VOL. XLVII.] VHILOSOPHICAL TRANSACTIONS. ISQ 



force as c, then the sign of c is changed in these expressions ; and the direct 

 motion of the apsis will be a X (1 — V^TZl — )• 



2. And hence, if some foreign variable force, added to, or subtracted from, 

 the central force of attraction, produce a given motion of the apsis, retrograde 

 or direct ; it is easy to find a constant force as c, which should produce the 

 same motion. 



3. Let s represent the sun, at an immense distance, t the earth, (supposed, 

 for the present, at rest) p the moon's place in her orbit adbc, in which c, d, are 

 the quadratures, a, b, the syzygies : then if pk, parallel to ab, and cutting tc 

 in K, be produced till kl is double of pk ; and lm parallel to pt meet ab pro- 

 duced in M ; LM and mt will represent the disturbing forces of the sun, by which 

 the moon is urged in the directions pt, mt. See Princip. lib. i. prop. 66, and 

 lib. iii. prop. 25, 26. And if tr be made perpendicular to lm, the force mt 

 shall be resolved into two forces as rt and mr ; of which the latter, mr, taken 

 from LM, reduces the disturbing force, in the direction pt, to their difference lr. 



4. Put now pt (= lm) =: 1 ; pk, the sine of the arc PC =s: and then tm 

 (= PL = 3s): MR :: 1 : .«; that is, MR = 3*^, and lr, the disturbing force in 

 the direction pt, is as 1 — 3*^. When cp, the moon's distance from the quadra- 

 ture, is an arc of 35° 15' 52", in which case 1 — 33* = O, / and r coincide ; and 

 the disturbing force vanishing, the line of the apses becomes stationary. But if 

 the moon's distance from her quadrature be still greater, as at v, then jaj exceeds 

 (*x; and their difference xj is a force represented by — (1 — 3j^), acting in the di- 

 rection ttt. This force, at the syzygies, is double of to. 



5. Hence, and from § 1, it follows; that c being the sun's disturbing force 

 in the direction ct, at the quadrature ; at any other point, as p, it will be 

 + c X (1 — Ss"^)- And that writing for c the variable quantity c- x (1 — 3i^), 



and A for the fluxion of the arc cp, the fluent of a X V ^-—: ~ f,\ will ecive 



1 + c X ( 1 — 3«') ° 



the motions of the apsis. 



6. The quantity c being , } gf - ° ^ of the earth's mean attractive force at the 

 moon ; by computing as above, it will be found, that while the moon moves 

 from c to p, through an arc of 35° 15' 52", the total regress of the apsis is to 

 the arc cp, as .005404 {= n) to unity ; and that the sum of its direct motions, 

 while the moon moves from p to a, is to the arc j&A, as .0105707 (= n) to 

 unity. It will be found likewise, by the inverse operation hinted in § 2, that 

 putting k — .00362552, and k = .OO69611 ; + k and — k are forces, which 

 acting constantly, the one from c to p, the other from p to a, would produce 

 the same motions of the apsis. 



7. The quantities h and k might have been found, pretty near the truth, only 



T 2 ■ • 



