INVERSE METHOD OF DEFINITE INTEGRALS. 387 



Given fj,<t>{t, T)f(t, T, a) = F{a), 

 the forms F and J" being known to determine (p. 



By Art. 16. Sect. iv. let a function Q„ be formed which shall be 

 self-reciprocal, relative to double integration for t and t. 



Put ^(#,t) = Co Q„ + CiQi + C2Q2 + &C. 1 _rrri'i 



and/(^,T,«) = ^„Q„+^,Q, + ^,Q, + &c.r'''* ^^^ a»-i.^t^», 



hence F(a) = aoCo^o + aiCi^, +0203^2 + &c. 



Let ^„' be a function of a reciprocal to A„, 

 then faA,'F{a) = c^aJaA.A^, 



faA,'F{a) = c,aJaA,A„ 

 &c. &c. 



whence Co, Ci, &c. being determined, the function (p{f,T) is known. 



Equations of superior degrees must generally be converted into equa- 

 tions of superior orders to be easily solved, thus; 



Given f,(p{t) .fit, a) x [,cp{t) . F{t, «) = >/.(«), 

 the forms ^ F, and -^^f being given to find the function 0. 



Introduce another variable t having the same limits as t, then it is 

 evident that 



J,<p{t) . F(t, a) = /^«^(t) . F{t, a) ; 



.-. U^cp{t) .(pi-r) ./{t, a) . F{t, a) = f (a), 



and since y(#, a) . F{t, a) is a given function of t, t and a, the unknown 

 function (p{t).(p(T) will be determined as above, and representing it by 

 <p^(t,T), let a be a root of the equation 0(t) = 1, then since (p{t).(p{T) 

 = 0i(#, t), we get the required function (}>{t) = <pi{t, a), and again putting 

 ^ = a we get ^1 (a, a) = 1, from which equation a is known, and there- 

 fore <p{t) = <pi{t,a) is also known. 



49. In researches on the subjects of electricity, and the phaenomena 

 dependent on the molecular construction of bodies, the only data which 

 can be furnished by experience are the total actions, and consequently 

 Vol. V. JPart III. SE 



