192 
entire development of (y + jx) (z + kx), we shall produce the 
development of 7 yz. 
‘¢ Without stopping to consider the case where m or nis 
fractional, we may now proceed to establish the mode of inter- 
preting 7 y”2", whatever be the nature of m and n. 
‘* The coefficient of a” y™*z"” in the development of 
7 y™z", is equal to the coefficient of a *,DyD; in the develop- 
ment of e*/?:*", multiplied by 
m(m—-1)....(m-pt+1l)n(m—-1)....(m-v+1). 
But, in the development of the exponential, D:D; occurs only 
in the term 
ah ( 7D, ip k.Ds)" 
(u +yv)! 
and there has for its coefficient 
= (mv) 
(u as v) ha 
Consequently, the coefficient sought is 
m(m—1)....(m—ywt+l1)n(n-1)....(n-v +1) 
data tela ge 
“‘ But again, if we develop (y+ jx)” (z+ kx)" in the manner 
already mentioned, that is to say, preserving all the powers 
of j and k, and afterwards substituting mean products for 
ordinary products of these imaginaries; the coefficient of 
gievy™P 2 is plainly f 
m (m—1) no Seah 1)....(a-v+4 Dub, 
Bev. 
2 
or, since 
Boy op il 
Mikag tiene he 
©) ) Grpice”? 
to 
m (oe TY oe slowed Dias Dev ela) i 
Thus, we have demonstrated generally that the expression 
