650 
MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
we can easily find the limits of error produced by neglecting terms after a given one. 
Thus suppose all after P,. be neglected, then § 10 
1 1 1 1 
P« C + S P»-i Glfi-i 
whence it follows that 
v 1 ov i 
■+! i/x B ^ 1 _^ (x _ k y Tr 
Similarly it may be shown that, r being odd 
r 
Sr +1 (-)-/Pn> 
< 
y 
1 - 3 F- 2 F 5 
: i 
1+hf 
(1-k'f 
’Pr 
2k' 
f/l+/v ' 2 \ 2 
y i i 
1 + k' 2 
' 1 - 7 A/P, 
For the two cases of Jc'= sin 3° and Jc'= sin 6° (corresponding very nearly to R—lOr 
and 5r respectively) the ratios are '5171 and '2656. 
The ratio of the velocity of the fluid at the centre of a tore to that of the tore itself 
when it moves without cyclic motion, parallel to its axis, is easily found. The point is 
given by u= 0 v—tt which makes P n = 7 r. The velocity of the fluid =y ~ 
IT 
therefore ratio = — 16XT( — )" n -h n 
In the table below are given the values in two cases of a , A l5 A 2 , T' (the effective 
mass of the fluid measured in terms of the fluid displaced), and V', the ratio of the 
velocity at the centre, to that at an infinite distance when the tore is held at rest in 
the stream. 
k’ 
CL 
Aj 
r 
V' 
sin 3° 
— •00645 
-•00216 
-■ooooo 
•99995 
1 03456 
sin 6° 
-■01868 
-•00871 
-•00007 
1-09449 
1-13712 
Suppose the tore held in a uniform field of electric force parallel to its axis, 
potential of the field is 
= ^ ~*y(C—c)%nQ,„ sin nv 
The 
