306 THILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



the circuit of the circle vpr, that is, the motion of the superficies contained be- 

 tween the circle vpr and the spheroid, in the direction t/j, will be equal to the 

 circumference of this circle drawn into 



W X AB X PM ^^ ,TK- X PM' , TK^ X HM^ 2TK X HM X TH , o\ i 



— ^ X (--2TP^ ' 7F ~iv •■ "^^^ )' ^^' because 



HM : TM :: tk : bk, and th : hm :: tp : tk, by writing d for the circumference of 

 the circle BDrf, that motion will be = — —-^ — X (tk^ X pm'* -|- 2bk^ X tm^ 

 X PM"^). Then the sum of these motions in the whole circuit of the globe 

 collected, that is, the motion of the whole matter incumbent on the globe bdc?, 

 will be = "L^—^— X ^— ,— . When the planet is in the plane of the 



equator, make bk = 0, and then the aforesaid motion is = — — ^^^-^ — . But 

 the motion of the globe aPR about the same axis, it is easily demonstrated, is 

 i2i.IIiLl\ therefore the motion of the whole earth is t2LZ!JiJL ^ UL^^2L± 



X ~ ' . ~—- J which, since it must remain always the same, making v denote 



the velocity in the superficies of the terrestrial equator when the planet is in the 

 plane of the equator, it will then be 



K X TP X </% , MX 3aB X rf* tt X TP X rf' , « X AB X rf* ^ 3TP* — BK* , 



16 + '3-2 = 16 + -—-32 ^ tT^— ' h^"^^^ 



writing I for tp since it is the radius to the sine bk of the angle btk, it will be 

 V : u :: tp + 4ab — -i-ab X bk"^ : tp -j- 4ab; and hence, because the altitude ab 

 is extremely small in respect of the semidiameter tp, u — vivv.ab X bk^ : ^tp, 

 and u — v=v y. — . But for v it is evident may be written the mean an- 

 gular velocity of the earth, because differing from it by a very small quantity and 



* -p W TJ \C° 



drawn into the small quantity — , and because the times of the earth's 



revolutions about its centre are reciprocally as the angular motions w, v, then the 

 difference of the earth's revolutions between when the planet is in the equator 

 and when distant from it by the angle btk, is = 23*' SG'" x •^-^— . Because 

 then the horary acceleration is to the mean horary motion of the earth about its 

 centre, as ab x bk'^ is to 2tp, or (because the sine p of the inclination of the 

 ecliptic to the equator is to radius 1, as the sine bk is to the sine of the planet's 

 distance from the equinox, which sine call k), as ab x p' x k'^ to 2tp ; therefore 

 the horary acceleration of the earth's rotation increases in the duplicate ratio of 

 the sine of the planet's distance from the equinoctial point; and the sum of all 

 those accelerations, while the planet passes from the equinox to the solstice, is 

 to the sum of as many mean horary motions, that is, the total acceleration ge- 

 nerated in that time, is to that time, as the sum of all the quantities ab x /j^ x k"^ 

 in a quadrant of the circle, is to the sum of as many 2tp ; that is, because the 

 sum of all the h^ in a quadrant of the circle is half the sum of as many squares 



