DE. W. J. MACQUOEN EANKINE ON PLANE WATEE-LINES. 
379 
(14.) Trajectories of Normal Displacement , and of Swiftest and Slowest Gliding. — By 
the “ trajectory of normal displacement ” is meant a curve traversing all the points in a 
series of water-lines at which the directions of motion of the liquid particles relatively 
to still water are perpendicular to the water-lines; or, speaking geometrically, a curve 
traversing all the points at which the circles AC,, AC 2 , &c. of fig. 1, Plate VIII. cut the 
water-lines at right angles. To find the form of that trajectory, it is sufficient to make 
vr+v z u ^ 
c 2 ~ c ~ U ’ 
(31) 
employing the values of these ratios given by the equations (28). This having been done, 
it appears, after some simple reductions, that the equation of the trajectory of normal 
displacement is the following, 
x*—y*—l\ (32) 
being that of a rectangular hyperbola LM, fig. 1, having its vertex at L, and its centre 
at O. Hence that curve is similar for all oogenous and cyclogenous neoids whatsoever , 
being independent of the excentricity, and is identical for all oogenous and cyclogenous 
neoids having the same base l. 
By the “ trajectory of swiftest and slowest gliding ” is meant a curve traversing every 
point in a series of water-lines at which the velocity of gliding, s/ u 2 -\-v 2 , is a maximum 
or a minimum for the water-line on which that point is situated. To find the equation 
of that curve, it is necessary to solve the following equation, 
(33) 
A, 
iv? 4- vf 
\ j 
(u d V d\ 
\ | 
/m 2 + v 2 \ 
cdV 
) 
1 — ! 
" dx'c ' dy J 
1 1 
the expression employed for 
being that given by the third of the equations (28). 
After a tedious but not difficult process of differentiation and reduction, which it is unne- 
cessary to give in detail, an equation is found which resolves itself into three factors, viz. 
tf=0, (34) 
being the equation of the axis OY, and 
•J^+y'+y+V F+?=o, (35) 
being the equations of the two branches LN and LP of a curve of the fourth order. 
This curve, too, is independent of the excentricity, and therefore similar for all ooge- 
nous and cyclogenous neoids whatsoever , and identical for those having the same base l. 
It has also the following properties: — The straight line joining L with P makes an 
angle of 30° with the axis OX; there are a pair of straight asymptotes through O, 
making angles of 30° to either side of OX ; and the two branches of the curve cut OX 
in the point L, at angles of 45°. 
(15.) Graphic Construction of those Trajectories. — The curves described in the pre- 
ceding article are easily and quickly constructed, with the aid of the series of equi- 
distant lines parallel to OX, as follows: — In fig. 2, Plate VIII., let ST be any one of 
