1896.] 



of Optical deviation by a Prism. 



109 



These projected rays are refracted across the prism in accord- 

 ance with the law of sines, the index being not fi but 



H cos V/cos v), where sin r\ = /j, sin v'. 



If D denote the actual deviation PQ, and d the projected de- 

 viation pq, then, from the isosceles spherical triangle, 



sin ^D = sin \d cos 97. 



Thus, of all incident rays which have the same inclination 

 \tt — t] to the edge of the prism, that one has its projected devia- 

 tion d, and therefore also its true deviation D, least, whose pro- 

 jection passes across the prism symmetrically. This least value 

 is given by the equation 



. 1 . , . . a cos rl . 1 . 

 sm - (d + A) - - sin ~ A, 



2 cos 77 2 



where A is the angle of the prism. 



As ix cos ?//cos r) is greater than fi, it follows that d is greater 

 than D , the minimum deviation for an actual ray passing in 

 the principal plane : but as it is also greater than D, no inference 

 can be drawn in this way as to the relative magnitudes of D 

 and D Q . 



We may however find the absolute minimum of D by com- 

 paring with one another the rays, corresponding to different 

 values of 7], whose projections cross the prism symmetrically. In 

 the annexed spherical diagram E represents the edge, N, N' 



the normals to the faces, I, I' the incident and emergent rays, 

 and R the ray inside the prism. The pole of NN' is E, and 

 EI, EI' are equal ; the symmetry of the projection of the rays 

 on the plane ±VN' requires that ON, ON' shall be also equal, 



