72 Mr. A. Whitwell on the Lengths of the 



angles being reckoned positive in one direction and negative 

 in the other. The condition that must be fulfilled if two 

 parallel rays symmetrical with respect to the plane con- 

 taining the line oo' and the optic axis are to be refracted 

 so as to intersect in this plane, is 



^ sin 6 cos 6 _„ 

 2 J-—0. 



The principal focal lengths of the system are 



Fl= _l - 



1 -cos 2 6>' 



F 2 = r-g-H 



^ sur 6 



~f~ 



and the semi-lengths of the focal lines are 



6. To find the lengths of the two focal lines of two piano- 

 cylindrical lenses of focal lengths f and f 2 with axes 

 crossed at 90°, the distance hetiveen the lenses being = S. 



Figs. 11 and 12 are elevation and plan respectively of the 

 system of rays. 



(a) The line parallel to the axis of the first lens which is 

 vertical. 



Two rays diverging from a point uu\ at a distance — u 

 from the first lens, and incident thereon at symmetrical 

 points a, h, a, b\ will, after refraction by the first lens, 

 intersect in the point c' (fig. 11) on the line represented in 

 elevation by c'v{ (fig. 11), and in plan by the point v 1 in 

 fig. 12. 



Let fvi = Vi and fa' = h v 



Then we have j, „, 



/i 

 The two refracted rays on meeting the second lens in 

 points distant h s from the axis of m are bent downwards, the 

 point of intersection being at d\ the distance c'd' or the 



