204 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
integrals which define the coefficients of the other particular solutions must vanish, 
while the integral corresponding to the solution left has its value determined. These 
results are easily verified by actual integration. 
This remark may be used to find the solution for a hollow cylinder, which of course — 
might also be obtained directly by the above process. We shall illustrate the method 
by finding the value of F; corresponding to any given root « of sinh 2«h—2«h, when 
we have given u,, U,, Za, Zy- 
This value of F, we know is of the form AJ «p,;+ BG«p,. 
The complete values of wu and of zp/2u for p =p, are given by series which manifestly 
converge uniformly so long as b<p,<a. 
Multiply the series for zp Du. by —(a+cosh 2«h) cosh «z+ 2«z sinh xz, 
the series for u by «(3 —cosh 2«h sinh xz + 2xz cosh Kz) , 
add, and integrate with respect to z from —A toh. All the terms disappear except 
that associated with the given root x. We thus find 
(AJ q'kp; + BGy'Kp,)2(a + 1)x?h(cosh 2xh — 1) 
= { of (p=p,) (—a+ cosh 2«h cosh «z+ 2«z sinh Kz) + Kulp = p,) (3 — cosh 2h sinh «2 + 2xz cosh xz) } dz. 
This is proved for the case b<p,<a. Now take the limits of both sides for p,=a. 
The limit of the integral is found simply by replacing z and u (e=p,) in the integrand 
by zp and u (p =a), provided the resulting integral has a meaning, which will be the 
case if z and u (p =a) are integrable functions of z. Similarly we may take the limit 
for p;=b, and thus obtain two equations to determine A and B. 
It will be observed that by this method we avoid two difficulties which in problems 
of this kind are often introduced unnecessarily by physical writers, namely, (i) the — 
difficulty as to the convergence of the series for z and uw, when the value p,=q@ or — 
b is substituted term by term, and (ii) the allied difficulty as to the continuity of the 
series right up to p=a or b, even when it is known to converge. Judging from 
analogy, we may feel reasonably certain that the series will in fact converge at the 
limits, at least in the case of ordinary functions; but it is worth while noting that, 
whether they converge or not, the Fourier method of assuming the continuity and 
convergence, and determining the coefficients by integration, does give the correct values 
of these coefficients. 
On the other hand, while our ‘Green’s function’ method proves definitely that 
any possible solution has the form given above, it does not prove that a solution ws 
possible for arbitrary edge values of z and u. The investigation might be completed 
by verifying that the solution obtained does actually satisfy the conditions, which 
would not be difficult in the present case. Alternatively, we may rely upon physical 
considerations, or upon a general analytical existence theorem. ‘The proofs of theorems 
of this type in other branches of physical mathematics have been considerably improved 
within recent years by Poincaré and others, and their methods are equally applicable 
to the elastic equations. 
