580 PHILOSOPHICAL TRANSACTIONS. [aNNO 1779. 



axis of rotation, let l represent such a particle situated in the circumference of 



the circle ik; then by the precedin? article its effect will be ^ ^ l mj<_mt x_2_ 

 ■' '^ ° i6a X sT^ X (1— 2;() 



and by the same way of reasoning, when 2 equal particles l, /, are supposed to 



be disturbed by the sun's attraction, the space described by that point of the 



equator, which is at the greatest distance from the axis of rotation, or the 



common intersection of the plane cd and the equator a, will be equal to 



^— ? — ■ : : : and the same argument holds tor every other 



16ST' X A X (1 — 2/)) ° •' 



particle without the sphere. 



The sum of all the l X lm X mt -|- &c. must now be found; and for this 

 purpose Sir Isaac Newton's construction is, perhaps, as convenient as any that 

 has hitherto appeared. In the same figure n« is parallel, and xi/ perpendicular, 

 to CD; take lx = xl, and let m, n, represent the sine and cosine of the angle 

 CTP to the radius unity. It is easy to prove in his way that l X lm X mt + / 

 X lm X mi is equal to 2l X m X « X (lo.'^ — ix"), and the fluent of i^x'- mul- 

 tiplied into the fluxion of the circular arc ^x is easily found in the following 

 manner, without having recourse to tables of fluents, or the methods of conti- 

 nuation. From a known analogy, the fluxion of the arc lot is to the fluxion of 

 its versed sine, as the radius ix of the same circle, to ^^x the right sine, lx 

 nmltiplied into the fluxion of the versed sine is the fluxion of the area of the 

 semicircle l/, and calling ix, y, the fluent of lx'^ multiplied into the fluxion of 



the arc lx is evidently equal to ^, where a still represents the area of a circle 

 whose radius is unity: Ay is equal to the semi-circumference ik, and Ky X Tx* 

 is equal to the fluent of tx'^ multiplied into the same fluxion; and calling tx, v, 

 and substituting for li its equal py, the sum of all the l X lm X mt -|- &c. in 

 the annulus li, is equal to mnpA X {y* — 2?/V). This last quantity multiplied 

 into the fluxion of v, and the fluent taken by the common method, when v is 

 equal to Tp or unity nearly, comes out ^ , and twice this quantity gives the 

 sum of all the l X lm X mt, without the whole sphere vnp, and therefore the 

 space described by a particle of the equator in the circle of the sun's declination, 

 while the centre of the earth is carried through the space i of its orbit, is equal 

 to ^ pmnz^ ^^j ^^^ y^^ supposed to be equal to -^^ , the alteration pro- 



duced by the correction in art. 4, on account of the spheroidical figure of the 

 earth being too inconsiderable to affect the conclusion. 



^ 6. We are to observe, that the space ^~i-, described by that point of the 

 equator which is the intersection of the circle of the sun's declination, is gene- 

 rated by the perpetual attraction of the sun. This attraction may be reckoned 

 constant during the very small time of the earth's describing i in its annual mo- 



