CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
445 
We must, in the next place, calculate SH. We have 
z= 11 (P' Su' + U' Sv) {d + li) dll' (i(/) + 11 (Q' + U' Sm') dll dz 
+1 NgSw/a cZ</) + j c?z.(27). 
From the way in which SC has been calculated, we see from (8), (9), and (12), that 
r S«' (1+ h'/a) dh' = T, Sm + G, + 1 A»P 
= T, 8 m - G, + I nh^En ~ ■ 
Treating the other terms in a similar way, we find 
8a = I {t, 8« + M, 8m + N, 8» - g/^ + a - 8 m) 
+ |{m, S« + T,8m + N. 8» + - Sm) - H, 
2nh^ ^ dSK , ,™ c^SK] , 
+ 6 ^• ■ (28). 
Lastly, since the shell is supposed to be so thin that X, Y, Z, may be treated as 
constants during the integration with respect to li, 
W = p III' (X Su' + Y Sv' + Z Bw') (1 + h'/a) dh' c/S 
= 2p h |j'(X Sw + Y Sv + Z Sit;) dS 
+ ip h^E f[ + l - Z (8X + 8^)} c^S 
-t-|[{^x' + !(f|-^") + 2E8K},iS . . . (29). 
9. We have now obtained all the materials for the complete solution of the problem, 
and we shall proceed to work out the variation in the ordmary way. 
Let us denote the four terms of the expression for W given in (24), when 
integrated over a curvilinear rectangle bounded by four lines of curvature, by W^, Wg, 
Wg, W4. Then 
8W^ = 4?i/i I (^Scr^ + ^^ScTg fi- ^ ctSct) a dz dcj). 
