Focal Lines of Cylindrical Lenses. 71 



If the distance of the point p or r 2 (fig. 10) from the 

 point o or from the lenses be called v 2 , we have from fig. 10 



— = -, r , but from fig. 9 T r == — -. 



v 2 'h — h /l i — h »2 



^1 = 1 m 



«, Aj W 



From equations 7 and 8, by substituting the values of 

 & and Z with the condition (6), we get 



, , /cos2<9 2 , cos20A 



or ^v*^-^), 



an analogous result to that obtained for parallel light. 



(2) The tangential focal line. 



Let the line af be produced to meet the axis of x at r (fig. 9) 

 and the line of be produced to meet the axis of x in the 

 point v 3 (fig. 10). The semi-length of the tangential focal 

 line =or=/ 2 , and the distance of the line from the lenses 

 = ov d =v n . From fig. 9 we have 



and from fig. 10 we have 



5l = ~ (10) 



r 3 /*! 



From equations 9 and 10, by substituting the values of 

 k and I and putting in the condition (6), we get 



, , /cos 20 2 ^ cos20A 



-V< 2 (j^ F J 



5. TAtf lengths of the focal lines of a number of cylindrical 

 lenses in contact arranged with their axes crossed at any 

 angles. 



The results obtained in the last section are perfectly 

 general. Let the focal lengths of the lenses hefi,f 2 ,fto &c., 

 and let the angles between the axes of the lenses and any 

 fixed line such as oo' figs. 7 and 9, be U 25 @z, &c., the 



