226 Proceedings of the Royal Irish Academy. 



whether the coefficient of dk varies asyiaptotically as siu A (/' - a) or as 

 cos A {r - a). 



That the factors of the numerator in the first form of the left-hand 

 member are both odd or both even functions of A, and do not liecome infinite 

 at the origin, is perhaps best seen by transforming into •/ notation. 



The case of a zero need not be specially considered ; for, if (69) then 

 assumes a definite form, it becomes of a type already discussed. 



I conjecture that when the element of the integral is, as a function of r, 

 of type other than a multiple of 



{dida + k) [ Jn {\r) Jl,, (X«) - J.„ {\r) J,, (\a) J , 

 the expansion is not unique. 



Art. 15. Dv§-erentiatiun of Equations of Art. 1 under Integral Sign. 



I now proceed to consider the conditions under which equation (12), Art. 1, 

 may be differentiated under the sign of integration. On so differentiating, the 

 element of the integral is, as far as a; is involved, a multiple of 



{ .S' ( A) cos \oi -C{\) sin A« J dX . 



Xow, assuming that 0'(,''j satisfies the Diriclilet conditions, its actual expansion 

 in this fashion is given by 



'' 7wrvAx A /y/xN • A X f" 'S'(A)cosA'2i-C{A)siuAn , 

 ^^ d\ (>S (A) cos A.v - 6XA) sm A.';) CW) + SHX) ^ ^^'^ 



= ^{:c): " (70) 



for the statement of conditions to be satisfied by C, S, is unaltered by changing 

 C into S, and S into - C. Integrating by parts, the integral in u becomes 



>S'(A) .. _-A^S'(A)cosAi/:i - C'(A) sin A«i . , , ,, ,, 



the second term arising from possible discontinuities. This reduces to its last 

 term alone, if ^ {u) is free from discontinuity and vanishes when x is zero. 

 (Of course (p{vJ) vanishes at infinity; otherwise equation (12) could not hold.) 

 'JBut'the substitution of the last term alone in the left-hand member of (70) 

 renders the equation identical with that obtained by differentiating (12j under 

 the integral sign. 



Thus, if ^ (.';) is continuous and vanishes for ..',■ = 0, (p\.v) may be obtained 

 by differentiation of the type alluded to, if ^' be a Dirichlet function. 



Similarly for successive differentiations, if the conditions usual in the 

 Fourier case are satisfied. 



r-- ■" 



C"-(A) + ^^A) ^ ' ^ ^ C\\) + S' (A) 



A { (7(A) cos Xu + S(X) sin Xit 



