142 PHILOSOPHICAL TRANSACTIONS. [aNNO 1751. 



by h, substituting the products for c, in the formula, with the arcs cn, and ng, 

 respectively, and finishing the operation as for the circle, the regress in a peri- 

 odical month will be 5548".3, and the progress l6489''.8: whose difference is the 

 direct mean motion sought, 3° 1\" 24-'. 



13. But nearly the same conclusion maybe obtained, and with much less 

 trouble, as follows: In the circle cgd, take cm = 35° 15' 51", and through p, 

 the point where mk, perpendicular to tc, cuts the orbit, draw tpn meeting the 

 circle in n. Then, if r be the regress of the apsis in a circular orbit, 



Ry/ — will be the regress in the oval cpa. 



In like manner, having inscribed in the orbit the circle aw//, and made a 

 similar construction for the rest of the quadrant, p v^— t will be the direct mo- 

 tion in the oval, p being the direct motion in a circle. 



Thus, the angle of variation mtn being (in Dr. Halley's tables) 33' g", the 

 subduplicate ratio of cm to cn will be I.007927, and that of Am to xh, or of 

 GM to GN, will be .99499, And therefore r (in § 9) will be augmented to 

 1386*.6, and p diminished to 4123"; whose difference, multiplied by 4, gives 

 3° 1' 25f "; exceeding the former only by about A". 



14. The rule is founded in this, that if, from the centre t, a circular arc f/ 

 be described, including in the angle ctn the sector ft/] equal to the elliptic 

 sector CTP, the cube of tp, the radius of this circle, may be taken for the mean 

 of the cubes of the moon's distances in the arc cp. And because the area cpt 

 is to the sector cmt, as pk to km, or as ta to tc ; and to or te is a geome- 

 trical mean between ta and tc, it will easily appear that tf^ : to^ :: cm| : cn|. 

 And that p, found from the tables, being (nearly at least) the stationary point 

 in the oval, if the force h be increased in the sesquiplicate ratio of cm to cn, and 

 the arc cn substituted for a in the formula, we shall, by § 1, find the retrograde 

 motion of the apsis. 



Now when the constant force -f- A is given, the regress r is as the arc a ; and 

 when a is given, and A is but a little augmented, r is proportional to A: in general 

 therefore, if A be but a little augmented, r is as ^ X a. Write q for the regress 

 in the oval, R standing for that in the circle, already found; and it will be q : 

 R :: i X (— )^ X cn : A X cm, or a = r X -/ — , according to the rule. The 



CN CN 



like reasoning for the direct motion. 



Second correction for the excevtricity. Fig. 10. — 15. This equation is incon- 

 siderable, because, though the ratio of the disturbing force, when the moon is 

 at a greater than her mean distance, is more increased than it is diminished in 

 the opposite points of her orbit ; this increase is very nearly compensated by the 

 comparative smallness of the angular velocity. Let ado represent the moon's 

 elliptic orbit, whose centre is c, its axes Aa, Del, the mean excentricity ct, and 



