282 
PROFESSOR W. J. MACQUORN RANKINE ON THE 
equal to -f^, ancl yV of the first parameter; and thus were drawn the 
five cycnoid curves marked respectively 1 1, 2 2, 3 3, 4 4, and 6 6. The lines of uni- 
form current and of disturbance used in drawing these curves are omitted in the 
engraving. 
This last figure illustrates the fact that, with a given set of foci, and a given para- 
meter for the inner pair of foci, the cycnoid becomes leaner and more hollow at the 
bow as the parameter for the outer pair of foci diminishes ; also that, with large values 
of the second parameter, that curve is convex throughout, like the line marked 66 ; and 
that for some intermediate value the hollowness just vanishes, as is very nearly the case 
in the line marked 4 4. It is obvious that any degree of fineness may be given to the 
entrance by increasing the distance of the second foci from the first, and at the same 
time using a small second parameter. 
§ 8. Cylindric Cycnoids. — Forms and Velocities of Streams . — The equation of a system 
of quadrifocal stream-lines in two dimensions is as follows, 
4/=y+A;^tan _1 — tan -1 T^tan -1 ^—- — tan -1 =b, ■ . (38) 
in which Jc and Id are the parameters for the inner and outer pairs of foci respectively, 
a is the excentricity of the inner pair of foci, and a! and a" are the distances of the 
outer pair of foci from the origin in opposite directions. The equation of the cycnoid 
curve, or trace of the surface of the cylindric solid which generates the series of stream- 
lines, is \f/ = 5= 0. If that solid is symmetrical-ended, we have a' : =a'. The components 
of the comparative velocity of a stream at a given point (x, y) are given by the following 
equations, in which, for brevity’s sake, the following notation is used : 
(x — ; (x-j-a) 2 -j-y 2 =r% : 
(x—a') 2 -\-y 2 =r 2 3 ; {x-\-a!'f- \-y 2 =r\. 
At the extreme breadth of the space bounded by a given stream-line we have ^=0; 
and when the cycnoid is symmetrical-ended, the longitudinal component u at the same 
point takes the following value, found by making a?=0, 
u n = 1 - 
2 ka 
2 fa 1 
a + y\ ' ^+vl 
(39 a) 
where y n denotes the greatest ordinate or “ midship half-breadth ” of the stream-line 
under consideration. 
§ 9. Cylindric Cycnoids.- — Extreme Dimensions . — The extreme length of a cylindric 
cycnoid is made up of the distances of its two rounded ends, where it cuts the axis of x , 
from the origin of coordinates. Let l be one of those distances ; in the expression for 
