IN PHYSICAL ASTRONOMY. 
341 
COS V 
sin v 
cos v 
ios v 
If the origin of t coincides with the instant of the perihelion passage, by 
Lagrange’s theorem 
= co S nt-esmnP-^.^^ - -fL 
2 d.nt 2.3' (d .ntf 2.3.4. (d .nt 3 
-smnt + e S mntco S nt +~ d - : sin ” 1 * * + ^ 
• .9 1 — cos 2 nt 
sin w t 2 = g , sm n = 
d . » t t 2.3 (d . n *) 2 
3 sin t? # — sin 3 n £ 
— &c. 
e* d 3 . sin nt* cos nt 
2.3.4- (d.nt) 3 
sin n P = 
d . sin « ^ 
d. 
d . sin n t* 
d.nt 
d . sin n t b 
d.nt 
d* sin n t 13 
(d . n if 
3 - 4 cos 2 n t + cos 4 n t . lOsin n t~5 sin 3 n t + sin 5nt 
8 ’ sm n $ = — 
3 cos nt — 3 cos 3 n £ 
4 
2 sin 2 n t — sin 4 n t d 2 . sin n t* 
2 j t y = 2 cos 2 n t — 2 cos 4 n t 
10 cos n t — 15 cos 3 nt + 5 cos 5 n t 
16 ‘ 
- 10 sin n t + 45 sin 3 n t — 25 sin 5 n t 
16 ~ 
d 3 . sin n t b 
(d.nt) 
— 10 cos nt + 135 cos 3 nt — 125 cos 5nt 
16 " 
= cos rc £ — • — cos 2w/ - y| 
3 cos nt — 3 cos 3nt 
}-o{ 
2 cos 2nt—2 cos 4 
j 
g* f — 10 cos + 135 cos 3 n/ — 125 cos 5 n/ } 
~ 2.3.4 \ 16 J 
If the origin of the time does not coincide with the perihelion passage, n t 
+ £ ® must be substituted for n t, but as s always accompanies n t, it may 
be suppressed at present for convenience, and afterwards replaced. 
fi 3 „ 5 e* 1 , „ e e f 
\ 8~ e 192 j C0S ( W — + ~2 | 1 3 ~ j-COS(2w/ — 2 sr) 
+ T 1 - e 2 jcos(3rc*-3sr) + ^cos(4w^-4 n r)-|-~e 4 cos(5w/-5 n r) 
sin ». / 2 nos n / d • sin « t 3 cos n t — cos 3 n t d . sin n t- cos nt — sin n t + 3 sin 3 n t 
3 d.nt ~ 4 ? d.nt “ 4 
