The Torsion ProUem for Bodies of Bevolutiori. 167 
Substituting this value of in the expression for ^'g, we get 
where . . . (21) 
Substituting the value of + (a + r)^ from the last equation, we have 
/e=^.(^)t|(2-F)K-2E} - - (22) 
Instead of finding the stream-function il^'g by means of equations 9 
and 22, we shall rather find it by making direct use of the stream-function 
for a point source. 
From equation 17 we have for a point source at the origin 
2z^ + 
= C. Z . : —.. 
Hence the stream-function at the point (z, .t.,, o, o, o) due to an element 
^x'~^, Sx'j^, Sx'~ at the point (0, Xo, x'.^, x^, x'~) will be given by (as in the 
case of (/), x- may, without the loss of generality, be taken zero) : 
25'2 + 3 Uxn - x'^) + 2 ^ '2 _|. ^,'_2 1 ^-^'^ ^/y^ 
CUJ Q.Z '- '- '- • 
{z^ + ('^o - ' + + =^7 + •^'5' }^ 
Hence, for a circular plate of radium ctx we shall have 
{x,f + X'./ +x',^ 
By applying the four-dimensional polar transformation the integral is 
transformed into 
^ ^ A f* f I* '^'A^^' ^ ^^^^ 
^ ^ J J J J {^2 ^ ^ _ 2a)' COS ai}t 
a — 0 a\ — o 02 = 0 03 = 
— TTC^z J J -\- r^ — 2arcosa}t 
a = o ai—o 
where r = x.-, 
For a ring source of radius a the stream-function will therefore be 
sin^ a^{2z'^ + 3 (a-2 -|- ,-2) _ 6ar cos a^} . da^ 
{z^ + a'^ -f — 2(ircosa^}t 
sin^ ai da^ 
bar 
I /' SlU" ai aa, 
/ 
sin'^ «! cos a, da^ ] 
[^^ + . . . - 2«/- cos i 
