262 REPORTS ON THE STATE OF SCIENCE, ETC. 



To find the coefficients of the powers of [z{x — 1)]"' requisite torepresent any 

 function A, 2 -\+ A. 51+ • • • + A,.^ -1-^+ . . . 



Letflr denote the coefficient of lx{x — 1)] "''. 



Then the following table will furnish the linear equations giving the values A,, 

 etc., in terms of a,, . . . ; and their solutions will determine Oi . . . in 

 terms of Ai, . . . 



The table is to be read vertically. Thus 

 2ai = Ai 



2a| + 4a.2 = A.) 



2a I + 8a.2+ 60^5 = Aj 

 etc. 



whence 



:A, 



2ai 



402 = Ao — Ai, or, symbolical]}', A A] 

 60;, = A..,— 2A2+A1 = A^A,, 

 after which the results are not quite so simple. Thus 

 203+ 804 = A4— 3A3+3A2— Ai=A-'A, 

 1 6a.i + 1 Oo, = A, — 3 A4 + 3A3 — A2 = A'^Aa 

 30% + 1 2ae = Ae - 4 A.5 + 6A4 - 4A;, + A.2 = A^ A2 

 and beyond this : 



32a;+ 64ae+ 14^7 = A'A, 

 64a6+ 98o7+16a8 = A"A:, 

 126a,+ 144a„+ 18a,, = A«A,,. 

 But ordy the earlier coefficients are as a rule needed. 



The coefficients required for >" — are Oi = -; 02 = - 

 _, o, = 0; a2=-; %=- 



3 '^ x^ 5 A x'^ 1 A x'' 



1 

 4' 



0.1 = 



6' 



for 



for 



11 

 '24' 



77 + 



aj=- 



1 



6' 



1 . 

 30" 



5.7.9 



1 

 3.5.7' 



32 

 2310' 



and if we require the sums of the inverse odd powers, starting at x-\- 1 instead 

 of X, the same coefficients are needed, but a; (*+ 1) takes the place of a:(a;— 1), and 

 now X may be any positive number. 



