SHEWING IT TO BELONG TO PHYSICAL OPTICS. 145 



D^ . 07-5'= supplement of the angle of deviation of any ray (this 

 angle is frequently itself called, incorrectly, the deviation) 

 0„, and A, the values of and B corresponding to the minimum 

 deviation, or the maximum of its supplement D. 

 Let O be the centre of the sphere and origin of polar co-ordinates, 



^ POT=e; 

 let QO^rbe the ray incident perpendicularly on the sphere, 

 cjPSjjT any other ray emergent at aS", 

 OS= radius = r, 

 also z OST= /. of incidence. 



= 0, by property of a refracting sphere; 

 let also z OPT=y = Tr-{D + 6), 



then Op = jo. sin 7 = r. sin 0; 



r .sin (b 



■ ■ P = — ^ J-. 



sin y 



Now when the point P is the intersection of consecutive rays, « and 6 

 remain constant, whilst (p and 7 vary; 



.._ f) _ y sin 7 cos - sin cos yd^y 

 sin'' 7 ' 



and d^y=-d^D; 

 .". tan 7 = tan (pd^y, 



or tan (D + 0) = tan 0rf^Z) (jj 



But as shewn in elementary treatises on Optics, 



Z) = 2(20'-0), 

 also we have sin (p = /i. sin </>' ; 



.: d^D = 2{2d^<p'-l) 



\n ' cos (p' 

 Vol. VI. Part I. T 



_ /2 cos d> \ 



=H-,-^-') (2). 



