199] NOTE ON THE THEORY OF ELLIPTIC MOTION, 
217 
or substituting for H its value, the equations of motion are 
dr 
Jt ~ P ' 
d6 _ q 
dt r 2 ’ 
dp = t + £ 
dt r 3 r~ 
dq 
di 
= 0. 
Putting, as usual, fi = n 2 a 3 , and introducing the eccentric anomaly u, which is given 
as a function of t by means of the equation 
, , du 
so that = 
dt 1 — e cos u 
nt + c = u — e sin u, 
the integral equations are 
q = na? Vl — e-, 
P 
nae sm u 
1 — e cos u ’ 
r = a (1 — e cos u), 
V1 — e 2 sin u 
6 — v* = tan -1 
cos u — e 
where the constants of integration a, e, c, tb denote as usual the mean distance, the 
eccentricity, the mean anomaly at epoch, and the longitude of pericentre. 
Suppose that q 0 , p 0 , r 0 , 6„, u 0 correspond to the time t 0 (q is constant, so that 
r/ 0 = q), and write 
V=na? (u — u 0 + e sin u — e sin u 0 ) ; 
joining to this the equations 
. r = a (1 — e cos u), r 0 = a (1 — e cos u 0 ), 
0-0, = tan-. - tan- ( VI 
cos u — e 
e* sm u. 
cos u — e 
u, u 0 , e will be functions of a, r, r 0 , 6, 6 0 , and consequently (n being throughout con 
sidered as a function of a) V will be a function of a, r, r 0 , 0, 6 0 . The function V 
so expressed as a function of a, r, r 0 , 0, 0 O is, in fact, the characteristic function of 
Sir W. R. Hamilton, and according to his theory we ought to have 
C. III. 
d V= \n~a (t —1 0 ) da +pdr + qd6 —p 0 dr 0 — q o d0 0 
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