380 
DE. W. J. MACQUOEN EANKINE ON PLANE WATEE-LINES. 
those lines. With the distance SL in the compasses, lay off SH on that line ; H will be 
a point in the hyperbola LM. Also from S lay off, on the axis of y , SI and SJ, each 
equal to the same distance SL. About the centre O, with the radius 01, draw a circular 
arc cutting ST in K ; this will be a point in the branch LN. About the centre O, with 
the radius O J, draw a circular arc cutting ST in 1c ; this will be a point in the branch LP. 
(16.) Properties of the Trajectory of Swiftest and Slowest Gliding. — The branch LN 
traverses a series of points of slowest gliding, where the water-lines are furthest apart ; 
the branch LP traverses a set of points of swiftest gliding, where the water-lines are 
closest together ; from O to P the axis of y traverses points of slowest gliding, and 
beyond P, points of swiftest gliding. 
Hence every complete oogenous neoid which cuts the axis of y between O and P, 
contains two points of swiftest and three of slowest gliding; and every complete ooge- 
nous or cyclogenous neoid which cuts the axis of y at or beyond P, contains only one 
point of swiftest and two of slowest gliding. 
(17.) Water-Lines of Smoothest Gliding , or Lissoneoids. — At the point P itself, situated 
at the distance j 
op =vs 
from the centre, two maxima and a minimum of the velocity of gliding coalesce ; and 
therefore not only the first, but the second and third differential coefficients of the 
velocity of gliding vanish ; from which it follows that the velocity of gliding changes 
more gradually on those water-lines which pass through the point P, than on any other 
class of oogenous or cyclogenous neoids. 
It is proposed therefore to call this class of water-lines Lissoneoids (from \iaa6c,). 
The oval neoid whose length is to its breadth as 3 : 1 is itself a lissoneoid ; and 
every series of water-lines generated by an oval more elongated than this contains one 
lissoneoid; for example, in the series of water-lines shown in fig. 1, the lissoneoid is 
marked PQ. 
The excentricity of the oval lissoneoid is computed by solving equation (24 a) of 
article 9, when y 0 —-^= ; and it is found to be 
a=-732l, or nearly (^3—1 jl (36 a) 
By giving the excentricity values ranging from *732£ to l, there are produced a series of 
lissoneoids ranging from the oval PL in fig. 3, Plate IX., whose focus is at A 0 , to the 
straight line PN, whose focus coalesces with L. J ) Q 1 , PQ 2 , and PQ 3 are specimens of 
the intermediate forms, having their foci respectively at A 15 A 2 , and A s . For a reason 
which will be explained in Section III., those curves are not shown beyond the trajec- 
tory of slowest gliding. 
The greatest speed of gliding, for a lissoneoid, is found by making yl=^ in equation 
(28 a) of article 13 ; that is to say, 
u 0 _ W 
c 3 a 2 + Z 2 
( 37 ) 
