404 H. 8. Uhler — Deviation Produced hy Prisms. 



positive so that the deviation experiences a minimum value. 

 Since cos (H-^) = cos ( — 17,) there will be two minima at grazing 

 incidence corresponding to © = 0. Moreover, since the cosine 

 of any angle may not exceed unity in magnitude it is evident 

 that these minima will exist when, and only when, tan ft < 2 tan c, 

 that is, simultaneously with the aforementioned maximum in 

 the principal plane. The case where tan ft = 2 tan c requires 



special investigation, for then (-1—?) vanishes. Keeping c or 



\ <*Vi / e= vi =0 

 n constant and allowing ft to vary in the equation cos rj 1 = 

 \ cot c tan ft it is seen that when ft = 0, rj 1 = -| 7r, and that as ft 

 increases, v l decreases and approaches the limit as ft approaches 

 the value tan" 1 (2 tan c). Since two minima are moving in 

 symmetrically towards one maximum of fixed position, it seems 

 reasonable to expect a resultant minimum, of higher order than 

 the first, when the limiting condition tan ft = 2 tan c is fulfilled. 

 By comparatively laborious but elementary analysis it can be 



shown that —=- - vanishes for rj 1 = 0, and that 



d Vi 



when 



4^? = 12 cos 2 p cot 2 p esc D 

 €(rj 1 



® = -iy t . = 0, i. e., 



tan p = 2 tan c (48) 



This result being positive the expectation of a minimum, of 

 higher order than the first, is realized. The expression under 



the radical of ( -^— - ) may also be written 



1 — cot 2 c + 2 cot c cot j8 



hence, (assuming 2 c obtuse so as to permit ft to become obtuse 

 and still satisfy the necessary condition ft<^2 c), as ft increases 

 from a very small value up to tan" 1 (2 tan c), exclusive of this 



limit, ( -1— j- ) maintains a negative value so that a maximum of 

 V^i J Vl = 



deviation persists. When /3 = tan -1 (2 tan c), (which implies 

 /3 acute), it has been shown that the deviation is a minimum. 



The instant ft exceeds tan -1 (2 tan c\ (-y-r) becomes positive 



and remains so while ft increases to ^7r, passes through |7r, and 

 continues obtuse up to the limit 2 c. To sum up, there is one 

 minimum for y 1 = \it when 2 > ft> tan -1 (2 tan c), whereas 

 there are discrete minima and one maximum when 



