The Hon. J. W. Strutt on BessePs Functions, 335 



number of values of k satisfying the pair of boundary conditions 

 written above, whatever may be the values of M^, M^, Nj, Ng 

 for any assigned radii r^ and rg. In certain cases it may happen 

 that either A or B vanishes, and that only one of the component 

 functions is required. It is scarcely necessary to say that, from 

 the more general case of a hollow cylinder, the formulse appro- 

 priate to a complete cylinder may be derived by simply putting 

 ri = 0. 



In the formation of equation (14) we have tacitly assumed 

 that, after a complete circuit from ^==0to ^ = 27r, the functions 

 <^, <^' recur. But if w be fractional, this will not be true ; and in 

 applying Green's theorem, we must take account of the fact by 

 including the surface-integrals over the planes ^=0, ^ = 27r, 

 which now no longer destroy one another. However the addi- 

 tional integrals are, as is easily seen, symmetrical in k and k^j 

 and therefore disappear from equation (15). Accordingly the 

 results remain precisely as before. 



Previously to applying (17) and (22) to the expansion of an 

 arbitrary function of r in a series of BessePs functions of order 

 n, a certain restriction should be noticed. The necessity of it 

 will appear from the consideration that if r is indefinitely small, 

 J„(/cr) contains no lower power of r than r«, showing that the 

 arbitrary function must possess the same peculiarity. This cir- 

 cumstance, however, does not interfere with the generality of the 

 expansion of an arbitrary continuous function of two variables 

 within the circle r = l. Let the function be expressed by rect- 

 angular coordinates x and y, and expand it by Taylor's theorem 

 in rising powers of those variables. On substitution of polar 

 coordinates, any power of r, say r^, will be accompanied by 

 powers of sin 6 and cos 6 not higher than the nth j or, when 

 these again are expressed by sines and cosines of multiple arcs, 

 the coefficient of 6 will not arise above n. It follows that when 

 a continuous function is expanded by Fourier's theorem in the 

 manner supposed, the coefficient of cos {n6-\-u)j considered as a 

 function of r, will contain no lower power than r". 



Suppose now thaty(r), subject to the above restriction, is ex- 

 panded in the form 



/(r)=2A,J„(v), ...... (23) 



where Kp is a root of (18). If we multiply both sides by 

 Jn(Kp7').r, and integrate with respect to r from r = to ?' = 1, 

 we have by (17) and (19), 



2 (\drfir)J^{K,r)=Ap V2rdr[5,XPJ)Y 

 Jo •^o 



=a,{[J'»W]^+(i-J)lJ«W]^}; 



