MATHEMATICAL THEORY OF STREAM-LINES. 
277 
In the particular case Z» = 0, this equation has two roots ; viz. 
y= 0, representing the axis OX, ancl 
y=k&, representing the oval of which L B in fig. 2 is a quadrant. 
That oval is the trace of the cylindrical surface of a solid which will disturb a uniform 
current in such a way as to produce the whole series of stream-lines ; and it is the only 
one of those lines which is closed and finite, all the others being infinite and having 
asymptotes. When the two foci coalesce into one, that oval becomes a circle. 
The component comparative velocities are as follows : 
d-\> k(pc — a ) , k[x + a) 
U dij [x— ay + tj' 2 ' {x-\-a) 9 -\-y 9 ’ 
dS ky ky 
V dx (x — a) 2 + y 9 ^~ (x-\- ffl) 2 + »/ 2 ’ 
In the previous paper already referred to, the parameter here denoted by k is 
denoted by f\ and the comparative velocities here denoted by u and v are denoted by 
~ and The origin O is taken midway between the foci for convenience. Should it be 
placed at unequal distances, let x=-\-a' for one focus, and —a" for the other ; then in 
the equations, a 1 is to be put for —a, and -f -a" for +«. 
Let l denote the half-length O L of the oval stream-line ; then by making u~Q, y=0, 
and X—I in the first of the equations (26), it is found that the following relation exists 
between the half-length l, the excentricity a, and the parameter k, 
r=cr— , 2ka=0 (26 a) 
Let y 0 be the greatest half-breadth O B of the oval stream-lines, then we have by 
equation (24), 
y 0 — 2&tan“ 1 - = 0 (26 b) 
Vo 
§ 6. Stream-line Surfaces of Revolution. — To obtain by the first method mentioned in 
§ 3 the equations of stream-line surfaces of revolution, the form of the function % is to 
be taken so as to represent a series of longitudinal planes cutting each other at equal 
angles in the axis of x. Hence we have the following expressions : 
(26) 
^ = tan 1 
£. d X 
dx 
= 0 ; 
d X -g . (/ X V ■ 
dy y~ + z l ’ dz y 2 + z~ , 
d^x d~x —2 ys 
drj 2 dz 2 ( y 2 + z 2 ) 2 ’ 
d q x z 2 —y 2 
dydz (y° + z 9 ) 2 
2 Q 2 
( 27 ) 
