Mr. R. J. Sowter on Astigmatic Lenses. 243 



The power of the combination in any direction OQ at an 



angle (j> to OA is 7*02 wnere OR is the radius vector of the 



inscribed ellipse, and is also equal to A cos 2 <£ + B sin 2 </>. 



If the lenses are equal, A = B, the ellipse becomes a circle, 

 the power in any direction is A say, and the crossed lenses are 

 equivalent to a spherical lens of power A. 



The point R on the ellipse may easily be determined as 

 follows : — 



Draw OP 7 at right angles to OP and make OP' = OP. 

 Join Q,F, and make FP"=OF = OP,and drawP"R parallel 

 to P'O. Then R is the point on the ellipse, and 



_L JL 1 f 

 OR 2 ~ OP 2 + OQ 2 



PQ_jvp^_OP m 1_ • 0F + OQ 2 

 OQ " OR " OR * ' OR OP OQ ' 



If P and Q coincide as at C, then the point F on the ellipse is 

 found by drawing the semicircle OEC and making OF = OE, 

 where E is the extremity of the radius perpendicular to the 

 diameter OC. 



III. Cylindrical Lenses crossed obliquely. 



Let the two lenses A and B be crossed obliquely, and let 

 the acute angle between their axes be (£. 



The curve of equal thickness or natural aperture for the 

 combination is easily determined. The axial thicknesses of 

 both representative lenses are equal, and the equation to 

 the curve for a thickness equal to that axial thickness is 



A sin 2 (j> x 2 + B sin 2 cf>y 2 =l, 



the curve being referred to oblique co-ordinates coinciding 

 with the axes of the lenses. 



That is, the natural aperture is an ellipse, and the crossed 

 cylindrical lenses are equivalent to an ellipsoidal lens repre- 

 sented by the ellipse inscribed in the parallelogram as shown 

 in fig. 3. The lines OC, OD are semi-conjugate diameters of 

 the ellipse. If the major and minor axes of the inscribed 

 ellipse are 2a and 2b respectively, then the equivalent 



ellipsoidal lens is one with focal powers -5 and j^ 



The magnitudes of the axes may be geometrically deter- 

 mined as follows : — 



Draw OC at right angles to OC, make OC' = OC, join 



