150 



2 TT . u d u 



, , TT . u u 



r -f u, and the expression becomes - - - -f 



jf d f x / &amp;lt;*/ F wit h (r + w) (r M), substituted for /, 

 when f results from this integration. Then let F = 



or the attraction be that of gravitation; the expression 

 2 TT . u d u r* ^ d f 2 TT . u d u 



becomes / fdfxf = &quot;~ x 



r *J J J j fi r 



/ 2 12 TT . u d u (r + u)(r u) _ 



2-7-7- ~2~ 



- - - x u = 2 TT u 1 d u x ; and the coeffi 

 cient of d r, taking the differential with r as the variable, 

 is + - 2 - consequently the attraction is inversely 



as the square of the distance of the particle from the 

 centre of the sphere, and is the same as if the whole sphere 

 were in the centre.* 



f The First Book of the Principia concludes with some 

 propositions respecting the motion of infinitely small bodies 

 through media, which attract or repel them in their course, 

 that is to say, of the rays of light, which, according to the 

 Newtonian doctrine, are supposed to be bodies of this kind, 

 hard and elastic, and moving with such rapidity as to 

 pass through the distance of the sun from the earth, or 

 95 millions of miles, in seven or eight minutes, that is, 

 with a velocity of above 211,000 miles in a second. Sir 

 Isaac Newton shows that, if the medium through which 

 they pass attracts or repels them from the perpendicular 

 uniformly, they describe a parabola, according to Galileo s 

 law of projectiles ; but if the attraction or repulsion be 



* Mec. Ccl. liv. ii. ch. 2. The expression is here developed; but it 

 coincides with the analvsis in 8 11. 



