CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 155 
Case (ii.). Eo = ; we find, using (13), 
and therefore is a solution of (D^ + Z.^ = 0, which is not at the same time a 
solution of 
(D2 + ^■•^)Z, = 0. 
The possible values of Z^ are 2 cos Jcz, z sin kz. 
Hence the possible sets of product functions satisfying the equation 
are as follows :— 
// = A cos {kz + a) Ij {kr) -) 
B cos {kz + /3) (kr) 
C cos {kz + y) {kr) 
D cos {kz + S) rK,;, {kr) . 
Ez cos {kz + e) Ij {kr) ; 
F2cos(^'2+ ' 
(14). 
§5. Solution under given conditions of Surface-loc.iding ; the first problem. 
Let us now consider first the case of a circular cylinder under the following system 
of stress : 
rr/g — a given even function of z{= f{z) ) over the curved surface r = o , 
rzjp = a given odd function of z { = xfj {z) ) over the curved surface r — cc, 
zz = 0 over the plane ends 2 = dc • 
Since dujdz, div/dr are both solutions of (8) we may have them composed of a series 
of terms as follows : 
Aj cos {kz + a^) li{k}-) + Cj^ cos {kz + yfi 
-{- EjX cos {kz T]^ {kr) i 
(15). 
(hu 
dr 
j Ao cos {kz -f- ttg) Ij^(/^’/’) -{- do cos {kz -fi yj) 
+ E,j2 cos {kz + €0) Ii {kr) 
. . (16). 
No K-functions have been introduced in this case, as they lead to infinite terms at 
the axis. 
