438 
MR, A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
Cylindrical Shells. 
4. Before we can obtain the equations of motion or the potential energy, it will be 
necessary to ascertain the values of the first and second differential coefficients of the 
displacements with respect to r when r — a. We shall, therefore, proceed to calculate 
these quantities. 
Putting X, IX ; X', p,' for the values of {dcrjdr), [da^^Jdr ); {dra- ildr~), {d ^cry dr-) 
when r = a, we have 
Pt' zir {m + n) o-'g + {m — n) {a-\ + crh) 
= (m + n) a-Q (in — n) (a-^ + cr^) 
But from (2), 
whence 
q- j(^ _p + (m - tt) (X + ^)|/^' 
+ 2 |('^ + ^'0 ~ 
B' = A + A,h' + iAJC + . . 
A = (-/w + n) cTg + (m — n) (cr^ + cr^ 
Ai = (m + n) J^(m-n)(\-^ix) 
Ao = (m + J?) 
dcr. 
a 
d~(T 
dd 
(3) . 
(4) , 
(5) , 
+ (rn — n) (X' + /x') J 
where A, A^ do not contain any lower powers of h, than Id and h respectively, and 
Aj does not contain any negative powers of h. 
If It, v , IV be the component displacements of any point of the substance of the 
shell in the direction z, (f>, r, the equations connecting the displacements and strains 
are 
, did 
^ ^ dz 
(7 , 
<T 
1 uW 
r \d(p 
did 
dr 
-f- tv' 
, dv v' 1 dio' 
^ ^ — rr - 1 --— 
dr r r dcf) 
( 6 ). 
^ o 
d u/ d id 
dz^dr 
1 d;id dv' 
^ 3 = “ ZT “T 7‘ 
whence if 
E 
r d(f) 
in — n 
111 + n 
Iv = CTj + (To 
(O- 
