CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
447 
8^ & # = - i {{dF + &K + E (a + 8w)} dz dj, 
= --Jz^Sudf- i [(Ed? 3. - a 8» + d? ‘^) dz 
and 
l[jpSpcZ 2 c?(^ = ^.\\p 
dSv dSu d“Biv \ y ,, 
a — -—— — 2a ) dz d(p 
d(f) dzdcpj 
dz dcf) 
dz d(j). 
The last integral can be evaluated in two different ways, according as we integrate, 
first, with respect to and secondly, with respect to z; or first with respect to z, 
and secondly with respect to (j). The proper way to deal with such a term is, to 
evaluate the integral in both wmys, and then multiply the two values by /3 and 1-/3 
and add, where ^ is a quantity which must be determined from the conditions of the 
problem in hand. We shall thus find that the value of /3 is ^ ; we therefore obtain 
lb dz d4. = i [(y Bv + I Sw - p d4 
^ ^ Sw) dz d(f) 
+ ^Jl(# “ S dzc 4 -J 
. . (35). 
Collecting all the terms together from (33), (34), and (35), we finally obtain 
8 W, = M + (5 + i f - B.)}ad^ 
+»\{-L t) 
-(P /dSw ^ \ d8w I , 
-!.-i?+!(3 + a» + wU 
dz^ a \ d(j)' 
d^p 
dzd(j) 
8iu 
a dz d(f) (36). 
