VOL. L.} PHII/OSOl^HrCAL TRANSACTIONS. 305 



the motion of the moon's apogee in 100 years will be = l6' nearly in conse- 

 quentia; if it be c = -rhn ^^^^ will e = ,3^]^^^ , and the motion of the 

 apogee = 20'.7. By this quantity then is the mean motion of the moon's apogee 

 to be diminished as determined by observations, to obtain the motion generated 

 by the sun's force. 



Of the Variation of the Earth^s Diurnal Motion, 



If the figure of the earth were quite spherical whatever might be the axis of 

 rotation, the same quantity of motion remaining in the globe, the velocity of 

 rotation would remain the same; but it is otherwise when, by reason of the forces 

 of the sun and moon, the earth takes the form of an oblong spheroid by the 

 ascent of the waters. For here I do not consider the oblate figure of the earth 

 from the redundant matter at the equator, but I suppose it spherical, unless so 

 ar as it is changed to spheroidical by the elevation and the depression of the 

 waters. Now in a spheroid of this kind, though the quantity of motion remain 

 the same, by the transverse axis changing its inclination to the axis of rotation, 

 it is manifest that the velocity of rotation will be changed also ; and since the 

 transverse axis passes always through the sun or moon, it will every moment 

 change its position in respect of the axis of rotation, because of the motion by 

 which these two planets by turns recede from the equator and approach to it. 



Prob. To Investigate the Variation of the Earth's Diurnal Motion arising 

 from the aforesaid Cause. 



Let the oblong spheroid Ancd (fig. 12) represent a fluid earth, whose centre is 

 T, transverse axis ac joining the centres of the earth and sun or moon, nd the 

 less axis, eo the equatorial diameter, and xz the axis of diurnal motion. With 

 the centre t and radius td describe the circle bdc/ cutting the transverse axis ac 

 in B, and draw bk perpendicular to te; then from any point p of the circle 

 having drawn pm perpendicular to the axis xz and cutting ta in h, let vpr be 

 the circumference of the circle which the point p will describe by its diurnal rota- 

 tion, to any point p of which draw Tp, and produce it to meet the spheroidal 

 surface at q\ then demitting j!;g perpendicular on pm, and gp on ta, if through 

 the points a, 9, c be conceived to pass an ellipse similar and equal to the ellipse 

 ADC, from the nature of the curve, and because our spheroid differs but very 

 little from a sphere, it will be p^ = ab X —; very nearly. Now let u denote 

 the velocity of a particle in the earth's equator revolving by the diurnal motion 

 about the axis xz at the distance of the semi-diameter tp, then will 



— — be the velocity of the particle p describing the circle ppr; and since tp = 



^~ "^ X tk -f- th, the motion of the whole lineola pq will he = pq X -^—^ 



X —^LLl X -—^ X TK^ 4- th ; therefore the sum of these motions in 



TP* TP 



VOL. XI. "^ R R 



