INVERSE METHOD OF DEFINITE INTEGRALS. 121 



Then by the theory of equations, the right-hand member of this 

 equation is equivalent to 



c {(x + 1)' - s,{x+iy-' + SA^+iy-' - SA^+i)"-', &c.| 



whence c = l, ^i = aS',, ^2=, „, ^3= J -^, &c. hence the required 

 function is 



8. It has been proved, that the function thus obtained (which we 

 shall denote by L„) in common with all others which possess the 

 property that ftj'(t) .f = 0, when x is any integer from to n — 1 in- 

 clusive, is of the form 



d\ {ft'" V) 

 dt" ' 



to verify this in the present case, we must sum the preceding series 

 which is represented by Z/„. 



First, by the nature of multiplication, we have 



hr + SJi^-' + S.h"-"- + +S„ = {h + \){h + ^) {h + n), 



and the development of an exponential gives 



i+7.h.l.(^+-A^ + + i.a.3..,:, +&c.=/-, 



the coefficient of h" in the product of both the latter series is iden- 

 tical with that by which Z/„ is expressed. 



But since that product =^(A + 1) (/« + 2) (A + w) 



df 



= ^{r(l+Ah.l.^4-^^l^^&C.)|, 



it follows that the coefficient of h" is also expressed by 



d" [f {h.\. ty\ 

 1.2.3 ndf' 



