218 Prof. J. D. Everett on the General Laws 



Again, since si and y' are functions of the two independent 

 variables £ and 77, and vanish with them, we have, to the first 

 order of small quantities, 



But by (4), 



da? V d 2 T da? V d 2 T 



(6) 



d% /jl x dx dg dt] /*! dx drf 

 Thus equations (5) reduce, by the help of (3), to 



*=^b^# y= 7XWdvr ' ' (7) 



Hence x J has a constant ratio to £, and y' a constant ratio to rj. 

 The product of these two ratios will be the ratio of the area 

 traced by the point x'y' to the aiea traced by %r), that is, it 

 will be the ratio of the solid angle cn l to the area A 2 . We 

 have accordingly 



Wi= U"[^|][^rJ A2 (8) 



By applying similar reasoning to rays from the perimeter of 

 A 1 to the centre of A 2 , we shall find (with similar notation) 



M Vr d 2 Tl . Vr d 2 Tl /0 , 



Pas SLS5J^ ^sito* • • • (9) 



/vyr rf2T ~ir ^i A 



^ = uwy Ai ' • • • (io) 



By comparison of (8) and (10) we have 



/*i 2 Ai«i = ^2 a A 2 fi>3 (1) 



The rays considered in the foregoing proof may undergo any 

 amount of either gradual or abrupt bending by refraction, 

 and may be reflected any number of times ; but there must 

 be no abrupt difference between the histories of rays which 

 come from consecutive points. 



We now proceed to apply this theorem to the investigation 

 of brightness. 



The brightness of an object as seen from any point is 

 measured by 



fa cu) 



A denoting a small area at the point, sensibly perpendicular 

 to the rays which reach it from the object, co a small solid 

 angle of arbitrary magnitude, and q the quantity of light 

 from the object which converges to A (and also diverges 



