THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
725 
Since Mo is derivable from M x by writing y / +^7r for therefore it is also derivable 
by writing x+2 77 f° r X- Hence — M 3 is the same as M 1? save that sines replace 
cosines 
Again M 3 =sin l f sin y = sin l cos a sin y -f- cos l sin a sin y 
But sin a sin y=sin i sin N= 2 PQ sin JV 
And cos a sin y=sin i cot a sin c=sin i (cot A sin B+cos c cos B) 
= cos i sin j + sin i cos j cos N 
= 2 pq(P' 2 ^-Q' 2 )-\- 2 PQ(p 2 —q 2 ) cos N 
Therefore 
VL ?> = 2 PQ\_p 2 sin q 2 sin (l—N)\-\- 2 p>q(P 2 —Q 2 ) sin l . . . (21) 
For the sake of future developments it will be more convenient to replace the sines 
and cosines in the expressions for the M’s by exponentials, and for brevity the \/—l 
will be omitted in the indices. 
Then 
2 M 1 =e x 1 N [_Pjp — Qqe N ~f-\-e x+l+N \Qp-\-Pqe A ] 3 + the same with the signs of the 
indices of the exponentials changed, 
— 2M 3V / —1= the same with sign of second line changed, 
M gv / — i=e l+N [Pp — Qqe~ N ^\ \_Qp-\-Pqe~ y ~\ — same with signs of the indices of 
the exponentials changed. 
Now let 
Tx—Pp — Qqe N , K—Qp-\-Pqe N 
777 = Pp> — Qqe~ N , k = Qp + Pqe~ N J 
( 22 ) 
From these definitions it appears that ttt and k are two imaginary functions, which 
oscillate between the real values cos and cos i>(i—/), and sin and sin 
as the node of the orbit moves round. 
Also let 6 =l-\-N, the true longitude of Diana measured from the autumnal equinox. 
Strictly speaking, when longitudes are measured from a fixed point in the ecliptic 
6 =l-\-N — \Jj, but in the present investigation nothing is lost by regarding xp as zero ; 
in § (12), and in Part III., we shall have to introduce if/. 
Then 
2 M 1= 
2 M 3V /^T= 
M 3V /—1 = 
77r 3 e x °-{-K~e x+e - \-zPe x+0 -\-K 2 e 
— m 2 e x ~ B — K 2 e x+0 -\- m 2 e~ x+e -\- K 2 e 
7XK6 e - 77>K€ 
(23) 
The object of the present investigation is to find the following spherical harmonic 
functions of the second degree of M l9 M 3 , M s , viz. : 
MDCCCLXXX. 5 A 
