38-4 



HAMILTON'S CHARACTERISTIC FUNCTION 



where 



=? 



a, % .. 



., 



' 



rfA 



2z,z,A 



=- 



n n 



.c? _ i-i- 



z. 



6,= - 



_Mi_ Mi 

 '-z, z,'A 



.(11). 



This ia the most general form of Hamilton's characteristic function for a pair 

 of points, each of which is near the principal ray. It is a homogeneous 

 function of the second degree in x,, y lt x t> y t , the coefficients of which are 

 functions of z, and z,. 



By turning the axes of re, and y, about z,, and those of <c, and y t about z,, 

 we may get rid of two of the ten terms, and so reduce the expression to 

 eight. 



We may, for example, get rid of c, and c,, and so of Sj and g,; but since, 

 in the theory of pencils having two focal lines, terms may enter which must 

 be added to 6, and ( this transformation is not of much use. 



It is better to begin by getting rid of q and r, by turning x l and y, 

 round z,, through an angle V such that 



and also turning a^ and y t round z,, through an angle #,, such that 



(13). 



For these new axes the values of q and r are reduced to zero. 



As an instance of the use of the characteristic function, let us find the 

 form of the emergent pencil when that of the incident pencil is given. 



The general form of the characteristic function of a pencil, whose axis is the 

 axis of z, is 



