242 Proceedings of the Royal Irish Academy. 



in which / denotes any polynomials whatever, and J„{Xh), J-J\h] may he 

 replaced by similar functions FziXb), Fi(Xh), where 



F,{\b) = ^Fp{X){d/db)PJ„{Xb), (47) 



F,[\b) = ^Fp{X){d/db)P.F,.{Xb). (48) 



The equation then becomes 

 TT 1 2 sin vitt)-! 2 ^ I a [ j;^ ^xr) F^^Xb) - ./_,. (Xr) F, (Xb) j 



X \Jn{Xp)FJXa) - J_n{Xp)F,{\a)]p<p(p)dp 

 J « 



- {#2{A«)i^3(A&)-i^,(ArOi^,(A5); 



= i\ip{r- t] + cp{r + e)], a < r < b. (49) 

 The approximate forms of the admissible values of A are 



X = {m7r + y){b -ay\ (50) 



w^here m is integral, and y is a constant which might be complex or imaginary. 



The proof proceeds on the same lines as before, by considering first the 

 equivalent integral form, and expressing the integrand in terms of the K 

 functions. There is, however, one slight modification involved. The inte- 

 grand is no longer necessarily an odd function of X, so that the value of the 

 integral cannot be obtained by considering only one-half of the infinite 

 contour and doubling the result. The aspnptotic equations (35j, (36) may 

 now be applied directly between the arguments - o7r/'2 (exclusive) and + 7r/2. 



By means of the differential and recurrence equations, the F's in (49) 

 may be expressed in a variety of forms. For example, Fi(Xa) may be Jm{Xa) 

 or any of its derivatives, where vi -' n is integral ; (this may require the 

 numerator and denominator of the fraction to be multiplied or divided by 

 a power of A.) 



Art. 4. Case in ichich a = 0. 



In the case in which a is zero, in order that the equation should then 

 assume a definite limiting form, J-n{Xp) and J^JXr), or else t/„(Ap) and 

 '^..(Arj, must disappear from it altogether. The problem then becomes that 

 of expanding (j){r) between the limits 0, b, in a series of the form 



SC/,.(Ar), 



A. ^ ^ 



where the admissible values of A are determined by the equation 



F^iXb) ^ ^Fp[\) {d/db/ J„{Xbj = 0, (51) 



