382 DE. W. J. MACQTTOEN EANKINE ON PLANE WATEE -LINES. 
the water-lines are endogenous, the orbit of each particle of water forms one loop of an 
elastic curve. 
The general appearance of such an orbit is shown in fig. 6, Plate VIII. The arrow D 
shows the direction of motion of the solid body. The dotted line AC is supposed to 
be at the distance b from the axis of x . The particle starts from A, is at first pushed 
forwards, then deviates outwards and turns backwards, moving directly against the 
motion of the solid body as it passes the point of greatest breadth, as shown at B. The 
particle then turns inwards, and ends by following the body, and coming to rest at C, 
in advance of its original position. 
When the water-lines are oogenous, the equations (39) and (40) show that the orbit 
is of the same general character with the looped elastic curve in fig. 6, but differs from 
it in detail to an extent which is greater the greater the excentricity a\ and the 
difference consists mainly in a flattening of the loop, so as to make it less sharply curved 
at B. 
When the excentricity increases without limit, the orbit approximates indefinitely to 
a “ curve of pursuit,” for which 
*=*,! = !“• (40B) 
§ f 
(19.) Trajectory of Transverse Displacement. — Of Speed of Gliding equal to Speed of 
Ship. — Orthogonal Trajectories. — The trajectories described in this article differ from 
those described in articles 14, 15, and 16 by being dependent upon the excentricity, 
and therefore not similar for all sets of oogenous neoids. 
By the “trajectory of transverse displacement” is meant the curve traversing all the 
points at which the liquid particles are moving at right angles to the axis OX, rela- 
tively to still water. It is determined from the first of the equations (28), by making 
- — 1 = 0 ; 
c 
from which is easily deduced the following equation, 
x>-f=a\ (41) 
being that of a rectangular hyperbola, with its centre at O and its vertex at the focus A. 
The trajectory of the points where the speed of gliding is equal to the speed of the 
solid body, is found from the third of the equations (28), by making 
Its equation is 
w 2 -fv 2 
- 1 = 0 . 
(42) 
being that of a rectangular hyperbola, with its centre at O and its vertex between A and 
L, at a distance from O equal to half the hypothenuse of a right-angled triangle whose 
other sides are equal to the base and the excentricity respectively. 
Let q= constant be the equation of one out of an indefinite number of orthogonal 
