of certain differential Expressions , &c. 277 
since, after this reduction, it would be necessary to compute 
E f-^===r from E{ -d~- + d 2 i~i . — &c. } 
These observations are, however, digressive ; the problem, the 
expansion of (a 2 -f 6 2 — Qab . cos 0 )“t-, is, I conceive, completely re- 
solved in the preceding pages, whatever be the ratio between the 
radii of the planets’ orbits.* 
What I have advanced, on a former occasion, concerning the 
independence of analysis and geometry, is confirmed by the pre- 
sent reasonings and results, fdx v / / 1 ■ ) , f.dx ( —— 7), 
-, have been computed, without the introduction 
r d* 
J V(i- 
■X*) (i— e 2 x*) 
of an ellipse, an hyperbola, an oblique cylinder, or a pendulum 
* In the ease of the new planets Ceres and Pallas, whose mean distances from the sun 
are nearly equal, the series (8), and the expression 
( x -f- 'b) ( i (x +."'&) 
— . /. — 1 — (F), will be very convenient, on ac- 
2 m ( m )b v ' 1 
count of the rapid convergency of the quantities, "'b, &c. ; and, in general, in 
estimating the disturbing forces of z planets, since e' is 
mean distance of nearest planet ^ ^ 
mean distance of the more remote planet’ 
putting b—e’, e~\Jz — x = . 4142, &c. 
1 + e ' 
hence, if d be greater than .4142, &c. the series of terms y b, " b , See. decrease more 
rapidly than e ", e w , Sic. and, consequently, the series (8), page 248, and the series 
(i-p'6) (i-f-"6) ... ( 1 + {’”)£>) ,4 , , , . , ... . . 
• L - are t0 be used in determining the perturbations, 
when the planets are Mercury and Venus, £'=20.535 16076 
Venus and Earth, £'—0.72333230 
Venus and Mars, £'—0.47472320 
Earth and Mars, £'=0.65630030 . 
Jupiter and Saturn, £'=0.54531725 
Saturn and Georgium, £'=0.49719638 
Ceres and Pallas, 
