378 Mr MURPHY'S THIRD MEMOIR ON THE 



containing t only, the first and second of these functions being given, 

 it is required to find the third so as to satisfy the definite integral 

 equation 



!,<l>{t).f{t,a) = F{a), 



the limits of t being given. 



Suppose (p {t) expanded according to any given class of self-reciprocal 

 functions as P„, that is, 



^(^) = CoPo + CiPi + C2P2 + C3P3, &c. ad infinitum, 

 where the coefficients Co, c,, Ca, &c. are unknown. 



Let J^{t, a) be expanded according to the same reciprocal functions, 

 f{t, a) = AoPt, + A^P^ + A2P2 + A3P3, &c. ad infinitum. 



Then j?P„P„ = 0, and fiPnPn = a„ a known numerical quantity depend- 

 ant on n, and on the particular species of reciprocal functions which 

 are employed. 



Multiply both series and integrate between the given limits of /, 

 and the proposed equation gives us 



F (a) = Aoao.Co + Ai ai.c^+ A2 as . C2 + ^303 . C3, &c. od infinitum. 



Now An being a known function of a and n, we can by Art. 23. 

 Sect. VII., find another function of a and n, as An such that fiA„A„' = 0, 

 when m and n are unequal integers. 



Multiply the equation successively by Ao, A^', Ai, &c. and take the 

 definite integrals relative to a, hence 



jaA(s-P\a) = CoOojaAaAt, ', .'. Co ^ C A ' A ' 



f.A,'F{a) = e,aJ.A,A,'; •.: c, = ^^4^, 

 and generally c„ = r'^'j • 

 Hence <b(f) = ^ /^^°'--^(«) + ^ fgA^Fja) ^ P. fa-A^'Fja) ^ ^^ 



^ ' ao ' faAo'Ao a,' faAi'Ai aj ■ faAa'A^ 



