The Torsion Problem for Bodies of Revolution. 169 
= (8^)(K-E)-^.K 
_ (8 — 5P ) K -'(8 - /g^) E 
Substituting in the value for Iq, we have 
-r _^ 3 ( 8 -^5^^)K - (8 - ^'^)E 
We therefore find for the stream-function of a ring-source : 
Sirca'z fro^,o/ "..T , rx s (8-5^^)K-(8-7^-^)E-]) 
^irca^z f^^ X.-. . (8-5A;)K-(8-fc2)E| 
Putting in the value of I^, we get 
^'6-4.c. -|3-.(-^)^j/\(^,r)K -f/,(^,r)El - - (23) 
where {z, r) = F (2 - k^) [2z'^ + 3 (a + r)^] - Ur (8 — 5A;2) 
/o r) =- 4ar (8 - F) - 2^3 [2^2 + 3 (ti + r)2] . 
Differentiating 22 with respect to r and z respectively, and substituting 
1 .Sic Sk ^E . , ^, 
tne values 01 ^ , — , — ~, given by the equations: 
or dz dk ok 
Sk _ z"^ a'' — r2 
— ^k' . 
Sr ^ ' ar^ 
dk .70 ^ 
Sz ^ ' ar 
^ - ^ -'E - (l-k^)K - 
Sk ~k{\-k'')\ ^ ^ \ 
we get for the components of velocity due to a ring-source the expressions : 
where A, . + o 
Ao = h^^ . ^ ' ^ +1(1 + 1^'% 
4ar 
k' being the complementary modulus 
= - 
r 
