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



The intercepts of a constant deviation locus on the axes of \ 

 and 77/ are given respectively by (21) or (25) and 



cos 77/ = (cos 2 c sin 2 -J/3 + sin 2 c sin 2 -JZ)) esc c esc p esc ^D. (29) 



Defining f by the equation 



tan £ = tan c esc iP sin -j-Z), (30) 



the last equation changes to the logarithmic form 



cos 77/= cose sec -J/Scsc 2£. (31) 



Finally, the loci of constant deviation intersect both coordinate 

 axes at right angles. 



Before passing to the discussion of the properties of the 

 <U)J) surface, a proof of a theorem relating to the internal 

 ray (OS in figure 1) will be outlined. This theorem seems to 

 be new, and it may be stated thus : " When the deviation of 

 the projected ray is kept constant, the internal ray generates a 

 right cone with an elliptical base" In figure 1, let a plane be 

 drawn tangent to the sphere of reference at the point L. 

 Also, let L be taken as origin of a system of rectangular 

 coordinates. The axis of x shall be the intersection of the 

 principal plane with the tangent plane. The y-axis shall be 

 the line common to the plane of minima, LOH, and the tan- 

 gent plane. Finally, let a denote the radius of the sphere. 

 It can now be shown that 



and 



where 8 



Substituting in equation (20), and performing elementary 

 transformations, it results that 



sin 2 i (E+P) [cos 2 c cos 2 £( E±/3) + sin } Es'm (£+£ E)]x 2 



+ cos 2 csin 2 i(^+ i 8) GOB*i(E+p)y % =z 



[sm 2 csm' i i(E+/3)- sin 2 iff] cos 2 i(E+P)cf (32) 

 Now 



E + fi = 7l - y 2 ' 



and I y x | ^ fa, | y„' | ^ fa, y 1 — y a ' < tt 



hence E + j3 < tt, 



for all values of Z? having physical significance. A fortiori 



p + !#<7T, 





v 2 



sin 



2 A = 













v 2 



cos 2 



\ = 



a 2 

 : S 2 



> 







(* 2 



+ 



a 2 ) 



sin 2 



c — 



2/' 



cos 2 



c. 



