374 
DE. J. W. MACQUOEN BANKINE ON PLANE WATER-LINES. 
for all values of od between two arbitrary limits. Let 2 a denote the distance between 
those limits: the most convenient position for the origin of coordinates is midway 
between them, so as to make the limits 
ou 1 = + «, sd——d respectively. 
Then the following is the integral sought : 
ydod , , a — x , , , a + x 
— = tan -1 
V 
£ 
_ „ ( x-df+y c ‘ 
tan' 
y 
(18) 
This quantity evidently denotes the angle contained between two lines drawn from the 
'point (x, y) to the points (+«, 0) and (— a , 0). For brevity’s sake, in the sequel that 
angle will be occasionally denoted by 6 ; the points ( + a, 0) and (—a, 0) will be called 
the foci ; and their distance a from the centre will be called the excentricity. 
Substituting this integral in the general equation (10), we find, for the water-line 
curves now under consideration, the following equation, which is the general equation 
of oogenous neoids : — 
.... (19) 
*=y-/ fl =y-/( tan '‘ ~ + tau " 
The coefficient/ denotes an arbitrary length, which will be called the parameter. 
(8.) Geometrical Meaning of that Equation . — The equation (19) represents a curve at 
each point of which the excess ( y—b ) of the ordinate (y) above a certain minimum 
value ( b ) is proportional to the angle (0) contained at that point between two straight 
lines drawn to the two foci. Except when b— 0, the curve has an asymptote at the 
distance b from the axis of x, and parallel to that axis. Since the value of b is not 
altered by reversing the signs of x, and is only changed from positive to negative by 
reversing the sign of y , it follows that each curve consists of two halves, symmetrical 
about the axis of y ; and that there are pairs of curves, symmetrical about the axis of x. 
In Plate VIII. fig. 1, therefore, which represents a series of such curves, one quadrant 
only of the space round the origin or centre O is shown, the other three quadrants being 
symmetrical. A is one of the foci, at the distance OA=« from the centre; the other 
focus, not shown in the figure, is at an equal distance from the centre in the opposite 
direction. BL is one quadrant of the primitive oval ; and the wave-like curves outside 
of it are a series of water-lines generated from it, having for their respective asymptotes 
the series of straight lines parallel to OX, and whose distances from OX are a series 
of values of b. 
The equation (19) embraces also a set of curves contained within the oval, and all 
traversing the two foci ; but as these curves are not suited for the forms of ships’ water- 
lines, no detailed description of them needs be given. 
(9.) Properties of Primitive Oval Neoids . — When in equation (19) b is made =0, so 
that the equation becomes 
y-f6= 0 , ( 20 ) 
there are two solutions; one of which, viz. y=. 0, represents the axis of x, agreeably to 
