IV — XXV : 



FOUNDATIONS OF THE KINETIC THEORY OF GASES. 81 



But by definition we have 



u = v cos VOC = v(cos a cos y -f sin a sin y c )S <p). 

 v=v 1 cos V 1 OC = ^(cos ai cos y + sin a x sin y cos 0) ; 



and by the Kinematics of the question, as shown by the dotted triangle in the 



figure, we have 



v cos a — v l cos a 1 = v , 

 vsin a — ^sin ai = . 



Thus, as indeed is obvious from much simpler considerations, 



U — V = V COS y , 



so that 



lvv^v sin /3 d{3 u(u — v) cos y sin y e?ye?0 



/wjVq sin /3 rf/3 cos y sin y rfycZ^ 



/w/^oSin/StfySv (cosacosy + sin a sin 7 cos (f>)v cos 2 7 sin 7 dyd<f> 

 /vviV sin /3 d/3 cos 7 sin 7 d<yd<p 



where each of the integrals is quintuple. 



The term in cos <f> vanishes when we integrate with respect to <f> : — and, 

 when we further integrate with respect to y, we have for the value of the 

 expression 



^ /vv x v sin /3 d/3vv Q cos a 



Ivv-Pq sin /3 d@ 

 where the integrals are triple. 

 Now 



2vv cos a = v 2 + v 2 — v-f , 

 and 



Wi sin (3 df3 — v dv , 



so that the expression becomes 



It will be shown below (Part VI.), that we have, generally 



/• y 2 ^^ _ Wl _ Vj (ft + fc)— 



J VVl vv x 2n + l "" 4 (M)^ 1 ' 



and that it is lawful to differentiate such expressions with regard to h or to k. 



Hence 



d d 



U' — UV : 



iVfi-U-a)v» 1 



4 I 3 /3 7. 



Thus Clerk-Maxwell's Theorem is proved. 

 VOL. XXXIII. part 1. 



