of Edinburgh, Session 1867 - 68 . 
241 
Let v x be the distance of the focus of incident light from the 
prism, v. 2 that of the emergent light, and u that within the prism, 
all measured to the right, then by the ordinary formula, 
v x u 
cos 2 <f>i ~ cos 2 #i 
u = 
/ x cos 2 0 2 cos 2 c p 2 
V x COS 2 0 X COS 2 d> 2 = v 2 COS 2 $ 2 COS 2 cf) l . 
Taking the differential coefficient of the logarithms of these quan- 
tities, 
1 dv x _ 2 sin 6 X dO x 2 sin cf>. 2 dcj> 2 1 dv 2 2 sin 0 2 d$ 2 
v x d/x cos 6 X d/x cos <£ 2 d/x ~ v 2 dfx cos dfx 
1 dv x 1 dv 2 2 sin <p, sin a 2 sin 2 0 X 2 sin 0 X sin 
v x dfx v ± dfx ~ cos 6 X cos'- <p, 2 fx cos 2 0 X [x cos 6 X cos 
Substituting for these angles their values in terms of a and 8 we 
find— 
1 dv x 1 dv 2 
v x d/x v, 2 dfx 
4 sin a 
(cos a + cos 8) 2 
— l)sin a — si n 8 (1 + f cos a — 8) 
1 - | fx 2 (l - cos (a - 8)) 
The quantity on the right of this equation is always positive, 
unless the value of 8 exceeds that given by the equation 
(/a 9 - 1 ) sin a = sin 8 ^1 + f cos (a — 8 )) 
If fx - 1-5 and a = 60°, then 8 = 22° 52', that is, the ray within 
the prism makes an angle of 11° 26' with the base, which corre- 
sponds to an angle of incidence 82° 50', or the incident ray is in- 
clined 7° 10' to the face of the prism. 
If two lenses are used, of the same material with the prism, we 
may correct the defects of the prism without turning it so far from 
its position of least deviation. 
Let a be the distance of the slit from the prism, and b that of 
the screen from the prism. Let f x be the focal length of the lens 
between the prism and slit, and / 2 that of the lens between the 
prism and the screen, then the condition of a fiat image is 
2 i 
VOL. VI. 
