a Problem in physical Astronomy. 537 
and consequently, A will be determined. But, with this value 
of v, the terms £, 7, &c. lose all convergency by the geo- 
metrical progression v, v\ v s , &c. and the computation of the 
value of the series, by the common method, would be more 
laborious than the computation of the quadrantal arch of the 
circle, by the series 1 4 - — ! 1 M — , & c. Here then 
* J s 2.3 1 2-4-5 2. 4. 0.7 
we are stopped. But, by contemplating this series, expressed 
in terms of a and c, as it stands below, a very different method 
of obtaining the value of it is suggested. 
3 - 5 - 7 -9 
4.6.8. 10 
a . 
3 a/ (1 — cc ) 
4.2 
3.5 y' (1 — cc ) 
4.6.4 
3-5-7\/(i— cc ) 
4.6. 8, 6 
3 - 5 - 7 -9 V ( 1 — cc ) 
4.6.8.10.8 
4 
+ 
4 
4 
3 CC a . 
\- z 
3.5. 3cc */{x — cc) 
4 6.4.2 
3 . 5-7-5 CC - y/ ( I — Cc) ' 
4.6. 8. 6. 4 
3-5-7-9.7cg v / (i— cc) 
4.6.8. 10.8.6 
+ 
+ 
+ 
3-5-3 c** 
4.6. 4.2 
3-5-7 -5-3 cW{i- c c ) 
4.6.8.64.2 
3-5-7-9-7-5gV(i— gg) u 
4.6.8.10.8.6.4 
&C. &C. &C. &C. 
Here it appears, 1st. That the geometrical progression 1, cc, 
c 4 , &c. has place in the first, second, third, &c. columns of 
quantities on the right-hand side of the equation, the terms of 
which, when b is nearly = a, decrease very swiftly. 
adly. That, in the diagonal line of quantities in which a en- 
ters, besides this decrease of the terms, by the literal powers, 
before mentioned, the numeral coefficients are so simple that 
a considerable number of the terms may quickly be com- 
puted. 
■gdly. That, if this diagonal line of quantities be taken away, 
the first, second, third, &c. infinite columns of quantities which 
3 ^ 
MDCCXCVIIL 
