165 
= 
n!n! 
{@ - ”) \" part of the sum of all the 
m!n! 
different expressions which arise as the differently arranged 
products of the m+n factors of which m are equal to y +z, 
and n to z+ka. 
“The following reasoning demonstrates the truth of the 
proposition just stated. 
“« Let C be the coefficient of ay" z"” in the develop- 
ment of 7y™2". Then C will be equal to the coefficient of 
avy DD" in the development of 
gtIDgrkDs), 
multiplied by 
m(m—1)c.... (m-p+1)n(n-1)..... (n-v+1). 
But, in the development of the exponential, DyD: occurs only 
in the term 
ah” (7D, + kD;)"_ 
(u+v)! 
and there has for its coefficient 
= (uv) | 
(ut+y)l? 
the numerator = (u, v) denoting the sum of all the variously 
arranged products, into each of which enter wjs, and v ks. 
Consequently, we have 
m!n! 
~ (m=)! (n—-v)! (ut+v)! 
“‘ But again, C, the coefficient of x” y" 2" in the 
~ (u, v). 
(m+n)! 
m! n! 
! 
1S = ”) He part of the sum of the 
m!n! 
differently arranged products of the m+n factors, of which 
m are equal to y+jz, and n to z+ ka, will be equal to 
m!n! 
