On Ellipsoidal Lenses. 



181 



to the focal lines by #, then by resolving along two directions 

 at right angles and adopting the curvature notation, we get : — 



-(Ucos^ + Vsin^J+U'cos'^ + V'sin^^A . (i.) 

 -(Usin^ + Vcos^J + U'sin'^ + V'cos^^B . (ii.) 



he 



re 



u= 



1 



v.* 



V 



and U', V, §' define the focal lines in the refracted pencil. 

 Fig. 1 shows the orientation of the powers, or principal 

 curvatures, with respect to each other. Equations (i.) and 

 (ii.) give : — 



-(U + V)+U' + V'=A + B . . . (I.) 



_(U-V) cos 20 + (U'- V) cos 26 f = A-B . (II.) 



This last equation shows that the power parallelogram oabc, 

 shown in fig. 2, has its sides equal to (U— V), (A — B), its 



Fig. 2. 



angle 20, the diagonal (U' — V), and the inclination of the 

 diagonal 20' '. The parallelogram affords the equation . — 



U-V sin W K ' } 



Equations I., II., III., or the parallelogram oabc together 

 with the simple equation (I.), give a complete solution. The 

 signs to be adopted are those that are usual, and such as to 



make -(U+\0 + U' + V' = A + B. 



The usual lengthy solution by the use of the characteristic 

 function resolves itself into that given by Herman* employ- 

 ing Malus' theorem. 



If a parallel pencil fall upon the first lens of a crossed and 

 separated cylindrical combination, the equations (i.) (ii.) are 



-Ucos 2 + U' cos 2 fl'-fV'sin 2 ; = A, 



- U sin 2 6 + U' sin 2 6 f + V x cos 2 6' = 0, 



* Geom. Optics, § 174. 



