452 
MR. A. B. BASSET 0^^ THE EXTEHSIOH AXD FLEXURE OF 
wliich require that 
u — Ua, 
■w — 2 : 
(lY 
d(j) 
{«)> 
where U and V are functions of (f) alone. 
In this case the potential energy reduces to the second line alone, and from (15), 
(16), and (17) we obtain 
and from (24) 
W = 
II 
0 
1 
/ d~w 
a- 
1 
1 
9 , 
/ d.^ w 
ydzd(f> 
m 1 
(dho 
(m + n) ' 
~j“ w 
(h 
dz 
J 
(50) 
dho 
dzd(f) 
dv 
dz 
(51), 
which agrees with the expression obtained by Lord Rayleigh."^ 
Also from (14) 
(52). 
The values of the stresses Mj, M 3 may be obtained either from (43) and (44), or 
from ( 12 ) combined with (15), (16), (17), and (18) by introducing the conditions of 
inextensibility ; and the values of Tj, T 3 might be calculated by taking the variation 
subject to the conditions of inextensibility, and using indeterminate multipliers. 
Tliis process would not, however, be of much assistance, inasmuch as it would intro¬ 
duce two undetermined quantities into the values of T^, Tg, Avhich depend upon the 
boundary conditions; whereas in this case the values of T-, Tg can be obtained directly 
from the first and third of ( 11 ) combined with (49). The values of N^, Ng are given 
by the fourth and fifth of (11) combined with (50) and (52). 
11 . We must lastly consider the boundary conditions. 
Equations (43) and (44) determine the stresses on the line elements adcj) and dz 
respectively, which are produced by the action of contiguous portions of the shell; 
and it might at first sight appear, as was supposed by PoissON,t that when a shell 
* ‘ Roy. Soc. Proc.,’ vol. 45, p. IIG. 
t ‘ Paris, Acad, des Sciences, Memoires,’ 1829, vol. 8, p. 357. 
