DE. W. J. MACQUORN RANKINE ON PLANE WATER-LINES. 
377 
Then about each of the points in the axis OY thus found, with the outer leg of the 
compasses starting from the focus A, describe a series of circles (shown in Plate VIII. 
fig. 1) AC n AC 2 , AC 3 , &c. 
Each of those circles traverses the two foci ; and the equation of any one of them is 
— 'F=f0=nhy, (25) 
where 6 denotes the angle made at any point of the circle by straight lines drawn to the 
two foci, and n has the series of values 1, 2, 3, &c. Since F, as explained in article 4, 
is the characteristic function for the motion of the liquid particles relatively to still water, 
it is plain that each of the circles for which F= constant is a tangent to the directions 
of motion of all the particles that it traverses. 
The paper is now covered, as in fig. 1, with a network made by a series of straight 
lines whose equations are of the form y=n '! ty, crossed by a series of circles whose equa- 
tions are of the form f6=nby. 
Consequently any curve drawn like those in Plate VIII. fig. 1, diagonally through the 
corners of the quadrangles of that network, will have for its equation 
y-ft=(ri-n)ly=b, 
and will accordingly be an oogenous neoid, having for its asymptote the line y=b. 
The primitive oval is drawn by starting from the point L, and traversing the network 
diagonally. As many curves as are required can be drawn by the eye with great pre- 
cision, and the whole process is very rapid and easy (see Appendix). 
When the problem is with a given base and excentricity to draw an oogenous neoid 
through a given point in the axis OY, such as P, the process is modified as follows : — 
The axis OY must be so divided that P shall be at a point of division. Then, up to the 
describing of the circle about A with the radius AE, the process is the same as before. 
Then join AP (Plate VIII. fig. 2), and draw A g making the angle PAy= APO, and cut- 
ting the axis OY in a point (such as G 10 ) which will be the centre of the circle traversing 
A and P. Then on the circumference of the circle about A, from g towards F, lay off a 
series of arcs each =2ty; through the points of division draw radii cutting the axis OY 
in the points G 9 , G 8 , &c., and complete the process as before. 
(12.) Graphic Construction of Cyclogenous and Parabologenous Neoids . — When the 
excentricity vanishes and the oval becomes a circle, all the circles composing the net- 
work become tangents to OX at the point O. They pass through the points where the 
primitive circular water-line is cut by the equidistant parallel lines. Their radii are in 
harmonic progression ; the equation of any one of them is of the form 
( 26 ) 
•which is a larger multiple of the parameter than double, the length of the divisions being increased in the same 
proportion ; or the points on the axis OY may be laid down by means of their distances from 0, calculated by 
the formula 00= a . cotan 0. 
3 e 2 
