FOR A NARROW BEAM OF LIGHT. 387 



Since OR is measured downwards, Z 3 is negative in the ease represented 

 by the figure. It is manifest that X lt Y lt and Z^ may be found from X,, F 2> 

 Z 2 , by the same process. 



The relation between the quantities A, B, C and X, Y, Z is shewn in 

 Fig. 3. 



Let a be one of the first three or the last three of the ten coefficients of 

 the characteristic function Vpy. 



Since a is the reciprocal of a line, let BP represent the line - . 



GV 



Draw PT perpendicular to BP so that PT is to the unit line as to p. 



Then, if PA=A, the line ATC will cut off from the line EC perpen- 

 dicular to BA a part BC equal to X. For 



-- . 



p A.p p 



In this way X may be found when A is known, or A when X is known. 

 The same method gives the relations between B and Y and between C and Z. 



The geometrical process for finding the focal lines of the emergent pencil, 

 when those of the incident pencil are given, is therefore as follows : 



From the distances a t and &, of the focal lines, and the angle <^ l between 

 the first of them and the plane of x^, deduce, by the method given in a 

 former communication (Vol. iv., p. 337), A lt B lt and Cj*. 



From A lt B lt C t , find, by the construction of Fig. 3, X lt Y lf and Z^, 



From X lt Yj, Z lt find, by the construction of Fig. 2, X t , Y t , and Z 2 . 



From X,, Y,, Z,, find A t , JS 19 C,. 



From AS, B z , (7 5 , find a 2 , 6 2 , <f> 2 . 



Thus far we have been considering the most general case of a pencil 

 passing through any number of media between P and Q, through surfaces of 

 any form, the media before incidence at P and after emergence at Q being 

 isotropic. When some of these media are doubly refracting, there may be two 

 or more emergent pencils corresponding to one incident pencil. Our investigation 

 is applicable only to one of these emergent pencils at a time. Each emergent 

 pencil must be treated separately. 



* [Page 337, Vol. 11. of this Edition.] 



492 



