TRANSACTIONS OF THE SECTIONS. 181 
of such lines the primitive water-line is a particular sort of oval characterized 
by this property—that the ordinate at any point of the oval is proportional to 
the angle between two lines drawn from that point to two foci. bin fig. 1 of the 
Plate illustrating the author’s paper in Phil. Trans. 1864 (not yet published), 
LB represents a quadrant of such an oval, O being its centre, and A one of the 
foci; the other focus is at an equal distance from the other side of the centre.) 
Ovals of this class differ from ellipses in being considerably fuller at the ends and 
flatter at the sides, 
6. The length of the oval may bear any proportion to its breadth, from equality 
(when the oval becomes a circle) to infinity. (In the Plate referred to above, the 
length O L is to the breadth O B nearly as 17 : 6.) 
7. Each oval generates an endless series of water-lines, which become sharper 
in figure as they are further from the oval*. In each of those derived lines, the 
excess of the ordinate at a given point above a certain minimum value is proportional 
to the angle between a pair of lines drawn from that point to the two foci. 
8. There is thus an endless series of ovals, each generating an endless series of 
water-lines ; and amongst those figures a continuous or “ fair” curve can always 
be found, combining any proportion of length to breadth from equality to infinity, 
ie. any degree of fulness or fineness of entrance, from absolute bluffness to a 
ife-edge. 
9. The lines thus obtained present striking likenesses to those at which naval 
architects have arrived through practical experience ; and every successful model in 
existing vessels can be closely imitated by means cf them, from a Dutch galliot 
to a racing-boat. 
10. Any series of water-lines, including the primitive oval, are easily and 
quickly constructed with the ruler and compasses as follows. Parallel to the 
longitudinal axis OX, draw a series of straight lines at equal distances apart. 
Through the foci draw a series of circular ares AC,, AC,, &c., so as to contain a 
series of angles found by dividing those distances by 
OL?— OA?" 
20A 
Each of those circular ares indicates the direction of motion in still water of 
each of the particles that it traverses. Then through the angles of the network 
formed by the straight lines and circular arcs draw a series of curves; these will be 
the required water-lines Tf. 
The centre of curvature of the oval at L is the focus A. 
11. The following curves, traversing certain important points in the water-lines, 
are exactly similar for all water-lines of this class, and are easily and quickly con- 
structed with the compasses. 
LM isa hyperbola having a pair of asymptotes crossing the axes at O at angles 
of 45°. It traverses all the points at which the motion of the particles in still 
water is at right angles to the water-lines. 
LQN and LP are the two branches of a curve of the fourth order, having a 
pair of asymptotes which traverse O, making angles of 80° withOX. A straight 
line joining L and P makes an angle of 30° with LO. The two branches cross 
the axis O X at L, making angles of 45°. The branch L QN traverses a series of 
points, at each of which the velocity of gliding of the particles of water along the 
water-line is less than at any other point on the same water-line. The branch 
LP traverses a series of points, at each of which the velocity of gliding is greater 
than at any other point on the same water-line. 
12. The axis O Y from B to P traverses a series of points of minimum velocity 
o ore from P onwards it traverses a series of points of maximum velocity of 
gliding. 
13. Every water-line, complete from bow to stern, which passes within the 
* As a convenient and significant name for these water-lines, the term “ Odgenous 
Neoids” is proposed (from ’Qoyevzjs, generated from an egg, ov oval). 
t The first émployment of a graphic process of this kind is due, it is believed, to Pro- 
fessor Clerk Maxwell, who applied it to certain curves connected with electricity and mag- 
netism. 
