AND THE ACTION OF TWO VORTICES IN A PERFECT FLUID. 
511 
sponding axes of coordinates. Tire difference in the work will come in when we 
substitute for the quantities involved their expressions in terms of the time; if we 
denote corresponding quantities in this case by affixing dashes to the symbols, 
denoting them in the previous one, we easily find 
f-- 
h': 
r= 
m'= 
n — 
— c sin \e-\-v sin e.t 
— c cos ^e— v(l— cos e)t 
cos e cos 0' 
- sin 0' 
— sin e cos 0' 
■'i 
> 
velocity perpendicular to the plane of the vortex A B 
=y cos e+a sin e 
velocity along the radius vector = a cos e cos O'—ft sin 6'—y sin e cos O'. 
It will be seen that we can get the expressions for the quantities denoted by the 
accented letters from those for the quantities denoted by the unaccented letters by 
writing 27r—e instead of e, hence the value of f3' will be got by writing in the 
expression for j3 / (equation (30)) 27 t— e instead of e, and interchanging a and a. 
Hence the value of {3' when t= oo 
mV 0 
—— cos * 
VC 6 
mcfiaf 
v& 
cos 2 
Now this being negative shows that the parts of the vortex ring A B where cos O' 
is positive are tilted backwards, now cos 0' is positive in the upper half of the vortex 
ring A B, therefore the direction of motion of the vortex A B, which is perpendicular to 
the plane of the vortex, is turned away from the direction of motion of the vortex C D 
through an angle whose circular measure is rad cos 2 ^e/vc s ; but since in the case we are 
considering a=d, m=m, this angle is the same as that through which the path of 
the vortex C D is turned towards the path of the vortex A B. We may express the 
results we have obtained by saying that the direction of motion of the vortex which 
is in front when the vortices are nearest together, is bent towards the direction of 
motion of the one which is behind, that the direction of motion of the latter is bent 
through an equal amount in the same direction, and that the amount of this bending 
is rad cos 2 -ere/vc 8 . 
Let us now consider the effect the collision has on the size of the vortices. 
The equation giving the increase in radius is 
dot 
-^=the part independent of 0 in the expression for the velocity along the radius 
vector of CD. 
MDCCCLXXXII. 3 u 
