



Two analogous equations hold for yjr and M*. 



The further calculations are now purely algebraical trans- 

 formations of definite integrals, and are effected exactly as in 

 the first section of the already mentioned " Further Studies." 

 I will therefore only briefly indicate the method to be adopted. 



After substituting the values of ^/ and ^ ln 



equation (24), we have a term with 



tyfo • [+to 0<£O + X + *'> *)-*(*> 0<M X , 0]- 

 This, with the aid of an equation for ^analogous to equa- 

 tion (27), is to be transformed into a similar term, with the 

 factor — l<j>{x',t) before the square bracket. ^ Both are, by 

 means of equation (25) , to be transformed into two terms, 

 having the factors ty(X, t) and — /<£(# + X— x\ t) before the 



