206 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
Hence 
nh — 
ie Qaal® Toa | _,( ~ Sebo 8 a2Z,)ade. 
U(py,%)= —%Py ( By ee I 
J a « Qa 2 3 
to(py 5 %) = dp, + atl 4 | 
The ordinary approximate theory takes account of the term in zP, only. 
The solution for a hollow cylinder can be obtained by this method, or by taking, in 
the notation of (94), 
F = Ap,’ + Blog p, + C (4p,’ log p; — 4p;’); 
iD eas 
determining C from the value of | _, ou at either edge, and then determining 
A and B from the conditions that 
rh eat 
| (—8pzpp +3 —a 229) dz must for p=a or p=b 
—h 
have the same value in the assumed form as in the actual displacement. 
40. Expansions of arbitrary functions. 
When we attempt to apply the method of last article to the determination of the 
modes corresponding to the various roots of sinh 2«h + 2h, we are at once confronted 
with an apparently insuperable difficulty. The determination of any one mode is 
reduced by the application of Betti’s Theorem to the special problem of balancing the 
particular source solution involving a given root «. Now in similar investigations 
connected with Laplace’s equation, the equation of conduction of heat, and other partial 
differential equations of the second order which occur in physical mathematics, the 
analogous balancing problem can be solved without difficulty for certain simple forms of 
edge, and the balancing solution is of the same type as the particular source solution, 
that is, involves only the same root «. In the present problem, however, the balancing 
solution will in general involve particular solutions of all types, as will be seen below. 
Various theorems relating to the expansion of arbitrary functions may be found, 
similar to the theorem suggested at the end of § 38, but these do not help us, at all 
events immediately, to the general solution sought. One way of obtaining these 
expansion theorems may be indicated here; the method is of very wide application. 
On a circular cylinder p=a, within the infinite plate, let areal force be distributed, 
the components of its intensity per unit area being P cosmo, {sin mw , Z cos mw , where 
P, 2, Z are functions of z. 
The infinite plate solution for this distribution of force can be written down, and 
the components of stress pp, po, pz, calculated. These are given in different analytical 
forms according as p is greater or less than a. The expansion theorems are derived 
from the conditions of equilibrium 
L ~ a 
pee ' pp(p = — €) — pp(p= a+) ; =e 
with two similar equations for Q, Z. 
