172 On the Equilibrium of Vis Viva. 



is now constant. Finally, let the angle between p and the 

 last apsidal line be -ty, then 



x 2 =peo$ {e + yfr), t ?/ 2 = / osin(e + ^)> 

 in which yfr is a function of p, K, and L ; both the latter are 

 now constant, and therefore 



pdpde = dx 2 dy 2 . 

 Collecting all these equations we lind that 



dx x dy, dz, du, dv x dw 1 = n dK dL dK dp da) de ; 



and it will immediately be seen that, if coordinates and 

 velocities without suffix or affix refer to any random set of 

 fixed coordinate axes, we must have likewise 



dx dy dz du dv dw = n dK dL dE. dp da de. 



u mm" a 



Introducing this in equation (15) and remembering that for 

 constant values of u y v, and w, 



71V 



da dh dk= 7 — - — 777-3 du" dv" dw", 

 we find * ( m + ro ") 



ttt^ t — ; — TRa.wrrdx dy dz du dv dw du 1 ' dv" dw" 

 Jb7r (m + m"y&(jc 2 a 



for the number of doublets in unit of volume for which the 

 variables x . . . iv" lie between x and x + dx . . . w" and id" + dw." 

 But we have previously found for this number the expression 



F . dx dy dz du dv dw du n dv n dw n , 



and we then saw that F must be of the form Ae _ * mc2 , in which 

 A is a function of c", p, and <x" only. It follows then, if we 

 write 



F = Be -K(me2+Hl " c " 2+2< P ( ' ))) 



that B must be on the one hand a function of c", p, and a" 

 only, and on the other hand a function of K, L, Gr, and H 

 only. B must therefore be a function of these latter variables 

 which is quite independent of c, a, and /3, and only a function 

 of c" p, and a". 



If we put B=/(K, L, Gr, H), then this function must be 

 independent of c, a, and /3, for all values of c" and a". Sub- 

 stitute for the variables K, L, G, H their values from 11, 12, 

 13, and 14, and make c" = at first. Then 



K=pcsina, L = ^mc 2 + <f> (p) , Q=mc, H = 0. 



TYiCT 



B=/Ocsin«, — + <j>(p), mc > 0). 





