INVERSE METHOD OF DEFINITE INTEGRALS. 323 



therefore S = |/2 - 1 + A (1 - 2/)} -" 



R 



Knowing thus the generated function S, we can conversely find q„ by 

 taking the coefficient of A" in the quantity S, and substituting for t 

 its value in terms of <^. 



An exactly similar process applied to the function Q„ of Art. (5), 

 woxild give 



as the function generated, 



and observing that 



R'-{\-h{\-^t)\" ^ 4^h'tt', 



this quantity may be transformed to 



a III 



^\R^\-h{\-^t)}-"; 



so that Q„ is the coefficient of h" in the expansion of this function. 



8. From the theorems given in Arts. (5) and (6), we can determine 

 reciprocal functions relative to <p, which quantity may denote any 

 transcendant contained in the formula Jt{tt'y, from m—-<xi to »« = + x ; 

 circular arcs are amongst these transcendants, namely, when m = — ^, 

 and since both theorems are true simultaneously, when m is between 

 — 1 and + 1, we shall get in this instance the two species of circular 

 self-reciprocal functions, namely, the sines and cosines of the multiples 

 of the simple arc. 



I. To evaluate Q„ when «/ = — i- 



For the variable with respect to which the integrations must be 

 performed, we have 



^ = jXtty^ = l ^y^r^ -■= COS- (1-2^), 



neglecting the constant which is unimportant. 

 Vol. V. Part III. Uu 



