FOUNDATIONS OF THE KINETIC THEORY OF GASES. 85 



From the remarks above it is clear that this can be expressed as 



Sj^ffn d2 | *\ 1 ,f 2 d2 + d2 \ 1 ) 

 3 4 1 \ 6 dhdk * dkV h Jh+k + [dkdh dh*J k Jh +k) 



2JirZ/ k+3h 3k+h 



+ 



-1 



"3 4 2\h 2 (h+k)^k\h+k)i 



J* ( k s +Sk Vi) + (3k7i 2 +h z ) 

 4 h?k 2 (h+k)S 



- 4 (hkf ' 



The peculiar feature here shown is the making up of the complete cube 

 of k 4. k in the numerator by the supply of the first half of its terms from the 

 first part of the integral, and of the remainder from the second* On trial I 

 found that the same thing holds for I 5 and I 7 , so that I was led to conjecture 

 that, generally, as in § 21 



2«-l 



2n+l 4 ' (hk)»+ 1 



After the preliminary work we have just ''given, it is easy to prove this as 

 follows. We have always 



((x+y)^-(x-y)^)((x+yy+(x-y) 2 ) = 

 (x + y yn+s _ (p _ y yn+s + Qfl - y 2 y(( x+ yf n ^-(x- yf n - !) . 



Operate on this by 



Jz'^xdx J e'Wydy ( ), 



and on the same expression, with x and y interchanged (when, of course, it 

 remains true), by 



Jz ~ hx -xdx I g - tfydy ( J , 



and add the results. This gives at once 



~KdJ + Mr 2n+1=l2n+3+ \dTi~dk) l2 "- a ; 



which is found on trial to be satisfied by the general value given above. 



* Prof. Catley has called my attention, in connection with this, to the following expression 

 from a Trinity (Cambridge) Examination Paper : — 



{a + b) 2 " = (a+b) n (a" + b") 



+ (a + b)"~ i (na"b + nab") 



+ («+5J»-« ( n - n + 1 a"V+ V±+la'bA 

 ' \ 1.2 1.2 / 



+ (« + &) n - n + 1 2n - i (aPb--i +a n-i b n ) . 



1.2 .... n-1 



