53 8 Mr. Hellins’s improved Solution of 
remain will have the literal factors ^(1 — cc), cc-i/(i — cc) T 
c *\/ i 1 cc )> & c - respectively, which are in the progression 
before mentioned. 
4thly. That, if the sum of the infinite series of numeral co- 
efficients below the line, in each of these columns, can be ob- 
tained, then the original series, which had lost all convergency 
by the literal powers v, v\ v\ &c. may be transformed into 
two others, in which the literal powers will be cc, c 4 , &c. 
8. But the sums of these infinite series are attainable, and 
are as follows : 
3-5-3 
4.64.2 
+ 
i_L 
4.64 
+ 
3-S-7-S 1 
4.6.8.64 
3-5-7 
4.6.8. 6 
* 4.6.8.10.8’ lb 
= \ rf- H. L. 2 = a; 
s I H-SM Lg & : 
4.6.8.10.8 .6 * 4.6. 8. 10. 12. 10. 8 ’ ^ 
3-5-7-5-3 
4.6.8.64.2 
1 3-5-7-9-7-5 
' 4.6.8.10.8.6.4 
— tt + I’H.L.2 = p; 
3.5.7.9.11.9.7 
4.6.8.10.12.10.8.6 
, &c. is 
_ 17 + 51 H. L. 2 * 
64 — v ’ ‘ 
&c. &c. 
But these three sums are as many as are requisite, when the 
perturbation of the motion of either the Earth, Mars, or Venus, 
by the attraction of any one of the other, is to be computed. 
9. The sum of the coefficients in the first, second, and third 
columns in Art. 7. being now obtained, take | — — a for the 
value of the series « (1 + ~ , &c\ which will be 
exact enough for the purpose, and we shall have, by Art. 3, 5, 
6, 7, and 8, 
* See the Appendix. 
