304 PHILOSOPHICAL TRANSACTIONS. [aNNO 1787. 



3. Let V be the centre of gyration, and put m = the quantity of matter in 

 the earth : then the effect of the inertia of m placed at v, to oppose the com- 

 munication of motion, is the same as the effect of the inertia of the earth; and 

 hence, by the property of that centre, et^ :tv- (= ^et^) :: m : am, which is 

 the quantity of matter to be placed at e to have the same effect. 



4. Put m = the excess of the quantity of matter in the earth above that of its 

 inscribed sphere. Now by Sir Isaac Newton's first 2 lemmas, it appears, that 

 the action of the sun on the shell m of matter, to generate an angular velocity 

 about an axis perpendicular to cabd, is just the same as it would be to generate 

 an angular velocity in a quantity of matter equal to J-n placed at e. Let us 

 therefore suppose the sun's attraction, perpendicular to et, to be exerted oh a 

 quantity of matter at e equal to ^m, and at the same time to have a quantity of 

 matter to move equal to \m ; and then from this and art. 3 it appears, that the 

 eff'ect will be the same as the accelerative force of the sun to turn about the 

 earth. Hence that accelerative force is, from art. 2, equal to 



3EK X KT X P^ X 4-?« 3eK X KT X P* X ?« •*. U • -4. TVT -c 



•^ X i^i xy ^ 3-_ __ ^ ffravity bemg unity. Now, if te : tp 



:: J : 1 — r, then m : m — m v. \ \ \ — 2?*, therefore, m. : m :: I : 2r, hence 

 — = r, consequently the accelerative force = ^ — ^. 



5. Let i = the arc described by a point of the equator about its axis in an 

 indefinitely small given time, which may therefore represent its velocity ; and let 

 az represent the arc described in the same time by a body revolving about the 

 earth at its surface; then — - = the sagitta of the arc described by the body in 

 the same time, and consequently a^z^ = the velocity generated by gravity while 

 a point of the equator describes i. Hence, by art. 4, we have 1 : 



3ek X kt X p^ X /• 2-2 3ek X KT X ;?^ X r X a*i^ ,, i -4. r ..i • .. r i 

 ^ — :: a'z^ : ^^ the velocity of the point e of the 



equator generated by the action of the sun, while the equator describes z about 



its axis ; consequently the ratio of these velocities is as x kt x ?> x az ^ ^ ^ 



6. Let i/ be an arc described by the sun in the ecliptic to a radius equal to 

 unity, while a point of the equator describes z about its axis; then (as ap = the 

 time of the eath's rotation, and the arcs described in equal times to equal radii 

 are inversely as the periodic times) - '. — v. y : z =. — ; hence, if v and w be put 

 for the sine and cosine of the sun's declination, the ratio of the velocities in the 



last article becomes - - : 1 . 

 p 



7. Hence if tsl (fig. 2) be the ecliptic to the radius unity, p the plane of 

 the sun, ser the equator, pe the sun's declination, and we take ec : cd (cd being 



perpendicular to ec) :: 1 : JPIl^^ and through rf, e, describe the great circle 



