AND THE ACTION OF TWO VOETTCES IN A PEEFECT FLUID. 
513 
We can now find the force resultant of the impulse after collision. The impulse for 
a vortex ring with a very fine core equals the strength multiplied by the area. Let co 
be the small angle through which the direction of motion of the vortices is deflected ; 
let Sa be the alteration in the radius of either, for the vortex ring in front Sa will be 
positive, for the one in the rear it will he negative but of equal numerical value. 
The resolved part of the force resultant of the impulse along the line bisecting their 
original direction of motion after collision 
—7rm(a-{-Sa) 2 cos — co) +7 rm(a — §a) 2 cos ^~+g> 
= 2irma 2 cos \e 
the same as before collision. 
The Component perpendicular to the bisector of the angle between their directions 
of motion 
= 7 rm(a+Sa) 3 sin (Je— co) — nm(a—Sa) 2 sin (^e-ho)) 
=7rm(4aSa sin 2 coa 2 cos -Je) 
Substituting for Sa and oj the values ma 8 cos 3 \e/ 2 vc 8 sin \e, and ma 2 cos 2 ^e/vc 8 
respectively, we find that the component of the impulse perpendicular to the bisector 
of the angle between the directions of motion vanishes, as it did before the collision; 
hence we see that the force resultant of the impulse is not altered by the collision, a 
result which we know is true. 
We pass on to consider the terms oq and /3 £ . We know that 
d0tr 
^ = coefficient of cos 20 in the expression for the velocity along the radius vector of 
the vortex ring C D. 
Now the vortex ring C D itself, as we see from equation (17), contributes to the 
expression for the velocity along its radius vector the term 
2 m' 2ft 
— cos 20. , log —. 0 , 
ft 3 & 6 
The vortex ring A B contributes as we see from equation (29) the term 
say 
Thus 
Now 
d# 3 
dt 
m'cc'^a 
- cos 2g.j-r---- . - ■ r - i (Ff 3 +CTy + H7-f K') 
cos 2 Of it) 
da 
ml 
2 ft 
dt 
~ cd e • A +/(0 
the coefficient of cos 2 0 in the expression for the velocity perpendicular to 
the plane of the vortex C D. 
3 u 2 
