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entire development of (y+ jx) (2 +2)", we shall produce the 
development of 7 yz". 
*¢ Without stopping to consider the case where m or n is 
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 yz" in the development of 
a y™z", is equal to the coefticient of «,*,DD; in the develop- 
ment of e%/2*"», multiplied by 
m(m—1)....(m—p+1)n(m-1)....(n-v+1). 
But, in the development of the exponential, Ds D; occurs only 
in the term 
a” (7D, +k.D;)e 
(u =F v) ! 
and there has for its coefficient 
= (us, v) 
(u + v) Ly 
Consequently, the coefficient sought is 
ra (m= Us hei ih oe St ohee eee 
“‘ But again, if we develop (y+ jx)” (z+ kx)" in the manner 
already mentioned, that is to say, preserving all the powers 
of jand k, and afterwards substituting mean products for 
ordinary products of these imaginaries; the coefficient of 
gevyme 2" is plainly . 
men): - ons) eles as Oe ae 
pp! v! 
or, since 
M(j, k) = TE: A 1 = (us v); 
to 
1a (me Tie ot Nails We» 1 owt l) ae 
(u+v)! 
Thus, we have demonstrated generally that the expression 
