102 Proceedings of the Poyal Society of Edinburgh, 
and component velocities 
respectively. 
[Sess. 
and 
(Vo) 
27T 
_1 
+ 00 
- V 
y<i - Vo 
+ GO, 
V 
Vp - Vo 
-r ^ “-O 
-00 ^ ® 
-^of + iVi- 
■t/of 
-00 ^ P 
*o)^ + (2/p- 
+o)^' 
27T 
7) 
+ 00 
— 'y 
x,-x^ 
■ y' 
— Xq 
-^o)^ + (2/?- 
-y.>f 
7^(+- 
■^(y + {yp- 
-y,r 
27T , 
1 O / 
-V' 
Vq - Vo 
+ 00 
Vp - ?/o' 
+ 
(x„ 
-00 ' ® 
— XqY + [t/q 
-y,y 
7^(+- 
-y,'r 
+ 00 
— v' 
Xq - Xq 
+ 00 
V 
0 
|isj» 
1 
^ (^3 
-00 ^ 2 
— XqY+ {Uq ■ 
-Vaf 
- ^af + (%. 
- y^f 
(15) 
(16) 
(17) 
(18) 
Inserting the values for , Xq , , etc., in these equations, and 
neglecting terms of higher order than the first in the ^’s and /;’s, we obtain 
+ y“o=2/l 
5 + 
2g- 1 
% 
^[^2(j_i)2 + /,2p +2.% 
,,, V + V - - 2/ - 
^ [^^(2 - 4 )'^ + ^ \n'i - hf + ^>^] 
^ **0 ~ fc 
+ 00 72 / 
P(g-i)ii-;>^ 
-« j,2Z2 
+ “ _ 1)2 _ ;j2 
%n 
00 
+00 
(19) 
1 
-00 
nq- hf-h^ 
(3g-l)^, 
( 20 ) 
2:r A 
4“" + + 
+ '^0 
+co,yi _ 
Z.rpri, + i^2 + ;,2i2 ^ z. 
^[<2(^ + 1)2 + A2] 
+ 00 
z+ 
4 f“^/2(j, + i)2 + /j2]2 ^[i2(^ + |)2+;,2]2 
.00 L ■ z,/ J _oo ^2^2 _ 
(p + 4)+-A^ 4(2/>+l) 
1X272 J. 7.212 “Z. 
-CO 
+ GO 
[(p + J)2p + ft2]2 [(p + 1)2^2 + /,2J: 
( 21 ) 
7?p(2j>+l) 
f; 
+ » <;2(j, + J))2_;j2 
^P^(P + 4>+/<^? 
+ 00 ,^3 
7+.-1 
• ( 22 ) 
00 g 2^2 J 
The whole system of vortices when in their position of steady motion 
moves forward with a velocity 
h 
:^Z2 (jP + 1)2 + /72 
. (23) 
