386 Mb MURPHY'S THIRD MEMOIR ON THE 



47. Simultaneous Equations to Definite Integrals. 



Given l-^*^^^^ ••^^^' *) +-'^^(^) • ^^^' «) = ^^^H 

 \!i<p{t)Mt, a) +f,i.(t) . FM a) = EAa)i ' 



the forms of the functions j^ F, E,fi, Fi, Ei, being known, the forms 

 of (p and \{^ are required. 



Multiply the second equation by an arbitrary quantity \, and adding 

 to the first, put 



f(t, a) + X/ {t, a) = A,Po + A,P, + A,P, + &c. 



F{t,a) +xF^{t,a) = AoQo + A,Q, + A,Q2 + &c. 



(pit) = CoPo + c,P,' + c,P^ + &c. 



^^(0 = ^oQo' + e,Q, + e,Q: + &c. 



where P„, P,, P2, &c.\ « .. /? ^ i 



^ ^ .-k o f are functions of t only, 



Qo, Qi, Q2, &C.J ^ 



A^, Ai, Ai,\ known functions of a, X, and self-reciprocal relative 

 to a, 



PI, Qn reciprocal to P„, Q„ respectively, hence 



(putting /,P„P„'=;),„ j;Q„Q„'=^„) E{a) + XEM 

 = c^poAo + c,p,Ai + C2P2A2 + &c. + eoqoAo + e^q.A, + e^q^A-^, &c. ; 

 .-. faAoEia) + X faAoE^ (a) = {copo + eoqo)faAo\ 

 faA,E{a) + XlA,E,{a) = iCiP^ + e,q,)faA^ 



and giving to X any two values in each of these equations, the first 

 will produce two equations which determine Co, eo, the second will 

 similarly give Ci, e^, &c., and thence the functions (p{t), \l/t are known. 



The same method is applicable to any number (n) of simultaneous 

 equations involving n unknown functions. 



48. Definite-integral Equations of superior orders and degrees. 



Methods similar to the preceding are applicable in most cases of 

 the former class thus : 



