INVERSE METHOD OF DEFINITE INTEGRALS. 141 



22. When 0(a;) is given we may obtain f{t) in an infinite variety 

 of forms by means of the theory of reciprocal functions given in the 

 preceding Section. For instance, if we denote by S^ the sum of the 



products of the natural numbers 1, 2, 3. n when taken m and m 



together, and put 



i.=i+«.h.i,^.j«^.(h.M.+ ^.(h.M-+....+ r^.ch.ur 



,5!lM!!iM. (Art. 8. Section IV.) 



and \„ = l-?.2"/+^4^^.3"f- ±{n+l)''t 



= (-l)"A"{(A + l)''^*}, when h is put =0, 

 then L„ and \„ are reciprocal functions. (Sect. iv. Art. 14.) 



Put therefore y*(^)=aoZ/o + «iZ/i + a2Z/2+a3i3 + &c. 

 and observing that 



1.2.3....W 



ftKL„ = {-lY.- 



w + 1 



we have «„ = ( - 1)" . ^ ^"^ ^ . ftf{t) . X„. 

 But jl/{t) . \ = ftf{f) . ( - 1)'. A" . (A + 1)" . t. When h is put = 0, 



=(-i)"A"(a + i)»j;/(o.^ 



= (-l)».A".(A + l)«.0(A), since ft/{t).f = (p{x). 

 Hence «„= ^ ^^ — - . A» . (A + 1)" . (k), 



and therefore 



f(f) = Lo(p{h) + 2Li — ^ ^' r ^.gjr,^ — i 2 +^-^^- — i 2 3 ' 

 Vol. V. Part II. T 



