V 
a Problem in physical Astronomy. 535 
4. Now, to find the fluent of ■■■ - ■ , g r-, we have 7 — 
~ * bein S P ut = the cosine of in which 
expressions, while % increases from o to 3*14159, x will de- 
crease from 1 to — 1. Therefore, to obtain a more convenient 
a — bx 
expression, put vv 
a + b 
; then, while x decreases from 1 to 
— 1, vv will increase from to = 1 ; and we shall have 
7 a + b a-\-b 
the following equations : 
a — bx=(a-\-b) vv, (a — bx)”—(a-\-b) "v™, x = , — 
(a + b)zvi' „ , , a— (<* + &)»» + 
A ’ 1+X—l-j r = t- = 
b 
1 — .r 
a — ( a-\-b)vv b- 
-a-\- (rt-f b)vv 
* + b 1 
■ b — a 
b — 
b 
— b i 
{ a + b 
a—b 
a-\-b 
a-\-b 
(vv — cr), cc being put 
a—b 
And from 
b ~ *'*' """""'to r 
these equations, the three following are easily obtained, viz. 
v/(t+*) </(>— •r) = v / ( 1 ^(i— ■»*)) xv(^(w— fe)| 
— <I+i v / (( 1 — »») (w — c<0)> an< ^> 
6 
X 
V li— xx) 
and, lastly, 
(a + b) zv<v h ___ 2tKy 
b X (a + b) v' ( ( x - w) (vv— cc) ) ‘“‘V ( i— w) V (w—cc) *■ 
— x 2111? 
v'(i-xx) (a — b x) n y^i — vv) \/(vv — c c) (a -f- b) n v 
i*n 
—n 
X 
zw 
; the fluent of which may be found 
vq 1 — vv) >y ( vv — cc) 
when the value of n is given. 
5. Now, the values of n with which astronomers are most 
concerned, are J- and J-. Let, therefore, f be written for n, and 
the radical quantity s /( 1 — vv) be converted into series, and 
the last expression will be 
