Focal Lines of Cylindrical Lenses. 65 



(a) TJte axial focal line ; that is the line parallel to the axis 

 of the first lens. 



If /j be the semi-length of the axial focal line, 



/i T l'l ll\V\ 



l 1 = d'v' = /v / -c'd' = --^_ 



if 



•'• '-HH)- • ■ ■ •' (5) 



or the length of the axial focal line of two piano-cylindrical 

 lenses in contact with the axes crossed at 90° = the axial 

 aperture x distance of the line from the lenses X difference 

 of the powers of the lenses. 



(b) The tangential focal line. 



If the line aid! (fig. 5) be produced to meet the axis of x 

 in e' and if the line a'd' (fig. 6) be produced to meet the axis 

 of x in v 2 , l 2 = o'e' is the semi-length of the tangential focal 

 line and o'v 2 = v 2 is its distance from the lenses. 



From fig. 5 we have 



and from fig. b* we have 



If j^ =y 2 both ^ and Z 2 = 0, or two piano-cylindrical 

 lenses of the same focal length crossed at right angles are 

 equivalent to a spherical lens. 



4. To find the lengths of the focal lines of two piano-cylindrical 

 lenses in contact of focal lengths f\f 2 with o:xes crossed at 

 an angle 6. 



(a) First we will take the incident light to be parallel. 



(1) The axial line or that within the angle 6. 



Fig. 7 is an end view of the system looking from left to 

 right of fig. 8 which is a front elevation. 



Phil Mag. S. G. Vol. 20. No. 115. July 1910. F 



