240 



Mr. R. J. Sowter on Astigmatic Lenses. 



direction can be represented by the inverse square of a length, 

 that length being the semi-chord for the surface in the 

 direction considered. Lens properties may be established by 

 a consideration merely of the arrangement or distribution of 

 refracting material in the lens — by a materialistic method one 

 might say — and this material distribution is determined or 

 indicated by contours of equal thickness. A curve drawn 

 through all points on a lens where the material thickness is 

 equal or constant, maybe said to determine a natural aperture 

 for the lens. From the natural aperture of a lens the power 

 in any direction of the lens is at once determined, for the 

 power is as the inverse square of the radius vector from the 

 centre of the aperture to the edge in the given direction. 



An ellipse is the natural aperture for an ellipsoidal lens, 

 and may be taken to represent the ellipsoidal lens. 



In the ellipse shown in fig. 1, the semi-axes are a and b, 



Fig. 1. 



/I 



and this ellipse is the natural aperture for an ellipsoidal lens 

 with focal powers, A and B say, where 



A- 1 



B: 



w 



The power of the lens in the direction OR (r, <f>) is 

 R=- 2 =Acos 2 </> + Bsin 2 <£, 



since 



1 cos 2 (f> , sin 2 d>. 



+ "" T in the ellipse. 



It is obvious that the sum of the powers in any two directions 



