INVERSE METHOD OF DEFINITE INTEGRALS. 131 



16. Let «„, h„, c„, &c. be any functions of t, the reciprocal functions 

 to which for simple integration are «„', J„', c'„', &c. 



Let a„, &c. be any function of another variable T, and let a/, &c. 

 represent the corresponding reciprocal function. 



Put S„ = a„a^ + Kai + C^a^ + 



and S,! = an'uo + i/a/ + c„'a.2 + 



then S„, Sn are general forms for reciprocal functions with respect to 

 the double integration relative both to t and T. 



For if we put m for n in the latter series, and multiply the series 

 for S„ and S,„' together, the integral of the products of any two terms 

 which do not hold the same place in either series when taken relative 

 to T must vanish, since a„, a„' are reciprocal functions. 



Hence frSnSJ = a„a„' fj.aoaa + b„b,„' fraiai + c^cj frO^a./ + 



Integrate now with respect to t, observing that when m and n are un- 

 equal, then 



_^ «„«,„' = 0, ftKbJ = 0, ftC„c„' = 0, &c. 



Hence /_4*S'„«S',„' = 0, when m is not equal to n, 



and ftfrSuSn = ftfr {a„a^aoa^ + Kb„'aiai' + CnC^'a-^a^ +...]. 



Cor. 1. In the same manner reciprocal functions of any number 

 of independent variables may be formed. 



Cor. 2. The equation S„ = has n real roots or values of t lying 

 between and 1, whatever value be assigned to 7', when a„, b„, c„, kc. 

 are functions possessing the property ftaj' = 0, &c, x being any integer 



from to w - 1 inclusive ; for then «„ must be of the form — —jj-„ — - > 

 by Art. 5, and similarly 



, d\{t'-t"'V') _ d\{t^t"'F") 

 "~ dt" ' ^"~ df ' 

 and therefore 



Hence »S',=0 must have n real roots between and 1. (Art. 6.) 



