376 
DR. W. J. MACQUORN RANKINE ON PLANE WATER-LINES. 
circle of the radius l ; and the water-lines generated by it become cyclogenous neoids, 
already described in article 5. 
As the excentricity increases, the oval becomes more elongated. In Plate IX. fig. 3, 
PL is an oval whose length is to its breadth as \/ 3 : 1, its focus being at A 0 . The oval 
BL in Plate VIII. fig. 1 is more elongated, its length being to its breadth as 17 : 6 nearly. 
When the excentricity is infinite, the centre O and the further focus go off to infinity, 
leaving only one focus. The parameter f becomes equal to the focal distance LA. The 
oval is converted into a curve bearing the same sort of analogy to a parabola that an 
oval neo'id bears to an ellipse*; but instead of spreading to an infinite breadth like a 
parabola, it has a pair of asymptotes parallel to the axis of x, and at the distance +^/' 
to either side of it ; and each generated water-line has two parallel asymptotes, at the 
respective distances b and b+nf from the axis of x. The properties of thesp curves 
may be easily investigated by placing the origin of coordinates at the focus A, and sub- 
stituting, in equation (19), tan _1 -|- for 6; but as their figure is not suitable for ships’ 
water-lines, it is unnecessary here to discuss them in detail ; and the same may be said 
of a class of curves analogous to hyperbolas, whose equation is formed by putting — 
instead of + between the two terms of the right-hand member of equation (18). 
(11.) Graphic Construction of Oval and Oogenous Neoids. — For the sake of distinct- 
ness, the processes of drawing these curves are represented in two figures, — fig. 2 show- 
ing the preliminary, and fig. 1 the final processes (see Plate VIII.). 
The axis OY is to be divided into equal parts of any convenient length (which will 
be denoted by by in what follows), and through the divisions are to be drawn a series of 
straight lines parallel to OX. (It is convenient to print those lines from a copper-plate 
divided and ruled by machinery.) They are shown in fig. 1 only, and not in fig. 2, to 
avoid confusion. 
Suppose, now, that the problem is as follows : — The base OL and excentricity OA 
being given , it is required to construct the oval neo'id and the water-lines generated by it. 
Through the focus A (Plate VIII. fig. 2) draw AD perpendicular to OX ; about O, 
with the radius OL, describe the circular arc LD, cutting AD in D ; from D draw DE 
perpendicular to OD, cutting OX in E ; then (as equation (22) shows) AE will be =2 \f 
the double parameter . 
About A, with the radius AE=2/ thus found, describe a circle cutting AD in F. 
Then commencing at F, lay off on that circle a series of arcs, each equal to 21 y (the 
double of the length of the equal divisions of the axis OY). Through the points of 
division of the circle draw a series of radii, AGj , AG 2 , &c., cutting the axis OY in a 
series of points (some of which, from G 3 to G 12 , are marked in fig. 2) f . (These radii 
make, with the line AD, a series of angles, &c.) 
* This curve is identical with the quadratrix of Tschirnhatjsen. 
t When the parameter is small, it is sometimes advisable to use a circle (such as a protractor) with a radius 
