H. S. Uhler — Deviation Produced by Prisms. 399 



consequently the coefficient of x~ is essentially positive, as well 

 as the coefficient of y*. When E — e , 



sin 3 c sin 2 i(E + (3)- sin 3 £/3 = 0, 



by formula (19), hence the conic degenerates to the point 

 x = 0, y = 0. For larger permissible values of E, 



sin 2 c sin 2 i(E + /?) — sin 2 \fi 



is positive, thus giving a true ellipse and establishing the 

 theorem stated above. Furthermore, the coefficient of x" is 

 obviously greater than the coefficient of y% therefore the longer 

 axis of the ellipse lies along the y-axis, i. e., in the plane of 

 minima. A consideration of the way in which E increases 

 along a small-circle MY gives rise to the expectation of a cone 

 of a general oval base, but the simplicity of the result was not 

 anticipated. Conic sections were not found, by the writer, to 

 pertain to the other rays under the various conditions which 

 were tried. 



Now that the more important properties of the Xrj/ D sur- 

 face have been presented, it is appropriate to discuss the y x r) x D 

 locus. The problem of the internal ray was taken up first 

 because of the simplicity and symmetry of the results obtained. 

 In spite of the comparatively complicated nature of the 

 7^,7) surface it is the more interesting because it shows the 

 dependence of the general deviation upon the azimuth and 

 altitude of the incident ray. This added interest rests partly 

 on the fact that it seems natural to consider the incident ray 

 in all permissible positions, around a fixed point of incidence, 

 and to investigate the corresponding values of the deviation. 



From (2), 



cos E= cos (y l —P) cos y/ -f sin (y, — /8) sin y/. 



From (1) and (6), 



sin y/ = v sin (y/— (3), 



hence, using (5) and observing that y/^iir, it will be found 

 that 



sin y 2 ' = cos /3 sin y x — sin /3 V v 2 — sin 2 y, . 



Since 7 2 '>1tt, 



cos y 2 '= + Vl- sin 2 y/, 



therefore 



cos E= sin (y l — (3) [cos (3 sin y 1 — sin f3 V v 2 — sin 2 yj 



-t-cos (y x — (3)yi — [cos (3 sin 7l — sin /Vv 2 — sin a yj 2 (33) 



