ME. CLEEK MAXWELL OX THE DYNAMICAL THEOEY OE OASES. 
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1st, with respect to <p from <p=0 to <p= 2t. 
2nd, with respect to 5 from 5=0 to 5= oo . These operations will give the results of 
the encounters of every kind between the and dN 2 molecules. 
3rd, with respect to ^N 2 , or fa,(^ a ^a)d^ a d7] a d^ 2 . 
4th, with respect to (ZN„ or f^^^d^d^d^y. 
These operations require in general a knowledge of the forms of f 1 and f v 
1st. Integration with respect to <p. 
Since the action between the molecules is the same in whatever plane it takes place, 
we shall first determine the value of 1 (Ql—Q)d<p in several cases, making Q some 
Jo 
function of £, ?j, and £. 
(a) Let Q=£j and Q'=|', then 
sin a fl (4) 
(/3) Let Q=%i and Q'=^ 2 , 
/*2 jt 1VT r \ 
) t (?f-|!)^=p7+ls 5 5{(| s -|,)(M 1 |,+M s | a )&rsm a «+M 2 ((, 2 -> i , )•+(?.-?. )«-2(| ! -J,) ! ) I rsm a 2S}. 
By transformation of coordinates we may derive from this 
( 5 ) 
/» 2jt M f / 1 
j # M 2 )(| 1 ;? 2 +^ 1 )) 85 rsin 2 fl— 3 M 2 (f 2 — §,)(%— *)}, ( 6 ) 
with similar, expressions for the other quadratic functions of £, j?, £. 
(y) Let Q=£i(§?+*!+£?)> and Q^=#(g?+fl?+S?); then putting £?+3?+£?=V?, 
&§ 2 +Wa-Ki£»==U, ^+^ + ^ 2 =V 2 , and (£ 2 -£,) 2 + 0? 2 — *?i) 2 +(£ 2 -£i) 2 =V 2 , we find 
£©v?-S,v;)#= 5 i^m;4x sin" <{( 1 ,- 1 , )v;+ 2 |,(U-y;} 
+'(M*Mj(8Tsm a 0-3 5 r S m a 2«)2(&-J,)(U-V0 
( M \ 2 
(^sin^^siir^V 2 
( M \ 3 
( 8 * sin3 * - 2 *- sin * 2 <) M> - ^>) V 2 . 
These are the principal functions of £, ;j, £ whose changes we shall have to consider ; we 
shall indicate them by the symbols a, (3, or y, according as the function of the velocity 
is of one, two, or three dimensions. 
2nd. Integration with respect to b. 
We have next to multiply these expressions by bdb, and to integrate with respect to 
b from 5=0 to 5=co. We must bear in mind that Q is a function of 5 and V, and can 
only be determined when the law of force is known. In the expressions which we have 
I 2 
