of ui'ppiclis in Geometrical Optics. 117 



normal to the surface shall remain in the plane of: the 

 section as the point of incidence moves along the curve 

 of section. As will be seen later, the principal plane 

 selected must be the one in which the chief ray lies. 

 Suppose it to coincide with the plane of the paper, and 

 take the principal sections as two co-ordinate planes. 



Let be the point of incidence of the chief ray, taken as 

 origin, 

 P be the point of incidence of an infinitely near ray. 

 Take the axis of x along the normal at 0, 

 the axis of y in the plane of the paper, 

 the axis of z perpendicular to the paper, 

 the co-ordinates of P as x\ y 1 ', z'. 

 Then the equation of the refracting surface may be 

 written 



2x = by 2 + cz 2 + hig v her terms in y and z, 



b. c behw the curvatures at 0. 



In the neighbourhood of the origin we have 

 b c 



o = — x + y y~ ■*■ 2 z2 — ^' y> z )> sa ^' 



showing that x is a small quantity of the second order. 

 In what follows .small quantities of the second order are 

 considered negligible, hence x = 0. 



The direction cosines of the normal at a point x',y / ,z / 

 near are proportional to 



dF_ clF _ , dF_ , 



dx'~ ' dy'-^' dz'~ C '' 



Let the direction cosines of the chief ray at incidence be 



[cos-v/r, smyfr, 0], 

 and the direction cosines of the incident ray at P be 



[L; M, N], 



and use dashed letters to denote the same quantities after 

 refraction. 



The direction cosines of the normal at are [ — 1, 0, 0] 



and „ „ „ „ „ P „ [ — l,by',cz']. 



The general equations of refraction give, denoting the 

 refractive indices by /ll, fju\ 



fi'L'-fiL _ fi 'W-fiM _ //N '-yaN 



"=J— " by' - cz' ' ' ' W 



