221] ON THE SECULAR ACCELERATION OF THE MOON’S MEAN MOTION. 
523 
I. 
The inclination and eccentricity of the Moon’s orbit, and, a fortiori, the variation 
of the position of the Ecliptic, and the Sun’s latitude, are neglected; and the longitudes 
are measured from a fixed point in the Ecliptic. I write 
n, the actual mean motion of the Moon at a given epoch; 
viz., it is assumed that the mean longitude at the time t is e + nt + n 2 t 2 + &c. where 
e, n, n 2 , &c. are absolute constants; and, moreover, 
a, the calculated mean distance of the Moon; 
that is, n 2 a? is the sum of the masses of the Earth and Moon; a is therefore an 
absolute constant; and, in like manner, 
n', the actual mean motion of the Sun at the same epoch, 
a', the calculated mean distance of the Sun; 
that is, if it were necessary to pay attention to the secular variation of the mean 
motion of the Sun, the assumption would be that the mean longitude was e'+ n't+n 2 t 2 + &c., 
e, n', n' 2 , &c. being absolute constants, and n' 2 a' 3 the sum of the masses of the Sun 
and Earth; a would thus also be an absolute constant. But for the purpose of the 
present investigation the secular variation of the mean longitude of the Sun is neglected, 
or it is assumed that the mean longitude of the Sun is e + n't, e', n' being absolute 
constants; and that n' 2 a' 3 is the sum of the masses of the Sun and Earth, a' being 
thus also an absolute constant. 
I put also 
that is, 
m, the ratio of the mean motions of the Sun and Moon; 
n , 
m = — , or n = mn ; 
n 
m is also an absolute constant. 
The Sun is considered as moving in an elliptic orbit, the eccentricity where 
of is e' + he' or e! + ft, d and f being absolute constants; the longitude of the 
Sun’s perigee may be taken to be vs’ + (1 — c') n’t; so that the mean anomaly g' is = 
nt + e — [■©■' — (1 — c)nt] = c'n't + e — -zri; o', tx', being absolute constants; but d is in fact 
treated as being = 1. Hence, if r, v are the radius vector and longitude of the Sun, 
we have 
r' = a' elqr (f + he', g), 
v = tx' + (1,— c')n't + elta{e + he, g') 
= n t -p e + [elta (e -1- he, g ) g ], 
where 9' ~ c n '^ + 6 — ' sr • 
In the expression for the disturbing function the Sun’s mass is taken to be =n 2 a' 3 , 
or, what is the same thing, = m 2 n 2 a' 3 . 
66—2