FOR A NARROW BEAM OF LIGHT. 



385 



as in the Proceedings of the London Mathematical Society, Vol. iv., p. 337 (1873); 

 where A, B, C are lines from which, by the construction there given, a and b, 

 the co-ordinates of the focal lines, and <j>, the angle which the line a makes 

 with the plane xz, may be deduced. 



The quantities a 1} &c., which occur as the coefficients of the characteristic 

 function, are the reciprocals of lines. 



Let 



also let 



Jl] O^j 



1 1 



" T ' & = "2 + -R- > y, = 

 A t S, 



(15). 



The condition that the incident pencil defined by A lt B lt C l should be 

 conjugate to the emergent pencil defined by A,,, B.,, C t , is that the value of 

 the characteristic function must be the same for all rays of the pencil. Now 

 a particular ray of the pencil may be defined either by the co-ordinates aj,, y l} z t 

 of a point through which it passes in the first medium, or by ox,, y 2 , z 2 those of 

 a point through which it passes in the second medium. In the first case, the 

 coefficients of x*, a^y,, and y* will vanish, and in the second those of a;/, x^. 

 and y,\ If the one set of conditions is fulfilled, the other set will be fulfilled 

 also. Hence we may write the conditions either in the form 



or in the form 

 where 



da, 



-i- 

 da, 



A= 



dy 



,(16), 



A - cw 5 + 20,7,7-5 - 



2y,y 2 (ps + 2 r ) 

 (ps- 2 r) 2 





VOL. n. 



...(19). 

 49 





