AND THE ACTION OF TWO VORTICES IN A PERFECT FLUID. 
495 
Now from the equation to the axis of the vortex we have 
dx'=d(p cos 9) — d9%{—na u sin nO cos 9—a sin 9—a u cos nO sin 6) 
dy'=d(p' sin 9)=d9'£( — noc /l sin nd cos 9-{-a cos 9-\-oc a cos nO cos 6) 
dz!— — d9%(n/3 n sin nO) 
neglecting a 2 and ft 3 
and 
ds / =d9(a~\-ta /i cos nd) 
<M 
ds f 
ds’ 
dz' 
dd 
■ — sin 9 — %ncL tl cos 9 sin n9 
a 
: cos 9—~ Xna 4 sin 9 sin n9 
CO 
- 1\np n sin n9 
a 
If p, 5 +£ be the cylindrical coordinates of the point x, y, z 
r 3 =p 3 -f-p /3 -{-(£—5-ft cos n9y—2pp cos (9—xjj) 
say 
r 3 =p 3 -{-p /3 -|-£ /3 — 2pp cos (9 — xp) 
Let 
{p* + ^+ r-2 pp' cos (^_^.)}*= C 0+ C 1 cos i e -^)+ • ■ • C« COS n(0-4,) 
where the C’s are functions of p, p, and £'. 
Since p and £' are functions of 9, C 0 , C v . . . C« will be functions of 9, but since 9 
only enters into p and £ in the form a n cos n9, ft cos n9, the terms in the C’s which 
involve 9 will be multiplied by a u or ft and so will be small. 
If 
KV+f = v®U¥ ,=A o +Al eos • • • A « cos n 
then 
C m =A m J r %a u cos n9~j~ 
dec 
fJA 
+2 £2ft cos n9~l 
+ terms of higher dimensions in a,* and ft. 
We shall only require those expressions for a point nearly in the plane of the vortex 
where £ is very small, so that in this case 
Cm—A cos nd 
3 s 
rtAv, 
dec 
MDCCCLXXXII. 
