THE DYNAMICAL THEOUY OF GASES. 39 



the results of the encounters of every kind between the eZ-ZV, and dN t mole- 

 cules. 



3rd, with respect to dN^, or / ($ a i) t Q d&d^dt,,. 



4th, with respect to dN 1} or (fi^i) d^d^d^. 



These operations require in general a knowledge of the forms of /, and /,. 



1st. Integration with respect to <f>. 



Since the action between the molecules is the same in whatever plane it 



P* 

 takes place, we shall first determine the value of (Q' Q)d<ji in several cases, 



) o 

 making Q some function of , 77, and . 



(a) Let Q = and <? = ? then 



Bin* .................. (4). 



08) Let <? = fraud <? = ?, 



Tif 



...... (5). 



By transformation of coordinates we may derive from this 



(V .Vi - ^o d $ = orao* KMif i1? * ~ Jfi ^ 1 + * (Jfi ~ M * } ( ^ 2 + ^ i)} 87r sinl ^ 



-SJf.tf.-f,)^-^] ...... (6), 



with similar expressions for the other quadratic functions of t], . 



(y) Let <? = (fr + V + .')> and Q^f^V + '?? + ?); then putting 



fr + ^ + tt = V?, & + ^.1?. + O. = ^ 

 and (-)' + fo - ^) 2 + & -,)'=F', we find 



+ f ^^V ( 8ir 8 in' - STT sin' 20) 2 (-)( Z7- F, 2 



\M 1 + MJ 



(M \" 

 s a F J(8 7 rsi 



+ U-rVr ) (Sir sin s - 2ir sin' 20) 2 ( - ) F a 



