108 



Mr Larmor, On the absolute Minimum [Feb. 24, 



We may also obtain similar relations connecting Q and R with 

 S. Thus we have 



P 2 : Q* : R? : S* 



— 1, COS l/r, COS /3 

 COS ^, — 1, COS 



cos /3, cos 0, — 1 



— 1, cosy, cos/3 

 cos 7, — 1, cos a 

 cos /3, cos a, — 1 



— 1, cos a, cos <£> 

 cos a, —1, cos^ 

 cos <f>, cos ty, — 1 



— 1, cos #, cos <£ 

 cos 0, — 1, cos 7 

 cos <f>, cos 7, — 1 



(3) On the absolute Minimum of Optical deviation by a Prism. 

 By J. Larmor, M.A., St John's College. 



When a ray of light crosses a prism in a principal plane, it 

 is of course well known, and easy to verify graphically, that the 

 deviation suffered by the ray is least when it crosses the prism 

 symmetrically. It seems also to be recognized that no deviation 

 smaller than this can be obtained when the ray does not pass 

 in a principal plane ; though I have not met with any valid 

 demonstration 1 of this result. The following proof may therefore 

 be worth recording. 



The inclinations rj and ?/ of an incident and refracted ray to 

 any plane normal to the refracting surface, for example their 

 inclinations to the principal plane of the prism, obey the law 

 of sines 



sin 11 = fi sin ?/. 



Hence, after passing across a prism, the emergent ray is in- 

 clined to the refracting edge at the same angle ^ir — rj as was the 

 incident ray. 



When directions are projected on to a spherical surface, let 

 E represent the direction of the edge of the prism, and P, Q 

 those of the incident and emergent rays. Then EP and EQ 

 are each \tt — r\ ; and if the arcs EPp and EQq are each a 

 quadrant, p and q will represent the projections of these rays on the 

 principal plane of the prism. 



1 The proof quoted by Czapski, Treatise, p. 156, from Heath, Treatise, p. 31, 

 does not seem to be valid. 



