166 
MR. W. M. HICKS ON THE STEADY MOTION AND 
r 
We will clioose the constants so that the two integrals are 
also 
_4u 2 —1 
du 8 n 
g dQ _ 4:7V 1 — 1 
du 871 
(P«+i P«-i) 
(Q«—1 Q«+i) 
1 
I 
1* 
±)SP„ 
dT„ 
du 
(n*-i) SQ 
(2) 
The value of xjj is now, putting in the value of p, viz., p=aS/(C—c) 
i/j — aq_ c ^o (A,3«+ B, Z T„) cos (nv-\-a) 
and clearly R, T belong to the same spaces as P, Q respectively, that is, lit to space 
outside, and T to space inside a tore. 
It is easy now to prove from the value of Q„, viz., 
v^ dv 
that 
T*=— : ^/c\ 0 cos nv^/(C—c)dv .(3) 
The R, T are all positive, except R 0 . 
2. Cyclic constant .—The cyclic constant of xjj is the flow along any closed curve 
threading the tore once. We know that this must be independent of the form of the 
curve. To find it, choose the curve to be u=u a constant; the flow along this is 
then 
f 2ir I b4r du dn 7 [ 2n 1 b4r 7 
Jo pl>udndv € t ' = Jo p^u C V 
the velocity in the aperture being in the positive direction. Consider first the 
general term in R,*; the flow due to this is 
A„f 2 * 
«S 
cos nvdv 
2v/(C -c) 
=^£{(» a -i)S p , v / (C- cos v) cos 
C ) 
dv 
— _^x P —“R 0 
_ a 2 n u 2 
A hN / 2, 
d Q 
d P, 
die 
= ^S( p ,^-Q,,^) = -;V2A,. [T.F. 3i.fi 
1 
