VOL. LXIX.] PHILOSOPHICAL TKANSACTIONS. 57g 



resolution of motion tm or pl will represent the perturbing force of the sun on 

 the same particle. By the construction sl : sk :: sk' : sp", and by division 

 KL : SK :: sk^ — sp' : sp^ :: (sK + sp) X pk : sp', and pl or tm is nearly equal to 

 3pk, and as 3pk is to sk or st, so is the space described by p in any small time 

 in the direction pk, to the space described in the same time by the centre of the 

 earth in consequence of the sun's attraction. This last space is equal to 



— , where z represents the arc described by the earth's centre during any small 



motion in its orbit, and the former is equal to — ^~^-- This is the space which 

 would be described by p in the direction pk if the particle was at liberty to move 

 freely. Let us at present suppose that no other particle is disturbed by the sun's 

 attraction except this one, and then proceed to inquire into the effects of this 

 disturbance when p, by its cohesion, communicates a motion to the different 

 parts of the earth, which is further constrained to turn round an axis t, the 

 common intersection of the plane cd and the terrestrial equator. From the 

 laws by which motion is communicated, and the property of the lever, it easily 

 appears, as in the 2d article, that the space through which any particle of the 

 earth's equator is impelled at the greatest distance from the axis t, is to 



3l'Kz' 



— — —, the space which would be described in the same time by any particle at 



liberty, the magnitude of which is represented by p, as p X kt x the radius of 

 the equator, to the sum of all the particles of the earth multiplied into the 

 squares of their respective distances from the said axis. .',.5 ■:^c. 



To compute this sum in the easiest way, and by an approximation, which is 

 quite sufficient when the polar and equatorial diameters differ little from each 

 other; let dpe (fig. 3) be a sphere whose radius is unity, divided into an infinite 

 number of thin cylindrical surfaces, whose bases are the circles naq; it is obvious, 

 that all the particles in any one of these surfaces are at the same distance ca = ;r 

 from the axis of motion perpendicular to the plane of the circle naq. Call ap, 

 y, and a, the area of the circle dpe ; then the fluent of 4A.x^dy, or of — AKx^y^ij, 

 because xx = — yi), gives the sum of all the particles in the sphere multiplied 

 into the squares of the respective distances from the axis. This fluent corrected 

 is equal to -Va, and must now be diminished in the ratio of 1 to 1 — 2/), if we 

 suppose the earth to be an oblate spheroid whose equatorial diameter is to the 

 polar as 1 to 1 — f\ and, lastly, the space described by a particle at the greatest 



distance from the axis is equal to > ^ —^. 



^ i6a X st** X (1 — 2p) 



^ 5. In fig. 4, let pia/)Dk represent the earth orthographically projected on 

 the plane of the solstitial colure, p, />, the poles, ik a lesser circle parallel to the 

 equator, and "eape a sphere described with the polar radius pt: then, since the 

 particles without the globe only are concerned in changing the position of the 



4e 2 



