184 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
This is the same as a or (—) rate of variation of \’ in the direction perpen- | 
dicular to the force at its point of application. 
Similarly p= be ~) rate of variation of ¢’ in direction of force. 
Hence for ; et ES _ dy’ 
ae py doy Waele, | 
pee ae , and for Q,, §= 1a 
Ail | py eo 
__ __ lag" 
ve: dp, ar p doy 
We shall take separately the extensional and flexural parts of the solution. 
Also in the following u, v are the displacements along radius vector and transverse. 
I. Extensional terms. 
The following solutions occur. 
(i) wa (F582 F* hog p eae 
w=(3—a) zp7' cos w 
(ii) Same as (i) with cos w changed into sin o, and sin » into — cos w 
(iii) When m>1, 
eee | 8m —(m— 2) (a +1) pont 
Hf 4(m — 1) 
={ S(m — 2) — ma. soils) ie ag 
4(m — 1) erat 
w = (3 — a)zp-™ cos mw 
é = eT map? ; cos Mw 
3- : 
+— 3 mp sin mw 
(iv) Same as (iii) with cos mw changed into sin mw, and sin mw into — cos mw. 
(v) u=p-™ cos mw \ 
v=p-™*sin mw 
(vi) w=p-™ sin mo } ' ae 
v=—p-™} cos mw 
For the force with components P,, 2,, Z,, the coefiicients of the above solutions 
are the following, in each case divided by 327uh. 
g) My 
i) P, cos, —Q, sin wo, =X 
1 1 1 1 1 : 
ii) P, sinw, +, cos o,=Y 
1 1 1 1 1 
(iti) p,”~) cos Mw,P, — py" sin mo,Q, 
(iv) p,”-* sin mo,P, +p,” cos mw,Q, 
—8m+(m+2)(at]1 i a J 
(Vv) { m in - ne Me +) + (3 — a) (42,7 — 1h2)mp." ; cos mu,P, i 
8(m +2 +1 pet: 
+ | se m+] _ (3 = a) (42,2 a 122)mp, 1 \ an mu,Q, 
+ (a—3)p,""%, cos mw,Z, 
(vi) Same as (v) with cos mw, changed into sin mw, , and sin mw, into — cos mu, . 
