140 PHILOSOPHICAL TRANSACTIONS, [aNNO 1751. 



by summing the ordinates 1 Z. R? or 1 — 3s\ on the arc A : in which case we 

 should have had k = c X .648869 = .00370g'25, and k = c X 1.24018 = 

 .006939 : and the motions thence computed would not have been much dif- 

 ferent from their just quantity. This however is mentioned, not as if the me- 

 thod itself were sufficiently exact ; but to show that if hereafter, in cases where 

 the limits of the forces are incomparably narrower, we shall, instead of summing 

 the momenta, make use of a mean force determined in a like manner, there is 

 no sensible error to be apprehended. 



8. Hitherto we have considered the body t, round which p revolves, as qui- 

 escent ; and it is thus that authors have always considered it : though the case in 

 nature, to which they meant to apply Sir Isaac Newton's rule, is widely different. 

 The earth and moon revolve about their common centre of gravity; their dis 

 tances from which being inversely as their masses, and the forces, by which either 

 is attracted by the other, as also the forces of the sun to disturb their motions, 

 being in the same ratio; it follows that the earth, in her motion round the 

 common centre of gravity, will suffer disturbances every way similar to those of 

 the moon. And the whole motion of the apsis of the moon's orbit, resulting 

 from the two disturbing forces, will be nearly the double of what either of them 

 could produce separately, round a fixed centre. 



9. To determine this, we may conceive the earth as revolving in an orbit al- 

 ready in motion from the sun's disturbing force on the moon; the retrograde 

 motion of the orbit, while the earth moves from c to p, being n X cp; and the 

 direct motion, for the rest of the quadrant, being n X />a ; hence it will follow, 

 that the disturbing force, = k, affects the earth's motion through an arc of her 

 orbit equal to cp X (1 -f- w); and the force — k acts through the arc pA X 

 (1 -f n). And the motions of the apsis being in the same ratios, if r be the 

 regress of the apsis of the moon's orbit (determined as in ^ 6) and p its progress ; 

 the regress of the apsis of the earth's orbit will be r X (1 + ")> and its direct 

 motion, J!j X (1 — n). That is, the whole motions of the apsis, resulting from 

 the sun's action on the earth and moon together, will be (r =) r x (2 + n), 

 and {p =) p X (2 — n); and the motions to be ascribed to either arc, r X 

 (1 4- i-n), and /> X (1 — ^n). — Now p, found as above, being 2082'''.9 and 

 N = .0105707, p is 4143". 8. And the same way, R = 1375".7- whose differ- 

 ence p — R multiplied by 4, that is, 4 X 2768" = IIO72" = 3° 4' 32", is the 

 direct motion of the apsis in a revolution. 



First correction for the moons variation. Fig. Q. — 10. In the foregoing cal- 

 culation, it is supposed, that the moon's orbit is nearly circular, more nearly 

 indeed than it possibly can be, even abstracting from its excentricity. For though 

 the moon had been projected with a direction and force to make her describe 

 a circle round the earth, as eol, the action of the sun would have changed this 



