INVERSE METHOD OF DEFINITE INTEGRALS. 119 



or, putting l — t=^t', we obtain 



J^' (jo + l).(jo + 2)....(jp + «)" r/^" 



5. From this result it follows that if we put 



then shall ftj'{t).t'' = 0, provided x is any number of the series 



0, 1, 2 (n-l); 



Op representing any constant quantity. 



Now OpfP may be taken for the general term of an arbitrary function J^; 

 hence the most general function which satisfies the equation ftf{t)-t'' — 0, 

 is expressed by 



.,,. _ d^jtH'T) 



In fact we have (supposing the integrals to commence from ^ = 0,) 



f,f{t) . r = t^f, it) - xt^-'f, {f)\X.{x- 1) ./s it), &C. 



representing by fn {t) the ri^ successive integral oi fit), and putting for x 

 0, 1, 2....(w — 1) successively, it follows that 



Mt) = 0, f,{t) = .f„{t) = 0, when ^=1; 



that is,Jn{t) and its n differential coefficients vanish when t = and when 

 t=l; therefore y^ (/) contains a factor of the form ^".(1 — ^)", and con- 

 sequently the most general form of f{t) is 



d"(t"t'''r) 

 dt" ■ 



6. Hence we deduce the following general property: '' If /{t) he 

 any function which satisfies the equation [tf{t) . t* = 0, a; being any integer 



from to in — V) inclusive, then the equation f{f) = will always have n real 

 roots lying between and 1." 



For the equation r.^'°F=0 has n roots t = and n roots ^=1; and 

 therefore f{t) which is the n^^ derived equation must have n roots be- 

 tween and 1. 



q2 



