CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 439 







Thus 



X*+Y'+Z* = (m+m'YVf, 



and 



(15) (m + m') (m2lP + m'2U'*) = V,* + mm'V *, 

 writing V,' = X.'+Y.'+Z,* Also, we have 



(16) (m + m')U = X 1 -A'X, (m+m') V = Y.-JPY, (m+m') W = Z,-L J Z, 



(17) (m+m')U' = X, + /tX, (m+m')V' = Y l + /,-, (m+m') W = Z^fcZ, 



m + m" 

 Since the mass velocities of the two systems of molecules are the same, we have 



(m+m')(u'u) = (m + m')(U' U) = X, 



and two similar equations. 



Considering Q = *, it is easily seen that 



r$i 2 u a de = (u ia 3 u*)de 

 Jo 



772- r/ / \ / / t\ f\ ' Q / r TT a *\ f i \tt o ^ n 



-jr a [(u u) (mu + mu') Sir sin 8 x +m { V 3 (w M)*} TT sin* 2 X J, 



where we have expressed the result in terms of sin* x and sin* 2 X ; on making the 

 above transformations we obtain 



(18) rXu'd= m> 



/ ., I uTxr - -^ 



y (m + m )~ 



We next consider <$,., (*+v a +w a ) ; we write 18 = u + a a' cos (e w,), where 



a = 2&'(w' M) sin* x , a' = kf\/V^(u'uf sin 2 X , and similarly for ? and w ia . Since 



we have 



1 f 5 ** 



12 (ttu 1 + v ia 3 + w,/) (if = (M + a) 2 (u + a) a +% (u + a) 2' 2 + a'2 ( + a) ' cos w,. 



ZTT Jo 



It is easy to show that 



a'2 (u+a) a' cos w, = /^/w { (Y 8 +Z)-i'XY-wXZ| sin* 2 X , 

 and from the last three equations it readily follows that 

 (19) rs ia u(u a 



2m/S - 7T5 (3X2XX.-X.2X*) x sin* 2 X . 



(m + m')* 



