SMALL VIBRATIONS OF A HOLLOW VORTEX. 
167 
which is independent of u as it ought to be. The corresponding terms in sin nv 
evidently disappear. Similarly the terms in T n would produce 
Hence the cyclic constant is 
7T\/2 
a 
So A 
n 
( 4 ) 
3. In the paper on Toroidal Functions several examples were given of the deter¬ 
mination of the potential function (f> when <£ is given over a tore; but when the 
variation of <£ along the normal to the surface is given, the determination of the 
co-efficients becomes more difficult, and one case only, for the motion of a tore 
perpendicular to its plane, was given. It will be w r ell, therefore, to consider here 
the general theory for this class of surface conditions. The co-efficients are to be 
determined from the fact that <f> is (1) finite in the space considered, and at infinity, 
and (2) d(f)/dn has a given value over the surface of a tore u. Here I consider only 
the case where the motion is symmetrical about the axis, and therefore the normal 
velocity given by a function of v only, say f(v). Condition (l) is satisfied by space 
outside the tore by taking only functions P„. We put then 
<£= ^/(C — c)2 0 (A„ cos nv-\- B„ sin nv) P„ 
and determine A it , B„ from the equation 
/(*)=- 
b(f> du 
bu dn 
when u—u', for all values of v. 
Consider separately the terms in cos nv and sin nv. For the cosines we have 
b(f) 
bu 2y/ (C 
For shortness write — = P'. Then 
^ 0 A W jsP, t +2(C— c)^|cos nv 
b(f> 1 
bu 2^/(0—c) 
But 
Therefore 
du 
[(SP 0 +2CP / 0 )A 0 -A 1 P' 1 -A 0 P' 0 cos v* + {(SP„+2CP'„)A ;1 
— A n+ jP'^+j— A n _J ?' n _ x } cos nv] 
SP„+2CP / „=P' W+1 +P;_ 1 [T.F., p. 646] 
SP 0 +2CP' 0 =2F 1 
bu 
■ 2x /(Q - C ) [( 2Aq “ A i) p/ i ~ A 0 P' 0 cos?; + {(A„—A* +1 )P'* + i — (A*_j — A^P'*^ } cos nv] 
* [April, 1884.—This term was omitted in the paper as read. It has necessitated slight alterations 
in some of the results then given.] 
