362 
NOTE ON THE LUNAR THEORY. 
[465 
-iy 2 c 
sin 
c 4- 2g 
- i Ÿe 2 
33 
2c — 2$r 
-l|y 2 C 2 
33 
2c 4- 2p 
+ Á7 4 
33 
4p. 
y (Plana) = 
y _ 
ye 2 
— t 7 3 
sin 
9 
4- ye — 
t 76 s 
33 
c- g 
4- ye — 
f 7e 3 
~ 17 3 e 
33 
c+ g 
+ I ye 2 
33 
2c- g 
+ | ye 2 
33 
2c 4- g 
4-^ye 3 
33 
3c- g 
+ 1 76 3 
33 
Sc+ g 
_ 1 
2 4 7 
33 
3g 
+ h y 3 e 
33 
c —3 g 
- ïï 7 3 c 
33 
c+3g. 
To compare these with the elliptic values, it is necessary to write e (1 + £ y 2 ) in 
place of e. Making this change, or say reducing Plana’s (e, y) to the elliptic (e, y), 
I write down in a first column the transformed coefficients, and in a second column 
the elliptic coefficients, as follows: 
Plana, with Elliptic e, y 
1 
1 
Elliptic 
1 
+ e - $ e 3 
4- e 2 —| e 4 
+ Ä e 3 
+ t* 4 
-f y 2 e 2 
-f y 2 e 
1 
+ e — -|e 3 
4- e 2 - £ e 4 
+ I* 3 
+ fe 4 
0 
0 
cos c 
33 
33 
2 G 
3c 
33 
33 
33 
4c 
2 g 
c-2g. 
Plana, with Elliptic e, 7 
Elliptic 
V — 
2 
Z 
+ 
2 
e -1 e 3 
4- 
2 
e — ^e 3 
sin 
c 
4- 
5 
4 
e 2 ~ M e 4 — ^ 7 2 e 2 
+ 
5 
4 
e 24 e 
33 
2c 
4- 
13 
T2 
e 3 
4- 
13 
T2 
e 3 
33 
3c 
4- 
103 
W 
e 4 
4- 
103 
e 4 
33 
4c 
- 
i 
7 2 - re r ß2 + s 7 4 
- 
l 
4 
y 2 4- y 2 e 2 4-1 y 4 
33 
2g 
+ 
a 
4 
y' 2 e 
- 
1 
2 
y 2 e 
33 
c 
-29 
- 
1 
2 
y 2 e 
- 
1 
2 
y 2 e 
33 
c 
+ % 
- 
y 2 e 2 
4- 
3 
16 
y 2 e 2 
33 
2c 
-% 
- 
13 
TÏÏ 
y 2 e 2 
- 
13 
TÏÏ 
y 2 e 2 
33 
2c 4- 2<7 
4- 
1 
32 
7 4 
4- 
1 
32 
7 4 
33 
4,7-