H. S. TJhler — Deviation Produced by Prisms. 423 



and 



dc 





2 



d/3~ 

 d 2 c 



' 1 

 (1 



+ 3 cos 2 /? 7 

 6 sin 2/3 



dp 2 " 



+ 3 cos 2 /?) 2 



exist on the locus. At the origin T ~ == — and -^ = 0, hence the 



The first derivative shows that maxima and minima cannot 



de 1 -, d 2 c 



dr*™ w 



curve has the line OA as the inflectional tangent at this point. 

 For /3 = 60°, g = £ cos" 1 1 = 40° 53' 36" or n = | ^21 = 



1-527525, and ~\ = \\ tan" 1 « = 48° 48' 51". For £ = 90°, 



^ 



c = 90° orn = l, and ~ = 2 ; tan" 1 2 = 63° 26' 6". Also 

 ' dp 



d*c 



at this point — -, vanishes and inflection takes place. For all 

 r dp* 



points between the origin and the point /3 = c = 90° the locus 



is convex towards the axis of abscissas. Thus far the curve in 



question has properties like the line OV. Besides the origin, 



these two loci also intersect at the point fi = 120°, c = 



tan- : ( - \ y/ 3 ) = 139° 6' 24", which is to the right of the maxi- 



dc 

 mum on the locus OY. At this point -— = I and hence the 



dp 



tangent is parallel to the tangent at the point whose abscissa 



is 60°. For j3 = 180°, c = 180°, % = \, and ~ = 0. 



Thus, there is an inflectional tangent at the last point which is 

 parallel to the line OA. 



Let the point where the curve whose equation is tan /3 

 = 2 tan c crosses the line PH of figure 8, be denoted by R. 

 Then in the domain OAQRO there ai*e no discrete minima at 

 grazing incidence or emergence. In the region ORHPO 

 there are two discrete minima, with one maximum in a princi- 

 pal plane, when y 1 = 90° or y/= — 90°. The last area is sub- 

 divided by the curve OY so that a still more detailed classifica- 

 tion is possible. Since a segment of the line UH lies within 

 the region ORMO it is clear that 60° prisms can have discrete 

 minima at grazing incidence, etc. However, for this to be the 

 case, the critical angle must exceed \ cos -1 \ or the index of 

 refraction must be less than \ v/21. 



In conclusion, the author desires to thank Mr. Leigh Page 

 for having verified the analytical work in this paper. 



Sloane Physical Laboratory, Yale University, 

 New Haven, Conn., December, 1912. 



