I7'i 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1757. 



mentum of an equal quantity of matter placed in the circumference of the equa- 

 tor, as is very easy to demonstrate. 



But to show now the use and application of the general proportion here de- 

 rived, in determining the regress of the c^ quinoctial points of the terrestrial sphe 

 roid, let aeap (fig. 2) be the equator, 

 and p/> the axis of the spheroid : also 

 let HECF represent the plane of the 

 ecliptic, s the place of the sun, and 

 HAPNH the plane of the sun's declina- h 

 tion, making right angles with the 

 plane of the equator aeof : then, it 

 ak be supposed parallel, and okm per- 

 pendicular, to OS, and there be as- 

 sumed T and t to express the respective times of the annual and diurnal revolu- 

 tions of the earth, it will appear (from the Principia, b. iii. prop. 25) that the 

 force with which a particle of matter at a tends to recede from the line om, in 



consequence of the sun's attraction, will be expressed by - X — Xf;f denot- 

 ing the centrifugal force of the same particle arising from the diurnal rotation. 



3it A IC OK 



Hence, by the resolution of forces, — x — X — X f will be the effect of 



' •' TT OA OA «^ 



that particle, in a direction perpendicular to oa, to turn the earth about its 

 centre o. 



But it is demonstrated by Sir Isaac Newton, and by other authors, that the 

 force of all the particles, or of all the matter in the whole spheroid A¥ap, to turn 

 it about its centre, is equal to -l of the force of a quantity of matter, placed at a, 

 equal to the excess of the matter in the whole spheroid above that in the in- 

 scribed sphere whose axis is pp. Now this excess (assuming the ratio of tt to 1 

 to express that of the area of a circle to the square of the radius) will be truly 



4t 



represented by - x op X (oa* — op^) ; and consequently the force of all the 



3tt 



AK 



4~ 



matter in the whole earth, by — x — X — X rr X op x (oa*^— op^). Let 



OA 



OA 



15 



therefore this quantity be now substituted for p, in the general formula 



X OA^ X OP, and 4-, instead of their equals c and«; by which 



Put the given quan- 



— -, writing also 



3« 



OA* — OP* 



OA' 



X 



AK X OK 



OA' 



means we have here — r. = -^— X 



ncj 2tt 



<3// OA^ ^— OP* 



tity -— X — -^ — = ^ ; and let the angle eac represent the horary alteration 



2tt 



of the position of the terrestrial equator, arising from the force f here deter- 

 nined, and let the arch Ee be the corresponding regress of the equinoctial point 

 K : then, in the triangle EAe (considered as spherical) it will be sin. e : sin. ab 



