INVERSE METHOD OF DEFINITE INTEGRALS. M9 



and similarly, 



jlFit, 2) fit, 1) = 0, f^Fit, 3)/{t, l) = 0...f,F{t, n) .f{t, 1) = 0, 

 &c. &c., 

 we thus obtain the following equations ; 



j; {t, n) .fit, 0) = A, j,f{t, 0) . F (f, 0), 

 ^,<^{t, n) .fit, 1) = A,S,f{t, 1) . F{t, 1), 



^0 (A n) .fit, w - 1) = A._,!,f{t, n) . Fit, n-l); 



hence the imposed condition of reciprocity requires that the first n co- 

 efficients Ao, Ai...A„-i in the expansion of 0(#, w), may be each equal 

 to zero ; and therefore, 



0(^, n)=A„F{t, n) + A„^,F{t, n + 1) +A„+,F{t, n + 2), &c. ad inf. 



Multiply successively both sides by f{t, n + 1), f{t, n + 2), &;c., and 

 integrate; and since n + \, n + 2, &c. are each > n, the definite integrals 

 must vanish. 



Hence, 



AJ,F{t, n) .fit, n + l) + A„^J,Fit, n + 1) .fit, m + 1) = 0, 



AJtFit, n) .fit, » + 2) + An^,^,Fit, w + 1) .fit, n + 2) 



+ A„^2ftFit,n + 2).fit,n + 2) = 0, 

 &c. &c., 



from whence the coefficients An+i, ^„+2, &c. are known in terms of A^ 

 and ti, and therefore the required function ^ it, n) is known. 



24. To find the function which is reciprocal to t°. 



First, we must form a self-reciprocal function, of which the general 

 term is of the form /"; this has been already effected in Section 

 IV., namely, 



:,. n n + 1 ^ , nin-\) in + !)(» + 2) ^ 

 ^,,-1- j.-y- t+ ^^ . j-^ .t -&C.. 



z z2 



