oe 
Ti ee ee ee eT 
165 
(m+n)! 
m!n! 
m!n! 
! 
{fe Bias . part of the sum of all the 
different expressions which arise as the differently arranged 
products of the m+n factors of which m are equal to y+jz, 
and n to z+kz. 
‘“‘ The following reasoning demonstrates the truth of the 
proposition just stated. 
«« Let C be the coefficient of ay” z”” in the develop- 
ment of 7 y"z". Then C will be equal to the coefficient of 
av DD} in the development of 
EMID ADs), 
multiplied by 
m(m—1t)i...'. (m-p+l)n(n-1)..... (n-v +1). 
But, in the development of the exponential, DD; occurs only 
in the term 
at” (7D, +kD;3)"_ 
(uty)! ” 
and there has for its coefficient 
= (u, v) 5 
(u+v)!? 
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 
min! 
SE a ye A : 
a (m—)! (n-v)! (ut v)! (u, ») 
“ But again, C, the coefficient of x” y”™ 2"” in the 
(m+n)! 
m!n! 
{or st \ part of the sum of the 
differently arranged products of the m+ factors, of which 
m are equal to y+jz, and n to z+ kx, will be equal to 
m!\n! 
(m+n)! 
