Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 7)

465] 
NOTE ON THE LUNAR THEORY. 
363 
Plana, with Elliptic e, y 
Elliptic 
y= y = 
7 - 7e 2 -§7 3 
+ 
< 
Cb 
1 
-It 3 
sin 
9 
+ ye — 7e 3 — § y 3 e 
+ ye — 1 ye 3 
-f 7 3 6 
33 
c+ g 
+ ye — § 7e 3 + -} fe 
+ ye 
-f 7 3 e 
33 
0- g 
+ f 7e 2 
+ î 7e 2 
33 
2c- g 
+ 1 7e 2 
+ 1 7e 2 
33 
2 c + g 
+ Ü7* 
+ t 3 2 7¿ 5 
33 
Sc- g 
+ 1 7e 3 
+ 1 ye 3 
33 
Sc+ g 
-^T 3 
“A 7 3 
33 
Sg 
+ i y 3 e 
- £ fe 
33 
c-Sg 
- h fe 
- i 7 3 e 
33 
c + Sg, 
where, for greater clearness, I remark that the values called “ elliptic ” of e, y, c, g, 
refer to an ellipse, such that the longitude of the node, and the longitude (in orbit) 
of the pericentre, vary uniformly with the time,—viz., we have mean distance = 1, 
excentricity — e, tangent of inclination = 7, mean longitude = l, mean anomaly = c, 
distance from node =g. 
We have therefore 
8- = 
r 
- f 7 2 ^ 2 
COS 
— f 7 2 e 
55 
c —2g 
Sv = 
- Te 7 2 e 2 
sin 
2c 
>3 
*9 
+ f 7 2 6 
35 
c —2g 
- te 7 2 e 2 
3? 
2c — 2g 
Sy = 
- f 7e 3 +1 7 3 e 
33 
c- g 
+ t 7e 2 
33 
2c- 9 
+ f 7e 3 
33 
Sc- g 
+ f 7 3 6 
33 
c-Sg, 
viz., these are the increments to be added to the elliptic values of -, v, y, respectively, 
in order to obtain the disturbed values of v, y, attending only to the coefficients 
independent of m; they represent, in fact, the lunar inequalities which rise two orders by 
integration. 
The elliptic values of ^ and y are functions, and that of v, is equal l +, a function, 
of e, 7, c, g, and the foregoing disturbed values may be obtained by affecting each of 
46—2
	        
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