348 Mr MURPHY'S THIRD MEMOIR ON THE 



!,F(t, n) ./{t, 1) = 0, 



SF{t,n).f(t,n-l) = 0; 

 then the function F(t, n) will obviously be self-reciprocal. 



But if f{t, n) not containing arbitrary coefficients, but being abso- 

 lutely given as P, (cos^% &c. is proposed as a function to which some 

 unknown function is reciprocal, the discovery of the latter, which is 

 effected in the next article, is of a more difficult nature than the pro- 

 cess above mentioned; and in the particular cases quoted, as well as in 

 many others, this required function is transient, it is therefore in this 

 character that transient functions are here introduced. 



23. Given f (t, n) a Junction of known form with respect to the vari- 

 able t and the integer n, it is required to find another Junction of t and 

 n, as ^ (t, n), such that the definite integral jjf (t, n) ^(t, n') may always 

 vanish when the integers n and n' are unequal. 



Begin with forming a self -reciprocal function F{t, n), the general 

 term of which may be of the given form J{t, n) ; thus 



F{t, n) = a,f{t, 0) + a,f{t, l)+a,f{t, 2)+ +a„f{t, n), 



where the coefficients are determined in the manner indicated in the 

 preceding article. 



Suppose next that the required function {t, n) is expanded in an 

 infinite series of which the general term is of the form F (t, n), thus 



<p{t, n):=Ao.F(t,0)+A,F{t,l) + ...+A„F{t,n) + A„^,F{t, (n + l)}, &c. 



Multiply by f(t, 0), f(t, 1), f{t, 2) f(t, n - 1) successively, and 



integrate the products between the given limits of t, observing that 



f,F{t, 1) .fit, 0) = 0, f,F(t, 2) .fit, 0) = 0. ..J,F{t, n)f{t, 0) = 0, 



by the property of the functions F {t, n) ; 



