242 



Mr. R. J. Sowter on Astigmatic Lenses. 



sented by an ellipse such that the semi-axes are the reciprocals 

 of the square roots of the focal powers, and that a cylindrical 

 lens may be represented by a pair of parallel lines at a dis- 

 tance from an axis equal to the inverse square root of the 

 power of the lens. 



II. Cylindrical Lenses crossed at Right Angles. 



Let two lenses of powers A and B be crossed at right 

 angles, and let the lenses be represented by their natural 

 apertures as in fig. 2. 



Fio-. 2. 





— — \T ~~ 





c 



1 

 1 



1 



B 



Q 



/ ( 



\ 





^\ 1 



^1 1 



P t 



/ I 



\r^ — ^ 





A 1 



\ 

 \ 

 \ 



\ 





• 







Then the lens equivalent to the crossed lenses is one in 

 which the distribution of refracting substance is the same as 

 in the combination. This distribution is determined by the 

 natural aperture for the combination. 



Now taking the lens axes as axes of coordinates, the 

 equation to the curve of equal thickness, for a thickness 

 equal to the greatest thickness of either lens, is 



The natural aperture is therefore an ellipse, and the com- 

 bination is equivalent to an ellipsoidal lens with focal powers 

 A and B. 



In fig. 2 the lenses represented have powers in the ratio 

 9 : 4 and are equivalent to the ellipsoidal lens represented by 

 the ellipse inscribed in the rectangle of which the sides are 

 as 3 : 2. 



