VOL. LX 



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rHILO«OPHICAL TRANSACTIONS. 



75 



the progressive velocity of the earth's centre will be found to be no more, than 

 that with which a body would traverse about 4 of one millionth of one millionth 



Sf^d^ 



the progreattve force which would be impressed on 

 the earth, if all the force of each particle were ef- 

 fective. In what proportion the progressive force 

 is diminished, on account of the various obliquities of 

 impulse, is thus investigated. 



Imagine t to be the earth's centre, and acb 

 to be half a great circle of the earth, perpendicular 

 to that which separates the enlightened and the dark 

 hemisphere, and which shall be called the terminator. 



On the plane of this semicircle, suppose tlie terminator, and its parallels in the enlightened 

 hemisphere, to be projected, into right lines a b, kh, Im, no, kc. which are diameters of the 

 circles respectively, of which they are the projections. The sun's distance may be considered 

 as infinite ; and therefore the rays of light, i. e. the directions of the particles, when they 

 reach the earth's surface , are to be considered as parallel to each other, and all of them perpendicular 

 to the plane of the terminator. Now imagine the whole enlightened hemisphere to be divided into 

 innumerable little zones a b kh, hkml, mlito. Sec. by small circles parallel to the terminator. Let the 

 breadths of these little zones, measured on a great circle passing through the poles of the termina- 

 tor, that is, let the infinitesimal arcs Bk, km, no. Sec. be so proportioned to each other, that perpen- 

 diculars kd, mc, of, &c. being drawn from the extremities of these arcs, to the right line ab, which 

 is the common intersection of the great circle acb, and the plane of the terminator, the infinitesimal 

 segments of that line arf, de, ef. Sec. may be equal. Now imagine the particles of light which fall 

 upon any one of these little zones, for instance, nogp, to meet with no resistance from the earth's 

 surface, but to penetrate the globe, and to pass on without refraction or inflection, in the direction 

 perpendicular to the terminator, till they arrive at the plane of the terminator, and there suppose 

 them to stop, and each to lie still, in the place on which it falls. It is evident that the particles of 

 light that fall upon, and have been supposed to pass through, the spherical zone pqon, will, with their 

 proper interstices, cover that annular space on the plane of the terminator, which is the orthographi- 

 cal projection of the zone pgon, on that plane, and is comprised between the circumferences of circles, 

 of which the right lines Tg and x/'are the radii. Hence the number of the particles of light, which 

 fall upon the evanescent zone pqon, are as that evanescent annular space which they cover, that is, 

 as g/' X the circumference of the circle of which rf is the radius, that i.s, as gf x in the right line 

 rf. But that part of the force of each particle, impinging on the zone pqon, which is perpendicular 

 to the surface of the zone, is as of, if tb (the semidiameter of the earth) be put for the whole force. 

 For join to, and draw /k perpendicular to to. The particle impinging at o moves in the direction 

 of. Let the right line q/'then express its whole force, and this force o/'is composed of the two ok, 

 k/', of which OK is perpendicular to the surface of the sphere ato, and K^is parallel to it. But ok : 

 of=of : OT or tb. Again, through k, draw Kt perpendicular to of. The force ok is resolved into two 

 ot, tk, of which ot is perpendicular to the plane of the terminator, and is the only part of the force 

 ok, which tends to produce a progressive motion of the globe, in the direction of the impinging par- 

 ticles, that is, directly from the sun. The other part tv. urges the centre of the globe along the 

 plane of the terminator; but the forces tiL being equal and contrary on opposite sides of, and at equal 

 distances from the perpendicular ray, destroy each other's effects. Now it has been shown, that the 

 whole force of the particle impinging at o, is to that part of its force which is perpendicular to the 

 earth's surface at o, as tb to of. And it is manifest that ok is to ot, that is, the perpendicular force, 

 of the particle impinging at 0, is to that part of it which it eflt;ctive in moving the earth's centre, ai 



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