MATHEMATICAL THEORY OF STREAM-LINES. 
283 
u, equation (39), make x=l, y= 0, u — 0 ; then we have the following equation, 
0 = l-j&/— 
\l—a l + a 
■Id 
1 
1 
l — a! l + a 1 
ii b 
which by ordinary reductions gives the following biquadratic equation : 
0 = Z 4 — l 3 (a 1 — a") — l 2 (a 2 + a! a" + 2 Jca + k\a! + a")) 
+ l(a 2 — 2 ka)(a! — a") + cda'a" -j- 2 kaa'a" 
~{-k'a 2 (a' J r a'). 
(40) 
Of the four roots of this equation, the two greatest, positive and negative respectively, 
belong to the cylindric cycno'id; and the sum of their arithmetical values is its length. 
The two least, positive and negative, belong to an internal stream-line, which is also a 
closed curve. It passes outside and near to the inner foci, and inside the outer foci, 
and it is foreign to the purpose of the present investigation. 
When the two outer foci are equidistant from the inner foci (that is, when a"— a'), 
equation (40) becomes a quadratic equation in l 2 ; that is to say, we have 
0=Z 4 — l 2 (a 2 +a' 2 +2,ka+ZHa!)-{-a 2 a' 2 -\-2Jcaa! 2 -{-2k l a'a? (40 a) 
For brevity’s sake, let 
a 2 -\-2ka=. A. 2 , « ,2 -|-2&V =X' 2 , 
being in fact, according to equation (26 a), the values of l 2 for two bifocal oval neoi'ds, 
with the respective excentricities a and a!, and parameters Jc and k'. Then the solution 
of equation (40 a) is as follows: 
(40 b) 
The greater root is the square of the half-length of the cycno’id ; the lesser root 
belongs to the internal stream-line already mentioned. 
The method of finding the extreme half-breadth in a cycnoicl with unsymmetrical ends, 
is to make \]/ = 0 in equation (38), and ~ = 0 in the second equation (39), and, from the 
pair of equations so obtained, to deduce x and y by elimination. When the ends of the 
cycnoid are symmetrical, the extreme half-breadth is midway between the foci ; hence, 
making #=0 in equation (38), we have the following transcendental equation, 
0=?/ 0 — 2^tan 1 
a 
Vo 
— 2/i / tan 1 — ; 
Vo 
(41) 
from which y n is to be calculated by approximation. 
§ 10. Cycnoids of Revolution . — Forms and Velocities of Streams . — The equation of a 
series of cycnogenous or quadrifocal stream-lines of revolution is as follows : 
y 2 k 2 / x — a x + a\ k ' 2 / x — a! x + a" 
4 = ^ + 
=b; . 
• • ( 42 ) 
in which r,, r 2 , r 3 , and r 4 have the same meaning as in equation (38) ; that is, they are the 
mdccclxxi. 2 R 
