Oiiii — Extensions of Fourier's and the Bessel- Fourier Tfieorenis. 247 



If \{^'{a) - Jii\p{a) is zero, (and this is involved in the supposition that 

 ip, \p' are continuous), the lower boundary terms automatically disappear 

 from (68). And, if t^Xh] - Jh\fj{b) is zero, the substitution of the upper 

 boundary values reduces the fraction in (61) to the form 



A I J„(Ar) J_„(A&) - J-niXr) Jn{Xh)\, (71) 



whose residue at any point is zero. 



Thus, the second differential coeflicient of the term of (61) being at 

 most of order A"\ that of the original is at most of order A"^ 



Now the portion of (p which is due to the original \p is obtained, save 

 as to the numerical factor, by multiplying each term of the left-hand 

 member of (61) by the corresponding value of cos Ac^. In the series obtained 

 by differentiating this twice, term by term, with respect to r or t, a typical 

 term is asymptotically of the form 



. mir {r - h) mnct 



Am cos — r — — — - cos r , 



- a - a 



where the values of m are the integers. On replacing the product of two 

 cosines by a sum, it is seen that, at points within the range, failure in the 

 uniformity of convergence occurs only at points given by 



{>• - ri ± ct)l{h - a) = 2p, (>• + r, - 2b ± ct)j{b - «) = 2p, 



p being zero, or an integer, positive or negative, where ri is a point of similar 

 failure in the original series for i^'^* And it is thus seen that the series 

 obtained by differentiating once with respect to r or t are uniformly con- 

 vergent everywhere, and those by differentiating twice, everywhere except 

 near certain values of r or t, corresponding to discontinuities in -p" ^ 

 propagated with velocity c, and reflected as often as may be. 



It is thus legitimate to differentiate this portion of ^ once or twice, term 

 by term ; and it therefore satisfies the differential and boundary equations. 



1 next consider the part of ^ which arises from ^(j. When ^ is expanded 

 in a series by the aid of (61), the order of magnitude of a term can be ascer- 

 tained sufficiently by differentiating once with respect to r. Instead of (68), 

 we now use the simpler equation 



A>/„(Ap)x'»f^p= - 



\pJ'J\p)x{p) 



\b 



^^Y'Jn [Xp) X{P) + ^Pj'n {Xp) X' (P) J dp- (72) 



* The argument at the corresiJonding stage in Part I., Art. 19 (p. 231, II. 1-3), is stated in words 

 which imply that \p"' (and later that x") is a Dirichlet function. Slight alterations would avoid this 

 as is done here. See also the two final paragraphs heloM'. 



