VOL. XLVII.] PHILOSOPHICAL TRANSACTIONS. 141 



orbit into an oval, as oadbc; whose greatest diameter, passing through the qua- 

 dratures CD, is to the least, as 70Vt to 69^. The reason and determination of 

 which we have in Princip. lib. iii, prop. 26, 28. 



11. That this action of the sun, and the figure resulting from it, must lessen 

 the mean motion of the a{X)gee, is easily shown. For let p be the moon's 

 place in her orbit, w hen the apsis is stationary ; and eol the circle of her mean 

 motion, cutting the orbit very near tne octant o, and pt in 0; then the accele- 

 rating forces of the earth at p and o, being inversely as the squares of pt and ot, 

 and the sun's disturbing force at the points p, 0, being in the simple direct ratio 

 of the same lines; ot being given, the ratio of the sun's disturbing force at the 

 point p, to the earth's accelerating force at the same point, that is, the quantity 

 c in the theorem, will be as the cube of the distance pt; and, a fortiori, in 

 every point of the orbit, from the quadrature c to p, will exceed the mean force 

 at o, and its effect in producing a retrograde motion of the apsis will be greater. 



For the remaining part of the quadrant, where the motion of the apsis is 

 direct, the force c is indeed greater than its mean quantity from p to o; but, 

 through the whole octant oa, it is continually decreasing as the cube of the dis- 

 tance from T ; hence, on the whole, that force, and its effect, from p to a, fall 

 short of their mean quantities at o. Seeing therefore the direct motion is dimi- 

 nished, and the retrograde increased; their difference, that is, the direct motion 

 in the quadrant cpa will be diminished. 



But this mean motion will be diminished somewhat also from the inequable 

 description of the areas (in prop. 26, lib. iii) ; on which account, the cubes of 

 the distance pt must be every where increased, or diminished, in the duplicate 

 ratio of the moments of time in which a given small angle is described, to the 

 mean moment at the octant.* 



12. By computing from these principles, it will be found: 1. That the angle 

 CTP, which was of 35° 15' 52* in the circle, will, ift the oval orbit, be dimi- 

 nished to 34° 43' 34". 2. That the ratio of the mean of the cubes of the 

 moon's distances in the arc cp, to the cube of the mean distance, will be ex- 

 pressed by 1.0239] 6 {= g), and the like ratio, in the arc pa, by .9852467 

 (= h). 3. Multiplying therefore the forces k and — k, found in ^6, hy g and 



• To express the distance pt by « the sine of the angle ctp, in an ellipsis not very eccentric; 

 from any point p draw pk an ordinate to the axis cd, and meeting the circumscribed circle in m; 

 draw likewise m/' perpendicular to tp produced. Then putting to = 1, ta = d, 

 ■ ' = <i by conjoining the ratios of tp to pk, pk to pm, pm to pf, it will he tp = — 2_ : in 



which, for the variable numerator rf, we might, because of the smallness of the angle ptm, write 

 unity; but taking it rather of its mean quantity tn (= -9999^7 in the moon's orbit) the distances, 

 whose cubes are to be summed, will be ~ — ;. And the ratio of the moments of time to the mean 

 moment, is that of 1 10.23 to 109.73 + *», by prop. 26, lib, iii.— Orig. '» 



