82 On Ellipsoidal Lenses, 



where U is the equatorial curvature of the cylindrical wave 

 at the second lens, power A, is the angle of crossing of the 

 lenses, and U' V 6' define the refracted pencil, and conse- 

 quently the equivalent ellipsoidal lens. 



The sides of the parallelogram are A and U. 

 The following experiments were performed : — 

 Experiment I. Light from a small aperture having vertical 

 and horizontal cross-wires was brought to a parallel beam by 

 a convex lens, and the beam fell on a convex cylindrical lens 

 of focal length 18*8 cms. supported with its axis vertical. At 

 8 cms, from this lens another convex cylindrical lens of focal 

 length 25*3 cms. was supported with its axis inclined at 60° 

 to the horizontal. The focal lines were at 8*1 cms., and 

 140 cms. from the second lens, and the near line was inclined 

 at 21° to the axis of the inclined lens. Equation (I.) 

 gives : — 



U' + V'=--132, 



and equations II. and III., or the parallelogram, give : — 



(U'-V) cos 20'=A+ U cos 20= --0858 



(t 



V- 



V) sin 2d' 



= Usin20 





= — • 



0.1 



from which 





U'-V 

 tan 20' 



= -•118 

 = -935 









that is, 



U' 



= -•125 



(u' = 



-8 



cms 



•) 





V 



= -•007 



(„' = - 



143 



cms 



•) 





& 



= 21i°, 











values in agreement with those observed. 



Experiment II. A beam diverging from a circular aperture 

 having vertical and horizontal cross-wires was made parallel, 

 and fell upon a concave cylindrical lens (/=18'8 cms.) sup- 

 ported with its axis vertical. At 10 cms. from this lens, two 

 convex cylindrical lenses were placed in contact and crossed 

 at right angles. Their axes were at 60° and 30° respectively 

 to the horizontal, and their focal lengths were 18*8 cms. and 

 25*3 cms. The focal lines were at 23 cms. and 70 cms. from 

 the crossed cylinders, and the inclination of one focal line 

 to one of the C3 r linder axes was 41°. 



Equations (i.) and (ii.) give at once :■ — 



-U cos 2 + U' cos 2 0' + V sin 2 0' = A 

 -U sin 2 + U' sin 2 0' + V cos 2 0' = B, 



and the parallelogram, the sides being A— B and U, gives 

 at once : — 



(U'-V) sin 20' = U sin 20, 



