of Definite Integrals. 431 



if we actually make these successive substitutions we shall get 

 a,,„ in terms of <p' {a) and its differential coefficients, and the law 

 of the series is extremely simple ; 



for if Sj represent the sum of c„ c„....c^„, 



S„ that of their products two by two, 

 &c. 



then we shall have 



«.„ = <P'{a) - hS,.(p"{a) + h\ S„.<f>"'{a) - &c. 



Suppose now that c^ = - , c^ = -— c„ = — 



__ 1 __ Jl_ _ 1 



TT 27r TlTT 



then it is plain that when n is infinite, S^ = sum of the reciprocals 



Sin ( x\ 

 of the roots of the equation '^— = o, 



S„ = sum of the products of these reciprocals two by two, 



&c. 



that is, S, = o, ^, = __i_, .S'3 = o, ^, = -_-i__ , &c. 



h- h'' 



• •• ",„ = <p!{a) - -j-^ . 0"'(a) + ^^ g^^ .(f{a) - &c. when n= x 



2h \/ — I 



= , /-— - , evidently (l). 



2A. V — 1 



But the value of a„„ may be found in another manner, thus: 



.^ . h d.ct>'{a) h '.„ d[<t^{a)e'-^-'') _ '^'.^ djfCa)^!") 



Vol. III. Part III. 3K 



