J>/ew Algebraic Series. 



279 



l+a 



4-i 



Y+a(a+A;) 

 + 2a6* 

 +6(6+^) 



+3ab{a-\-k) 

 -^3ab(b+k) 

 -\-b{b+k){b+2k) 



+kc. 



1.2.3 



4- 



+ 



z z 



Here the co-efficient of t is a-^b ; that of ~J7^ may be 



decomposed into a({a+k)-{-b)=a{a-\-b-{-k), and 6((6+A;) 



+a)=6(4-«+6+A;), the sum of which is (a+6)(a-l-6 + A:). 



z^ 

 The co-efficient of % may likewise be decomposed 



into 



a^ (a+k){a-{-^k)-{-'2b{a-^k)+b{b-\-k) )>, and 

 b^ {b+kXb+-2k)-{-2a{b-i-k)-{-a{a-^k) > 

 But the multiplier of a in the first part is evidently that 



z" 

 which would become the co-efficient of "p^; when a is 



changed into a-^k, and the multiplier of 6, in the second, 

 is the same co-efficient that would result by changing b in- 

 to b-{-k. It follows, therefore, that, as the coefficient of 



z^ 

 yy, becomes {a-\-k){a+b+k), the multiplier of a, in the 



first part of the co-efficient of "Y"^, and that of 6 in the 



second, will become a(a-{-b+k)(a + b-\-'2k)+b{a+h-^k) 



{a+b-\-'2k), that is {a+b){a-\-b+k){a+h+2k). Hence, 



from the preceding, the law is evident for the four first 



terms. And to show that Ihe law will hold for an) number 



of terms of the product of the above series, we have only 



to prove that if the law holds as far as the co-efficient 



zp — 1 

 r-r — ^r- ^rinclusively, it will equally hold for the next co- 



z^ z^ 



* In the actual multiplication afeXY=2a6. -j-^, and 



z^ z^ 



ah{a-¥k)~:^=3ab{a-\-k)-Y^.^<^'^^^^''^' 



