;20 PHILOSOPHICAL TRANSACTIONS. [aNNO 1755. 



with which the sun urges the particle s towards the centre t, then 3sm will de- 

 note the force urging the same particle from the plane perpendicular to the 



plane of the ecliptic, therefore will denote the force pn disturbing the 



situation of the equatorial plane, and its effect to turn the equator is as pn x st, 

 that is, as the force pn itself. But the force st is to the force by which the 

 earth is retained in its orbit about the sun, as the earth's radius st is to the 

 earth's distance from the sun ; and the force by which the earth is retained in its 

 orbit, is to the centrifugal force at the earth's equator, in the ratio compounded 

 of the direct ratio of the earth's distance from the sun to the earth's radius, and 

 of the inverse duplicate ratio of the earth's periodic time about the sun to the 

 time about it sown axis : hence by composition of forces, writing s for the earth's 

 annual periodic time, and t for its diurnal, it will produce the force pn to the cen- 



3SM X MT t^ 



trifugal force at the earth's equator, as j — X -, to 1. And it appears that 



this is the efficacious force pn to turn the equator about the axis perpendicular to 

 the plane tasqb, that is, about the axis which lies in the common section of the 

 equator, and the plane qt perpendicular to the ecliptic. 



At equal distances from the point s, in the circumference of the equator, take 

 the two points f ; then because the force of each of these to turn the equator 

 about the axis perpendicular respectively to the plane tpb, the effect of both 

 forces compounded concurs with the said force pn, for turning the equator with 

 the earth about the axis perpendicular to the plane tasq. Draw the perpendi- 

 culars FG in the plane qt perpendicular to the ecliptic ; then the sum of the 

 forces by which these two particles fly from the plane of the equator, will be 

 6rG x_GT^^ as appears from above, of which that part which conspires with the 



said force pn, since it is to as ft to st, will be ^^ ^ "^ (the remain- 

 ing parts of those forces, because opposites, mutually destroying each other ;) 

 or, from the similitude of the triangles fgt and smt, this sum will be to the 

 force PN as 2ft^ to st*; and therefore, since the sum of all the ft^ through the 

 whole circumference, is half the sum of all the st^, the sum of all the actions 

 through the circuit of the equator, is half on the particle s : therefore that force 

 by which the jx)sition of the equatorial circle is disturbed, collected from the 

 forces of all the points in that circumference, is to the centrifugal force at the same 

 equator, putting the earth's radius st = 1, as — —^UU x ^ to 1 . q. e. i. 



Lemma 2. The force of all the particles without the interior globe of the 

 earth, viz. except that whose diameter is less than the earth's axis, on all sides, 

 to turn the earth about the side axis, is to the force of all the particles uniformly 

 disposed in the equatorial circle after the manner of a ring, for turning the earth 

 about the same axis, as 2 to 5. As is clearly proved by Newton. 



