Mr. R. J. Sowter on Astigmatic Lenses. 245 



In a similar manner it may be shown that 



OD f = a + b. 



Hence D'D" = 2a, 



D'D W = 26. 



Again, the circle ci^/ is the locus of the feet of the 

 perpendiculars from the foci on the tangents to the ellipse ; 

 therefore the points FF f are the foci, and the axes are 

 determined in direction by the line OF. 



Analytically a and b are easily determined in terms of 

 c and d from the equations (i.) and (ii.). Further, 



(iii.) *=Asin 2 (£, 



(iv.) j z = B sin 2 <£ ; 



and from the equations (i.) (iv.) a and h are readily deduced 



in terms of A, B, and <j>. 



Thus, since an ellipsoidal lens with powers « = ^ and 



/3=jt 2 is equivalent to a spherical lens of power a. and a 



cylindrical lens of power (3 — a. or jz & if C is the cylin- 



dricity of the sphero-cylindrical lens, it is easily shown that 



1 _ 1 _ \/(c 2 + ^) 2 -4c^ 2 sin 2 

 ° ~ W d l ~ c 2 d* sin 2 </> 



= v^ + A^ABa-Ssin 2 ^). 



.'. C = */A 2 + B 2 +2ABcos2<£; 



and a parallelogram-construction with sides A and B, and 

 angle 2(f), as was shown by Prof. S. P. Thompson *, gives the 

 cylindricity in a sphero-cylindrical lens equivalent to two 

 crossed cylindrical lenses, A and B, at an angle (f>. 



The angle of inclination of the cylindrical lens in the 

 sphero-cylindrical combination to the direction 00 is the 

 angle between the major axis of the inscribed ellipse and 

 the axis OC. 



The equivalence of a sphero-cylindrical combination and 

 an ellipsoidal lens follows at once from the method of contours 

 * Phil. Mag. Mar. 1900; Phys. Soc. Proc. 97, July 1900. 



