On the General Laws of Brightness of Images. 217 



the second is the same as that of the second as seen from the 

 first. 



Proof. 

 Let T denote the time of propagation of light from a point 

 xy z to a point % tj £, or from the latter point to the former. 

 If 8s be an indefinitely short line drawn from the point xy z 

 at an inclination 6 to the forward direction of a ray from £ t] £", 

 the increase in T will obviously be the same as for a line 

 8s cos 6 drawn along the ray. Hence, if v denote the velocity 

 of light at xy z, and /x the absolute index of refraction at the 



same point, we have dT=-8s cos 0, or 



dT _ cos 9 _ jju cos 6 



Ts~~^r~ v ' w 



where V denotes the velocity in vacuo. 



Take the centre of the small area A L as origin of the rectan- 

 gular coordinates x y z, and the centre of A 2 as origin of £ 97 £". 

 Let the axes of z and £ be tangential to the ray, and reckoned 

 positive in the outward directions. The axes of xy and of £17 

 will therefore be in the planes of Aj and A 2 respectively, and 

 we shall choose them so as to satisfy the two conditions 



E]=o. [SH ■:■•<*> 



the brackets signifying that the six variables x, y, z, £, 17, £ are 

 to be put equal to zero after the performance of the differ- 

 entiations. 



To express the solid angle formed at the centre of A x by 

 rays from the circumference of A 2 , suppose tangents of unit 

 length, forming prolongations of the rays, to be drawn from 

 the centre of A 1# The solid angle (being small) will be 

 numerically equal to the base of the cone or pyramid enclosed 

 by the tangents, or to the projection of this base on the plane 

 of xy. Let x' and y' be the coordinates of the projection of 

 the extremity of one of these unit-tangents, then the solid 

 angle will be equal to the area traced by the point x'y' in the 

 plane of A 1? while the point £ rj, from which the ray proceeds, 

 moves round the circumference of A 3 . The coordinates x' 

 and y' are obviously equal to the cosines of the angles which 

 the ray makes with the axes of x and y. 



Hence, putting cos 6 in (2) successively equal to x' and y' } 

 we have 



^ = V^T , = V^T . 



fa denoting the value of fju at the centre of A v 



