DE. W. J. MACQTJOEN EANKINE ON PLANE WATEE-LINES. 
381 
(18.) Orbits of the Particles of Water . — The general expressions for the components 
of the velocity of a liquid particle relatively to still water have been given in equation 
(11) of article 4 ; and to apply those to the case of oogenous neoids, it is only neces- 
sary to modify the equations (28) of article 13, by introducing the expression for 
- instead of that for - » as follows : — 
c c 
— c (P — a 2 ) . (a 2 — x 2 + y' 2 ) v — 2{P—cP)xy 
c ~ {{a-x) 2 + y 2 } . {{a + x) 9 -+y 2 } ’ c~ {(a— xf+if) . {{a + x) 2 + y 2 } ’ 
(a — c) 2 + v 2 (P — a 2 ) 2 
c — {{a—xY + y 2 } .{(a + xf + if} ' 
(38) 
From the last of these equations it appears that the velocity of a 'particle relatively to 
still water is inversely as the product of its distances from the two foci. 
The only other investigation which will here be made respecting the orbit of a par- 
ticle of water, is that of the relation between its direction and curvature at a given 
point, and its ordinate y. 
It has already been explained, in article 11, that the direction of motion of a particle 
is a tangent to a circle traversing it and the two foci. The radius of that circle is 
a a 
. y—b sin Q ’ 
Sin ~f~ 
and if <p be taken to denote the angle which the direction of the particle’s motion rela- 
tively to still water makes with the axis of x, it is easily seen that 
cos<p = cos0— | sin 6 (39) 
While that angle undergoes the increment d<p, the particle moves through an arc of its 
orbit whose length is ; consequently the curvature of that orbit at the arc in ques- 
tion is 
1 sin 
? 
d . cos f 
dy dy ~(f + l) s[nd +jL COS& = P^'^l^- sirid +y cosd }- ( 40 ) 
For cyclogenous neoids, we obtain the value of this expression by making 
sin 6- 
y-b 
f 
5 COS 0=1, 
substituting l 2 — a 2 for 2 fa, and then making a — 0 ; the result being as follows, 
(40a) 
that is to say, the curvature of the orbit varies as the distance of the particle from a line 
parallel to the axis of x, and midway between that axis and the undisturbed position of 
the particle. This is the property of the looped or coiled elastic curve ; therefore, when 
