Our — Extensions of Fourier s ami the Bessel-Fourier Theorems. 21 



Now the additional terms cause, in the expression for ^, jIIi to be replaced 

 by an expression which differs from it by terms whose order is less by at least 

 three than that of i^i(,uc, +n), (and only the term of the highest order is eitlier 

 unique or important). We may, indeed, and will, choose terms which cause 

 2 jfli to be replaced by 



+ Miuc,,,} (,,-1^(6) + n-^-xPXb) + fi-^rii) + c->-=x(5) + c-V-Y(S)i, 

 and similarly, mutatis mutmulis, for J^i- The added terms are not precisely 

 of the same type as (6), inasmuch as the numerator of their integrand includes 

 powers of fi with negative index. Ltenote by Ph, Pa the expressions which 

 would thus replace jIIi, afli respectively. 



We could add also to aY^, etc., terms which would permit of similar 

 differentiations. It is easily veriiied directly that, in the case of aY^, it 

 would suffice to replace jll, by Pj, and the numerator of the integrand of 

 the portion which involves Jli by 



■2,iTA,, 



e>^'''-'>F,{,,c,-f,) (/.-';/,(«) - fr^\a) + ^^^."{a) + cr^-'xia) - c'n'-^xi"')] 



And we may and will replace this last expression by 



2P„A,,/Au . {e^^''-'''Fi(jic,-fM) - 0^^''-"Fi(^,c,^)\ ; 

 for, bearing in mind that 



Fa{^ic,-ft)-F4,,c,f,)=2^,TAu, 

 it is seen that the difference of the numerators of the integrands concerned 

 contains the denominator of (,16) as a factor, so that the difference of the 

 integrals themselves reduces to an integral taken along a finite contour 

 surrounding the zeros of An and the origin, and may therefore be differen- 

 tiated under the sign of integration as often as is desired. These added 

 terms, as far as the portion of a^^ which involves oJIj is concerned, simply 

 change „n2 into PaA^nJAu. 



It would therefore suffice to establish that, if, in equations (7) et seq., T^ is 

 replaced by 



1 e>^''"-^Fa{fjiC, -n)- e-'^^^-'-^Fa (^f, ^) | (Pj - jH,) 



+ {eM(=^-*)Pj(juc,-/x)-r'"^-'jPi(^c,^)j(„n,-P„)| T Denominator of (16) 

 y. by (22) 



ei^^'du 



J2^7'^i,,(A-jn,)f(„n,.-P.^:..M„)(«^'"-''P6iMC,-i"l-e-'''^-"P*(^c,;.)ij 



-^ Denominator of (16) 



(23) 



and Ys, &c., by similar expressions, x being eventually replaced by «, then 



