AND THE ACTION OE TWO VORTICES IN A PERFECT FLUID. 
49 9 
Since the vortex rings always consists of the same particles if F(p, 6, (ft) — 0 he an 
equation to its surface, we must have 
dF .dF T dF 
dt^~ dp + d<f>' ~~ 
when the differential coefficients are partial. 
It is the velocity in the direction in which p is measured, 4/' the angular velocity 
round the axis of z, and <E> the angular velocity of C P round a normal to the plane 
containing the axis of z and 0 C. 
Applying this equation to the first of the equations to the core, we get 
'ta.a cos n\jf—’R—^na n sin -m/UP-fie cos </>.<!>=0 
or 
R=^a ;; cos n\\s — %na n sin cos (£.<& 
and in a similar way we find • 
w=%-\-'Z(/3 n cos mjt—nfin sin nxfj'F) — e sm <£.<h 
where w is the velocity of a point on the surface of the core parallel to the axis of z. 
Now ’F is zero when a n and fi H are both zero, and it will be small in this case since 
a n and fi n are both small, hence neglecting the squares of small quantities, these 
equations become 
R=2a, rt cos nxfj-\-e cos </>.<!>.(7) 
.( 8 ) 
w=%-\-'Zfi n cos n\p—e sin cp.® . 
But It = u cos ifj+v sin \fj. 
Substituting for u and v the values given in equations (4) and (5), we get 
R=-|wa{£A 1 +i2$» cos n\Jj((n— 1)A S+1 —(n+ lJA^)} 
Since the A’s are multiplied by the small quantities £, a n , fi n , we may suppose since 
we neglect quantities of the order a* 2 that the A’s are found on the supposition that 
a n and fi n are zero, or that the A’s are the same as if the vortex was undisturbed. 
Let us denote the value of the A’s for the undisturbed vortex by German letters. 
Equating the two expressions for R and putting 
we get 
cos n\p=e cos <f> 
l mat{(fi u cos mfj+e cos cos mjt((n— (w.+ l)^, J _ 1 )} 
= ta. n cos nxjj-i~e cos <£.<!> 
equating the coefficients of cos <f> and cos nxjt we get 
. 
^mcifin { %ti +i( (n — 1 )&» + 1 — (n +1 )&L-i)} = ot n . 
(9) 
( 10 ) 
