50 7 
And the action of two vortices in a perfect fluid. 
In this expression for a we have only retained the highest terms in each coefficient. 
If ft be the velocity parallel to the axis of y', we have 
/ 3 =- 
3 m'a'hog' sin 6 
¥ (p /3 +r 3 ) f 
m'a'-ct 
' . n 5ah sin B cos 0(h sin e + /cos e) n n ^ 
y 7 0N *, h sm 0 - 77—77 - \-a sm tf cos 6 sin e 
(/-+^)’\ p+h* ) 
The velocity perpendicular to the plane of the vortex C D 
/ AT 
m a * 
2 (/ 3 +a 8 j 
—y cos e —a sm e 
:{( 2 cos e —3 fh sin e} 
+ cos 6. 
m'a^ci 
(/ 3 +A 2 ) 
/ o 
m a*ar 
.{cos 2e(/ 5 —4/7V)+J sin 2e(7f 2 h—3lv i )} 
(24) 
+ cos f(/i sin e+/cos e)(/i sin 2 e-f/cos 2 e+ 2 /’)- 
3 S 
/ 3 (A sin e +/ cos e) 2 ' 
/ 3 - f /^ 3 
The velocity along the radius vector of the vortex ring C D due to the vortex A B 
=al-\-ftm-\-yn 
— cl cos e cos 6—(3 sin 9-\-y sin e cos 0 
or substituting for «, /3, y their values 
— 3 
o m'a' 2 a f 17 , (3A 3 —7A/ 2 ) . (/ B —4/A 3 ) 
5 C °S 2e d - Sill 2e y2_j_^2 
1 
/Oyj ^ ft 
+i cos 61 (7 » 7: A 3)> {( 2fea ~/ a ) sin e + 3 .A cos e) 
h (25) 
/ /9 
m & 
4_3 cos 26» _3/ i II cos 2e ffl 3 -7/ a /Q • 2 C/*" 4 / 7 ^)]. 
' 4< COb 2 COb y2_j_^2 I &m 73 0.7,2 [ 
/ 2 + /^ 2 J 
These expressions will enable us to find the effect of one vortex on another. We 
have, for example, expressed the velocity perpendicular to the plane of C D due to the 
vortex ring A B in the form A+B cos #+C cos 20+ . . . , in Problem I. we expressed 
the velocity in the same direction due to the vortex C D itself in the same form, hence 
the total velocity perpendicular to C D can be expressed in this form, but by formula 
( 8 ) the velocity perpendicular to the plane of C D is 
cos nO 
