318 Mh MURPHY'S THIRD MEMOIR ON THE 



SECTION VI. 



Method of' discovering Reciprocal Functions when the integrations are per- 

 formed with respect to any Junction of the independant variable. 



(l) When the limits of integration are arbitrary. 



1. The investigations of reciprocal functions contained in the Second 

 Memoir on the Inverse Method of Definite Integrals, are founded on the 

 supposition that and 1 are always the limits of the independant 

 variable, but it is often of importance to possess reciprocal functions in 

 which the limits of integration are different from those quoted. The 

 principle by which this is most easily accomplished, is to suppose the 

 integrations performed relative to a function of the independant vari- 

 able, which must be so chosen, that when the values and 1 are 

 assigned to the independant variable, the corresponding values or the 

 function may be the proposed limits of integration. 



2. Let Q„, R„, be functions of a variable (^), the limits of which 

 are arbitrary, as a and h, between which limits f^Q^Rm always must 

 vanish, except when the integers m and n are equal. 



Suppose that a function of <p, as t, is found such that when ^ = a 

 t = 0, and when (p = h, t=l, conditions which it is always easy to satisfy. 



We may now conversely regard as a function of t, and then the 

 preceding integral becomes fiQ„Rm-jr, the limits being now reduced to 



and 1. Suppose that -~ is separated into any two factors, X and X'; 



then since f,QnX x R,„\' = 0, except when 7n — n, it follows that Q„X, 

 R„\' are mutually reciprocal, and may therefore be found in an inde- 

 finite variety of modes by the principles explained in Section iv; and 

 dividing these functions respectively by X, X', and substituting in the 

 quotients the value of t expressed in terms of ^, the required functions 

 Q„, Rm will be obtained. 



