280 
PROFESSOR W. J. MACQUORN RANKINE ON THE 
y # 
finitely long, and having asymptotic cylinders expressed by To avoid con- 
fusion these infinite bifocal stream-line surfaces are not shown in fig. 3. They bear a 
general likeness to those shown in fig. 2. 
When the two foci coalesce into one, the disturbing solid becomes a sphere, whose 
stream-line surfaces were investigated by Dr. Hoppe (Quarterly Journal of Mathematics, 
March, 1856). 
As to the modification of the formulae required if the origin is not taken midway be 
tween the foci, see the end of § 5. 
The component comparative velocities are as follows: 
d-\> , A 2 f — x + a x + a 
'U-— ~T'-— 1 “T W! q “1 3 
ydy 2 + {(tf + a) 2 + ?/ 2 p 
1 . 1 
ydx 2 | {[x — «) 2 -f ?/ 2 } 2 {(a? + «) 2 + ?/ 2 
Let l denote the half-length O L (fig. 3) of the oval solid. Then by making, in the 
first of the above equations, ?s=0, x=l, and y= 0, the following relation is found to exist 
between the half-length, excentricity, and parameter, 
(1? — a 2 ) 2 — 2lc 2 la= 0 (36) 
Let y 0 be the extreme half-breadth OB, then by equation (33) we have 
yt-\-a 2 yt— 4#v=o (36 a) 
Chapter III. Special Theory of Quadrifocal Stream-lines , or Cycnogenous Neoids. 
§ 7. Quadrifocal Stream-lines in general . — A quadrifocal stream-line is the trace on 
a longitudinal diametral plane of a quadrifocal-stream-line surface, belonging either to 
the cylindrical class or to that of surfaces of revolution. The four foci are situated in 
an axis parallel to the direction of the uniform current which is disturbed by the solid ; 
and, as in the previous chapters, that axis will be taken for the axis of x, and the trans- 
verse axis in the plane of projection for the axis of y. 
The general equation of a quadrifocal stream-line may be expressed as follows : 
4 / =4 / o+4 / i (37) 
In that expression f 0 is the function representing the uniform current of the velocity 1, 
which is equal to y or to \ y 2 , according as the surfaces are cylindrical or of revolution ; 
expresses the convergence of certain currents towards one of the foci, f 2 the diver- 
gence of the same currents from a second focus, f 3 the convergence of certain currents 
towards a third focus, the divergence of the same currents from a fourth focus. 
The graphic construction of quadrifocal stream-lines is illustrated in figs. 2 and 3. In 
each of those figures, A is one of the first pair of foci, A' one of the second pair ; the 
other focus of each pair is supposed to lie at the other side of the origin O, beyond the 
limits of the drawing. 
The lines of disturbance expressed by being those due to the first pair of foci, 
