AND THE ACTION OF TWO VORTICES IN A PERFECT FLUID. 
515 
Thus the coefficients of cos nt and sin nt in this expression for ot. 2 will involve 
integrals of the type 
cos ntdt 
S 
2P + 1 
{cd + tf) 2 
I have not succeeded in evaluating this integral ; it is evident however that the 
more important part of these integrals will be produced during the time the vortices 
are nearly at their minimum distance apart. During the time they are far apart they 
will not contribute anything appreciable to this integral, so that soon after the vortex 
rings have passed their minimum distance the equation may without sensible error be 
written 
t, cos nt „ sin nt 
a 3 =P-— 
* n n 
where P and Q are constants and 
f + oo 
'= x(t) 
* —00 
cos nt.dt 
Q= x(<) 
sin nt.dt 
There will be a similar expression for ft. z . Thus the vortex rings are thrown by the 
collision into a state of vibration about their circular form. 
We can find the action of two unequal vortices on each other by means of work of a 
very similar character to that just given. The only difference is that instead of the 
former values for f and h we must substitute the values 
f— — c sin a— v sin e.t 
h=c cos a+ (v cos e—w)t 
where v is the velocity of the vortex which is in front when they are nearest together, 
w the velocity of the one in the rear; a is the angle between the line joining their 
centres when they are nearest together, and the direction of motion of the vortex in 
the rear, /3 is the angle between this line and the direction of motion of the vortex in 
front, e is the angle between the direction of motion of the vortices; a and /3 are given 
by the equations 
W COS CL=V COS ft 
« + / 3 =€ 
I shall not trouble the reader with the expressions for the velocities perpendicular 
to the plane of either vortex and along the radius vector, but confine myself to quoting 
the most important consequences to be got from these expressions. 
I find that after the collision the direction of motion of C D (the vortex which is in 
