H. S. TJhler — Deviation Produced by Prisms. 409 



32' 6", a 2 = 21° 18' 17", and a/ = 33° 1' 26"._ The a's may 

 be calculated by the aid of the following relations : 



COS a, = COS y x COS rj 1 (52) 



COS a 2 ' = COS y,' COS 77, (53) 



sin a/ = sin c sin a T (54) 



sin a 2 = sin c sin a 2 ' (55) 



^=^' = 30° 51' 57", 11/ = 20*, 7: = 49° 18' 2", 7/ '== -27° 

 29' 4ti", 7, = 7° 29' 46", 7/ = 12° 22' 12", D = 14° 31' 7", 

 and .# = 16° 55' 50". The absolute minimum of deviation, 

 which is attained when S coincides with L, e = 10° 11/ 46". 

 The partial minimum corresponding to the altitude LM, when 

 S falls on M, D = Z Q OP = 11° 16' 7". The associated 

 deviation for the projected ray, E, = 13° 8' 8". The devia- 

 tion at grazing incidence or emergence in a principal plane, 

 when P coincides with K or K, B x = 36° V 50". ZN x OK = 

 ZN x OT = 90°, and ZN.OR = -33° 52 / 10". The deviation 

 for simultaneous grazing incidence and emergence, when P 

 and S coincide respectively with T and C, B,=22° 23' 22". 

 ZKOT = 78° 37' 49". When referred to rectangular frames 

 the small-circle arcs C'AC and C'BC have been called respec- 

 tively the right and left boundary loci of the 'h.rj/B surface. 

 In like manner, the great-circle arc T'KT and the curve T'RT 

 have been designated as the right and left boundary loci of 

 the 7 1 ?? 1 i? surface, in the order named. T' being the point 

 homologous to T. The course of the curve TR does not show 

 very well in the photograph. When viewed from above, in 



-> 

 the direction HO, it is seen to recede more and more from H 

 as the tracing point starts from T and moves towards R. 



Figures 2 to 5, inclusive, show the projections on the Xrj/ 

 coordinate plane of certain constant deviation sections of the 

 X^/Z* surfaces. Only the first quadrant is plotted for each 

 locus since all of the sections are symmetrical with respect to 

 both of the axes OX and Or)/. The complete boundary loci of 

 the Xri/B surfaces are cylinders whose generating lines are 

 parallel to the axes OD and hence perpendicular to the plane 

 of the paper. For this reason any one quadrant of each 

 boundary locus projects on the X77/ plane as a single curve 

 which may be intersected by any number of loci of constant 

 deviation. The boundary loci arise from the physical con- 

 dition that total reflection is to be excluded. There is nothing 

 inherent in the mathematical functions to prevent the loci of 

 constant deviation, or any other sections of the Xrj/B surfaces 

 which intersect the boundary loci, from being plotted contin- 

 uously so as to include points in the region outside of the 



