Calculus of Operations to Algebraical Expansions. 359 

 will be found to be 



°- p + 1 A0 + (j, + l)a, + 2) AU 



8.(j» + 8) AS()M 



(»+«(n + 2)(o + 8) 



where g . (j» + g — 1) denotes the sum of the products of q indi- 

 viduals in every combination that can be made from the series 

 of natural numbers 1 . 2 . 3 . 4 . . . (p + g — 1). 



x n 

 This expression, therefore, is the coefficient of j—^ in 



(x \p 

 _-: - 1 ; and, this coefficient being, as we have seen, 



/■ io g (i+A) y „ 



we have 



|d£±B *"i fSec 65 1 



Now it is easily shown that the sum of the products now 

 under consideration is obtained by forming 



.,.S(i>20>Sp)).. 



(there being q operations denoting multiplication by p and sum- 

 mation with regard to p), and by writing p + q for p in the 



( — log(l— x)\ p . 

 K± ' \ IS 



' . . . S(( j? + g) S (jp + g)) • • • (g times) 

 (jP + l).O> + 2)...(* + 0) 



This last example suggests an obvious method by which we 

 can find the expansion of cj> log (1 + a?) wherever we are acquainted 

 with the coefficient of x n in <px. 



Thus, since the coefficient of x n -?-l . 2 , . n in (e*— l)-s- a? is 



— -y, we have A , 



log(l + A) "~ii + r 

 Therefore, if 



-jj^A^ =V 0+ V 1 A + V^ + ... +V„A»+ .... 



