Obliquely-crossed Cylindrical Lenses. 

 By equation (6) 



C 2 = 12-25 + 6-25 + (17-5x0-866) 

 = 33-65, 

 whence C = + 5*8 dioptries, approximately. 



By equation (8) 



323 



D 



3-5 + 2-5-5-8 



D = +0*1 dioptries (spherical). 

 By equation (7) 



2-5 



sin 2<£ 



b-l 



xO-5 



= 0-215, 

 </> = i sin" 1 (0-215) 



= 6° 13', approximately, beyond the position 

 of A. 

 Hence the equivalent combination would be 



+ 0-1 sph. O +5*8 cyl. ax. 26° 13'. 



5. The circumstance that the cylindrical part of the 

 resultant of two crossed cylindrical lenses varies from maxi- 

 mum to minimum when the angle between the axes of the 

 two lenses is varied from 0° to 90°, suggests a solution to the 

 practical problem how to make a cylindrical lens of variable 

 cylindricity. If two equal positive cylindrical lenses are used, 

 the value of the cylindrical part of their resultant varies 

 from their sum, when the angle = 0°, to zero when the 

 angle = 90°. But while the cylindrical part thus diminishes, 

 in proportion to the square of the cosine of the angle between 

 them, the spherical part of the resultant increases in the 

 proportion of the square of the sine of the same angle. One 

 never obtains a simple cylindrical lens, except in the case of 

 the coincidence of the axes. Similarly if two equal negative 

 cylindrical lenses are used, the resultant consists of a negative 

 cylindrical part and a negative spherical part, the one 

 decreasing, the other increasing when the angle is increased 

 from 0° to 90°. 



Ophthalmists are acquainted with a combination (known 

 as Stokes's lens) consisting of two cylindrical lenses of equal 

 but opposite powers (one a convex, the other an equal con- 

 cave), arranged to be rotated to various angles of obliquity 

 across one another. When their axes are in coincidence, 

 or = 0° , they neutralize one another completely. When 

 crossed at right-angles their resultant cylindricity is a 



