244 Mr. B. J. Sowter on Astigmatic Lenses. 



CD, C'D 1 , produce D'C to D", making CD"=CD, and from 

 CD' cut off CD'" = CD. Then 



D'D" = 2a, and D'D/" = 25. 



Fiar. 3. 



To determine the direction of the axes the following 

 procedure may be adopted : — 



• With as centre and a length a = JD'D' / as radius, 

 cut the parallelogram in the points d, c, e,f, and draw d¥ and 

 ^F perpendicular to the sides of the parallelogram to intersect 

 in F. OF is then the direction of the major axis. 



To prove the above constructions, let GG = c, OD=d. 

 Then, since c and d are conjugate, 



(i.) c* + d* = a 2 + b*, 

 (h.) cd sin <j> = ah. 



Now DC 2 = 0C 2 + OD 2 -20D . OC cos DOC; 

 .*. DC 2 = c*+d2-2cd sin <f> = {a-bf 

 by the relations (i.) and (ii.). Therefore 

 DC = a-Z>. 



