a Problem in physical Astronomy. 
54S 
that is, = y A7 r; and the negative part, by converting ^/(i — w) 
into series, will become — 2 ~ - x — ~ — ( 1 4- — 4- 3 4- 
b(a + b)i V{w—cc) V * 2 ' 2.4 ' 
3*5 
*yjy, &c.); the fluent of which appears, by Art. 5, to be 
(a + e + ^ + M^-j- &C.), which will vanish 
when v = c, and therefore needs no correction ; and, when 
v = 1, the series, without the factor, will be as follows : 
2 cc = 2 cc 
e = 
\/(l—CC) 
+ 
C C cc 
2 
■ 3 \/(i— cc) ' 
■4.4 
+ 
3.3 6-C /(I— CC) 
4.4.2 
+ 
3-3 
4.4.2 
LI £ 3-5v / ( I — cc ) , 3 5 . 5 ccv ' ( 1 — cc) , 3-5-5-3 cV(i— cc) 
4-6 4-6.6 < 4 . 6 , 6 . 4 . < 4 . 6 . 6 .a.z » 
, 3-5-7-7CC ^(i~cc) , 
*T r 
4.6 u 4.6.6 
3 - 5-7 __ 3 - 5-7 a/(i— cc) 
4.6.8 4. 6. 8. 8 I 4.6. 3.8.6 
8 V. &c. 
Now, the sum of the infinite series 
4.6. 6. 4.2 
3.5.7.7.5 c 4 y/( I— CC) 
4. 6. 8. 8. 6.4 
&c. 
T + Ti + rti + bein g = 2 H . L . 2 = f, 
r»jp . 3-3 1 3 - 5-5 f 3 5 - 7-7 i 3 - 5 - 7 - 9*9 7 3 tt t 
01 4-1.2 + T 6 M + + 4.6.8.10. .0.8 - &C - beln S = -f-H . L .2 = cr. 
of 
3 - 5-5 3 
+ 
3-5 7 7-5 
4.6.64.2 ' 4.6.8.8,64 
+ 
"64T + 
— H . L 
3 2 
2 
3 - 5 - 7 - 9 - 9*7 , 3.5.7.9.11.11.9 ~ 
4.6.8.10.10.8.6 "* 4.6. 8.10.12. 12. 10.8’ ^ C ' 
&C. &C. 
By proceeding as above, in Art. g, a sufficiently near value 
of the whole series will be obtained in this expression, 
16— 9 cc a + \/(i — (|3-(-<r^ + rc + ); and this, multiplied 
by its proper factor, gives 
