CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
443 
—((Ti + 0-3 + A/ 2 n)® -i- j (\ -j- /j. -j- A^/ 2 w)'^ 
27),3 1 
-i-j/i^(crI -i- 0 - 2 -j-A/ 2 n) (\ -j-/i' -j- A 2 / 2 n) -^(cr^ -{- 0-3 + A/ 2 ?i)(\+ ^ + Aj2n)^ ( 20 ) 
in which in the last three terms we may omit the A’s since they are multiplied by h^. 
Again 
(1 “i" h'la) = or^cTg + X/rA® + ¥ ^3 d" “h (^*^3 + iacr 2 ) A ^/a + . 
whence 
2n[ cr^Vp/ (1 + Aja) dli = 4,n]icr-^cr2 + ^nJdXfx + fill'd (XVp + /xV^) 
a' — A 
A-'il ll ^ 
+ ^^(Xa-3 + ^cr2) (21). 
Also 
( 0 - 1 ' 4- 0 - 2 ) 0 - 3 ' (1 4- A/a) = ( 0-2 4- 0 - 3 ) 0-3 4- A^ (X 4 - ^) -yf 4- lA^ (X' 4 - p.') m 
d(7.. 
_ dr 
4 - \ A ^{( t ^ 4 - 0-3) 4- ~ |(^ 4- /^) o's + (^1 4- 0 - 2 ) ^ ^ 
whence 
rh 
dr 
2 n{ {<71 4" o's (1 “h Aja) dli = 4,nli ( 0-2 4" o‘ 2 ) |-^ (‘^1 “t“ ^ 3 ) 
— |7l/<®E(X + p.)2 —|?l/i3E(o-2 + 0-2)(X'4-/x')— '-yyE(X + p,)(o-2 4- 0’2) (22). 
Lastly 
rA 2??//^ 
■|n zTTg''^ (1 + ^^V«) dA = nhrn^ 4- i nAjA 4* —^sP (23)- 
J —h oCC 
Substituting from (, 20 ), ( 21 ), ( 22 ), (23) in (19), it will be found that the term Ah, 
which is (or at any rate may be) proportional to A, disappears ; and thus the value 
of the potential energy per unit of the area of the middle surface is 
W — 2nh [ 0 - 2 ^ 4" 4" E ( 0-2 4“ d“ i 
+ I nA {X3 4 - 4 - E (X + ^)2 4 - I p2] 
4- f nA (^X' 4“ 4~ ^ 
7iA 
-4- A 
I 3 
(aX+i$/x+lr^p).(24), 
m which cr is written for ra-g, the sufBx being no longer required. 
This is the expression for the potential energy as far as the term involving A. The 
first term depends solely upon the extension of the middle surface; the second term 
depends principally upon the quantities by which the bending is specified, and the 
3 L 2 
