172 
MR. W. M. HICKS ON THE STEADY MOTION AND 
Q 0 =7r{l+iF+^+^ 6 + • • 
Qi = i 7r {i+t^ 2 +M^ 4 + • • -}^ 
Q ,r 
Q 3 =f(i+^ 2 + • • .)& f 
Q3=#‘+ ' ' ' 
Qi=o+ . • - 
Q 5 =0-f . . . 
(10) 
r 
Ro=-{iL-H-i(L+l)^+rk(L-i)^+5ir(L-|)F+ . . 
E 1 =i{l-f(L-i)F+A(L+i)^+ T f8(L-23)F+ . . .}^ § 
E 2 ={l-P 2 -i|(L-|)^+M(L+*)F+ . • .}k-* 
R 3 =|{1-1F-M^-M(3L-^)P+ . . .}*f* 
H — 8 1 .37.2. . 21 7.4 ,. 1 0 5 7,6 I \ 7.-f 
-•Aj.—5 l 1 4A 64 A 256^ “T • • - R J 
(11) 
T 0 —^7r(l+i^'+ ^4 ^ 4 +T56 : ^ C + • • -)^ * 
T 1 =y(l-i^-A^+ • ■.)» 
T 2 = ] | r ( 1 -P 3 + ., -W 
t 3 =M^«+ ... 
( 12 ) 
Section II.— Motion about a rigid tore which moves perpendicularly to its plane. 
As the motion of a tore throws some light on the analogous problem of the uniform 
translation of a vortex ring, and as the functions required in its discussion will be 
needed in investigating the latter, it will be useful to give a short treatment of the 
question, especially as the motion can be determined for any size of tore, whereas our 
methods, in the case of hollow vortices, will only apply when the cross section of the 
hollow is not large compared with the aperture. The stream function is necessary for 
the cyclic motion, and it will therefore be convenient to take the stream function also 
for the motion of translation. 
5. Stream function for cyclic motion. —If the tore he given by u—u' the conditions 
which xp must satisfy are that it must he finite for space outside the tore, and be con¬ 
stant for all values of v when u—u. Hence xp must be expansible in the form 
l, ' = v /(C-c) 
cos nv 
