218] A THIRD MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 511 
and thence also 
and 
d'A 
dt 
nZ i 2 k sin (v + int + e), 
d'A 
2nB —= — nZ (2i + Ï 1 ) k sin (v + int + e), 
so that the equation for By becomes 
d 2 By 
dt'■ 
i + n 2 (1 + f to 2 ) By + nZ (2i + i 2 ) k sin (v + int + e) = 0 ; 
or, taking as the independent variable v (= nt) in the place of t, this is 
+ (1 + f to 2 ) By + ^ Z (2i + i 2 ) k sin (v + iv + e) = 0; 
which is, in fact, Laplace’s equation, n being retained instead of being put equal to 
unity, and By being the part which depends on the variation of the ecliptic, of his s r 
IV. 
Conversely the equation 
d 2 By 
~di? 
+ n 2 (1 + f to 2 ) By = — 
2 Bn + 
d'A 
dt ’ 
3- 
may be obtained by a process similar to Laplace’s. Assuming that the M.oon and 
Sun are each of them referred to a fixed plane of reference and origin of longitudes 
therein, by the coordinates 
u, the reciprocal of the reduced radius vector, 
v, the longitude, 
s, the tangent of the latitude, 
for the Moon, and by the corresponding coordinates u', v, s' for the Sun, then we have 
d 2 s 
dtf 
+ s + 
ds dkl dfl oN dkl 
dv *-“Æ- (1+ ^dï 
n 2 v? ( 1 + 
2 fdfl 
dv 
z ï a: 
n 2 J U 2 1 
dv 
= 0. 
Here, as before, 
ß = ^(fcos 2 tf-i), 
or, as it is now to be written, 
« _ to'(1 + s 2 ) u' 3 f 3 /cos(v — v) + ss'\ 2 1 ) 
(l+s'ft,» rwi+iVi+W ;