H. S. Uhler — Deviation Produced by Prisms. 407 



location of the greatest deviation produced by a prism of spec- 

 ified constants. The equation 



cos f /? — cot c sin -£/?, (50) 



which is the necessary and sufficient condition for D 1 = Z> 2 , 

 shows that the deviation at grazing incidence in a principal 

 plane can never equal the deviation at simultaneous grazing 

 incidence and emergence for a prism having an angle greater 

 than -J7r, regardless of the value of the index of refraction. 

 For, let the curves whose equations are y = cos J-/3 and y = 

 cot c sin -J/3 be plotted on the same diagram, with the values of 

 ft as abscissae. As /3 increases from zero, the curve y — cos f/3 

 will commence with the intercept 1 on the y-axis and will have 

 decreasing positive ordinates until it crosses the /3-axis at the 

 intercept /3 = %ir. On the other hand, assuming c given, the 

 curve y = cot c sin -J/3 will begin at the origin and have 

 increasing positive ordinates, as /3 increases, until /3 —tt. 

 Therefore, the abscissa of the point of intersection of these 

 curves must be less than -J-7T, save in the trivial case where 

 cot c = or n = 1. In this extreme case the point of intersec- 

 tion is (-J-7T, 0). It is also evident that the larger the value of 

 cot <?, (or n), the smaller will be the value of /3 corresponding to 

 D 1 — Z> 2 . When c is given and the prism angle which will 

 make the deviations D l and D 2 equal is required, it will be 

 found convenient to transform equation (50) to a function of 

 -J/3 alone. Denoting tan -J/3 by t, the last equation may be 

 reduced to the following cubic 



cot c.t* + 3t 2 + cot c.t —I = (51) 



Equation (51) always has one positive real root because cot c is 

 positive and the constant term is negative. Moreover, it can 

 only have one positive root in accordance with the " Rule of 

 Signs" of Descartes. Since ir > ft "> 0, t must be positive so 

 that there is no ambiguity in the solution of the cubic. Writing 

 the left-hand member of (51) as 



t (1 -f £>) cot c — (1 — 3£ 2 ), 



and observing that t must be positive, it becomes clear that 

 1 — 3f must be positive to make the expression vanish. This 

 amounts to a second proof that {$ must not exceed ^ir in order 

 to make possible the equality of D 1 and D 2 . 



It seems appropriate, at this stage of the discussion, to direct 

 attention to the illustrative figures and numerical data. The 

 chalk lines reproduced in fig. 1 represent roughly the follow- 

 ing theoretical values : fi = 20°, n — 1*5, c = cot -1 Vn 2 — 1 

 =Z ^OA = ZNfiB = 41° 48 7 37", ^ = 55° 57 r 45", a/= 33° 



