260 



REPORTS ON THE STATE OF SCIENCE, ETC. 



APPENDIX. 



On the connection of the inverse powers of x (x—l) and x (x+ 1 ), toith the sums of the 

 inverse odd powers of the successive numbers x, x-\- 1, a;+2, etc. 



x{x—\) X^\ X x^ x^ / 



Changing x into a;+l, which is the same thing as changing the sign of x, 



a;(a;+l) a;* \ x x^ x^ ) 



.-. 1 _ 1 -9/^1 + 1+ \ 



x(x—\) x{x + \) \x^ afi / 



Hence, writing a series of lines in which each is obtained from the preceding by I 

 changing x into x+l, and adding, we get finally " 



1 1,1, J • / 



= —.+7 VTZ+ ... ad inf. 



2a; (a;- 1) a? (a; + 1)3 



+ i+.-L..+ . 



3fi (Z+1)5 



+ 



ad inf. 



which 



maybewritten2|,+ 2^4-2^+ • • • 



Also -_l_ = lA + 2+3 \ 



x^x-iyi x*\ x x-' } 



and therefore, by a similar process, 



and, in general, r being a positive integer, 



1 -^rV ^ ,r(r+l)(r+2) y 1 



2a;'' (a:-!)'' -4fa;-2'-i^ 1.2.3 Ax^r^s'^ " 



Hence, by choosing suitable coefficients, the quantities 5 ^' 5 — 



x^ 



any given combination of these, can be expressed as functions of { g in 



x(x — l)' 

 series of its powers. 



All this holds whether x is an integer or a fraction, so long as it is positive anJ 

 greater than 1 , 



