506 A THIRD MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. [218 
I. 
The position in respect to a fixed plane of reference and origin of longitudes 
therein, of the variable ecliptic and of the departure point or origin of longitudes 
therein, are determined by 
6', the longitude of node, 
a-', the departure of node, 
<jthe inclination, 
where, by the definition of a departure point, 
da-' — cos </>' d0' = 0, 
and then, in respect to the variable ecliptic and departure point or origin of longi 
tudes therein, the position of the disturbed body is determined by 
r, the radius vector, 
v, the longitude, 
y, the latitude ; 
and this being so, then (Supplementary Memoir, pp. 220, 227) the expression for the 
Vis Viva function is, 
T = | {r 2 + r 2 (Q 2 + R 2 )}, 
where 
Q — —ÿ + [ C0S ( v — <r') sin <f>' . 0' — sin (v — a) (j)'], 
R = COS y . V — sin y [sin (V — a-') sin (f)' . 6' + COS (V — a-') (j)'], 
the superscript dots being used to denote differentiation with respect to the time. 
The last-mentioned expressions may for shortness be denoted by 
Q— — ÿ + A, 
R = cos y . i) — B sin y. 
The equations of motion are of course, 
d dT _dT = dV 
dt dr dr dr ’ 
d dT_ (IT_ clV 
dt dv dv dv ’ 
d dT _ clT _ dV 
dt dÿ dy dy ’ 
where V +Î2, if f2 is the disturbing function, taken with Lagrange’s sign 
(Cl = — R, if R is the disturbing function of the Mécanique Céleste).