IN PHYSICAL ASTRONOMY. 
49 
In order to convert the coefficients of the inequalities of the longitude into 
sexagesimal seconds, they must be multiplied by 1 - 9 ^ Q0 ; the logarithm of which 
number is 5-3144251, the corresponding logarithm for centesimal seconds is 
5-8038801. 
d R \ z. , z — z. ] 
— — = m l i -] i L 
dz l r ' 3 {r*-2rr;cos(A-A' J ) + r, 3 }^ J 
If z = 0, and the products e tan i p e t tan /, be neglected, 
d R _ m, , 
^7 — ^5 tan i, sin (n t t + e / — v,) { 1 — b 3>0 — 6 3)I cos (ref — re f + e — e,) — b 3i oCOS ( 2 nt— 2 n l t+ 2 e— 2 e l )} 
a 5 d 9 S 7B . 
a- tan i, sin (re,f + e, — v,) { 1 — h Jj0 — & 3>1 cos (nt — nf + e — e,) — &c.} = 0 
s=-Hh 
lltj It U r 7 4 • 
pT (re -re,) (» + »,) V tan i( { 1 “ ^ o} Sin (V + *' " ^ 
+ i sin 1 +l)nt + £ — 
m , n- a- , 
~ T. 4 »V(2 » - 2,,) a? lan ‘‘ M ° (- 1 - 2 V + » ~ 2 «■ + >,) 
— pr-T- — ; — tan i, 5 3 „sin (2 re t — w,f + 2 s — s. — v.) 
/x 2 (re — re,) (3 re — re,) a, 2 1 3,2 v 1 in 
Vfl 7ft 
+ J a (» - 3 „,) (3 n - 3 .,) V *°° l ' i “ Si ° < 2 ” « ~ 3 V + 2 » ~ 3 », + >,) 
7W 72 “ (ft 
— 2 ; 2 »- 2 » / )( 4 »- 2 ,,,) V ta ” ’< S « ™ (3 » t - 2 »,! + 3 . - 2 ,, - v,) 
JYI ^ft (ft 
+ — 5-775 7 — 7-7-7 7 — 7 — tan 1, b 3 4 sin (3 n t — 4 re i + 3 £ — 4 s, + v,) 
/x 2 (2 re — 4 re,) (4 re — 4 «,) a, 2 ' 3,4 v 1 1 1 
m, re 2 a 2 , . . . „ . 
— — 75-75 5 — 5-77 5 — 7 — tan i, 6 3 4 sin (4 re < 3 n.t + 4 s — 3 £, — v,) 
/x 2 (3 re — 3 re,) (5 re — 3 re,) a, 3 1 3,4 v ' ' ' 
*(2 + Z) i = ^5 tan », & 3>I 
a, 
Expression 
for the tan- 
gent of the 
latitude. 
R«= - 
3 a . 3 a 
2a/ 2 + 2a^ 
^3,0 
2 a' 
b 3 tl -—b 3>i (Seep. 31 .) 
9 e 
3 a 3 a, a 2 , a, 3 . 3 o 5 J a , j , 1 
“ 2^~ 3 + 2 V 3 ’ 0_ 2^7 3,1 ~ ~2df 5,0 2 6,1 J 
+ 4 £ ( -*y - K ° - 4 *« } + 1 0 { t 6 « -**'■-* 6 « } 
MDCCCXXXI. 
H 
