592 Mr. S. H. Burbury on the 



Q, and so sacrificing the generality of the theorem. 1 will 

 assume for this purpose that 



Q = JWx (w x 2 + l\ 2 + W! 2 ) + m 2 (w 2 2 + V 2 + u- 2 ) + &c. 



-4- (m 1 + 7n 2 )& 12 (w 1 w 2 + ^iV 2 + iviiv 2 ) + &c. 



+ (m P + m q ) b Pq (v p u q + v p v q + w p w q ) -f &c. 



Here m 1 m 2 &c. denote the masses, u y v x w t &c. the component 

 velocities of the molecules, and b PQ or b qj> is a function of the 

 distance r pq at the instant considered between the molecules 

 m p and m q , which function is of negative sign, decreases in 

 absolute magnitude as r increases, and becomes evanescent 

 for values of r which may themselves be small beyond the 

 limits of observation. It is not necessary for the present 

 purpose to determine their form. 

 18. We first prove that 



on average. For 



dQ 



dQ, dQ . 



aui an 2 



du^ 



= 1m^u x + (7W! + m 2 ) b 12 u 2 + (m^ + m^)b n u % 4- &c 



The mean value of -=— , u± being supposed given, is 



ctu± 





\\\ ... du 2 . . * du n e~ 



Since Q contains no products of the form uv or uic, we need 

 not take the v's and w's into consideration in forming this 

 mean value. The result of the integration is as follows: Let 

 D denote the determinant 



J) = % m l (77? 1 + ?77 2 )&12 (™1 + ™ 3 )&13 • • - 



m 1 + m 2 )b 12 2m 2 (m 2 + m^)b 2Z ... 



and let D n , D 22 , &c.'be its coaxial minors. Then, given 



Mi, -— has for its mean value vr— u,, and u,-— has for it* 

 ; du x i) n ll du x 



mean value 7^— w, 9 . 



To find the general mean value ^-7-^ when u x also, 

 varies, we must write for u^ its mean value. But that is 



