136 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
These z-integrals of 1/r may be expressed in the form of definite integrals involving the 
Bessel function J, analogous to the integral forms for ~* and its 2-derivatives, 
got = [eS ur0de 
fe = | (—x)e-"*JI xpdx, ete., where z>0, 
¢ 0 
We may notice that the value of ii ead oxpd« follows at once from the remark that 
it is a potential symmetrical about the axis of z, and taking on that axis the value 
oO 
i Wait 5 ce eee 
| , ek = re We may use this idea to express u,, U,, Us Im similar form, 
“(eter dk ———— log = 
0 K (O? 
i 2 |I ee: ] a 
i OBES a TIKZE yea = #1085 -# 
For we have 
and 
[(es- 1L+«z- : cates) =- ie log = ar ; oe 
by integration with respect to z from 0 to z. 
Hence 
SG ae: = = a 
i (e-Joap - 1+ x«ze- = ee | = zlog "=r 
|, (edu l+xz- ped — poten = — 2(2 - SP ) log + Sa = Ve 
because in each case the functions equated are symmetrical potentials, taking the same 
value on the axis of symmetry. 
By putting z=0 in the first and third of these we obtain two integrals, of great 
importance in the following analysis, 
4 INGE p 
I, (ose a ye Sa eee 
4 1 dk oil 1 
i (Susp lle ge penne) = AP log 5— Qe -ZPs 
There is no difficulty in generalising the above results, but those given are all that we 
shall require. 
(1) With a view to indicating the broad lines of the treatment: of the elastic 
problem given in the succeeding pages, a discussion on similar lines may be given here 
of a simple problem in potential, in which the attention is not distracted from the 
principles of the method by any complexity in the calculations, 
The problem is to find the flow from a source situated between two parallel planes 
z= th, under the condition that there is no flow across these planes. 
