[163 
163] 
ON HANSENS LUNAR THEORY. 
19 
'%1’S 
• cf>) de|. 
Now 
jUos(/ 0) l + a( /_ e!) (oo S (/ <M l)} 
= a(l-e 2 )( 2 2cos ^ </>) + e sm/sm (/ 0)); 
therefore 
7 p sin</> pesmcf) 
dlp rsin f dlr + a(l-e>) {d/ d * ) = 
j p cos(/ !+„/_,,)(«»(/ « e!) 
. , . 1 , wae sin f 
psm( ^ ««ava-o^a-o' 
+ P '' . 1 cot/pm (/ <i) , dr. 
r 2 ^ 17 r wa 2 v(l — e 2 ) 
w) dR - 
So far the variations of the elements have, in fact, been treated as independent; 
but if we substitute for dna?\J( 1 — e 2 ), d-dr, their values in the disturbed 
motion, the equation becomes 
dlf, + all -t)W ^ )= |r Cos(/ « 1 + a(l- e »)( cos U Olva-^dK* 
^ f äR, 
Consider now the point in which the orbit is intersected by any orthogonal 
trajectory to the successive positions of the orbit, or to fix the ideas, the orthogonal 
trajectory passing through T, the point in question may, for want of a recognised 
name, be called the “departure point;” and the angular distances in the orbit measured 
) dQ 
from this point may be termed “departures;” the expression, “the departure,” is to be 
understood as meaning the departure of the moon. Write now 
^) dR }’ 
%, the departure of the perigee, 
v /} the departure, = f+ 
cr, the departure of the node, = % — w, 
®, the longitude in orbit of departure point, = 6 — a. 
? 2) 
It should be remarked that % is not properly an element, i.e. it is not a function 
of a, e, c, o), 6, i without t, and in like manner cr and © (which depend upon y) are 
not elements. 
/ 
*) 
We have from the geometrical definition 
r. 
d% = dco + cos i dd ; 
3—2