372 
DR. W. J. MACQTJORN RANKINE ON PLANE WATER-LINES. 
When the solid is symmetrical at either side of the axis of x (as it is in all the cases 
that will be considered in this paper), the axis of x itself, so far as it lies beyond the 
outline of the solid, is a water-line. Hence it is necessary that the equation of that 
axis, viz. 
y= o, 
should be one of the solutions of the equation 
• ( 13 ) 
b=y+ F(ff, y )= °, 
and consequently that F should vanish with y. 
The vanishing of F when y— oo , indicates that every straight line given by the equa- 
tion y —b either forms part of, or is an asymptote to, a water-line curve. 
The vanishing of F when x—oo , indicates that the further the water-lines are from 
the generating solid, the more nearly they approximate to parallel straight lines. 
Every water-line curve is itself the outline of a solid capable of moving smoothly 
through a liquid. 
(5.) Water-Line Curves generated by a Circle , or Cyclogenous Neoids . — Conceive that 
a circular cylinder of indefinite height, and of the radius l , described about the axis of 
z, moves through the liquid along the axis of x. Then it is already known that the 
general equation of the water-line curves is the following, 
*=y(i-??p). (14) 
giving a series of curves of the third order. When b=0 this equation resolves itself 
into two, viz. 
y— 0; x 2 -\-y 2 —l 2 ; 
the first of which represents the axis of #, and the second the circular outline of the 
cylinder. For each other value of b, equation (14) represents a curve having two 
branches : one of them is an oval, contained- within the circle, and not relevant to the 
problem in question ; the other, being the real water-line, is convex in the middle and 
concave towards the ends, and has for an asymptote in both directions the straight line 
y=b. 
For brevity’s sake, let x 2 -\-y 2 =r 2 . Then the component velocities of a particle of water 
relatively to the solid are given by the equations 
u db P , 2/V _ , IHyt-x 2 ) 
y (is) 
v__ db_ 2 Pxy [ 
c dx r 4 ’ J 
and the square of their resultant by the equation 
while the component and resultant velocities relatively to still water are given by the 
