Obliquely-crossed Cylindrical Lenses. 



From equation (1) we have 



B _ sin 20 



A "sin 2(0-0)' 

 Substituting this in the preceding gives us : 



° -cos 20 + sJn^^cos^-0) 

 A -cos 20 + sin2 (0_0) ' 



_ sin 20 



A~ sin 2(0-0)' 

 From this immediately follows the relation : 



321 



B 



C 



sin 2(0-0) sin 20 sin 20* • • • ( 5 ) 



This at once suggests that the three magnitudes A, B, and 

 C can he represented by the three sides of a triangle whose 

 respectively-subtended angles are 2(0 — 0), 20, and 20. Or 

 2(0 — 0), 20, and ir— 20. In short, the two given cylindrical 

 components A and B may be compounded to find their cylin- 

 drical resultant by means of a parallelogram in which, 

 however, the angle between A and B is drawn as double the actual 

 angle between the axes of the two given components. 



Hence we obtain the graphic 

 construction of fig. 4. Draw 

 the lines A, B to represent, 

 in magnitude only, the powers 

 of the two given cylindrical 

 lenses, and make the angle 

 A B between them double 

 the given angle 0. For posi- 

 tive (i. e. convergent) cylin- 

 drical lenses these directions 

 may be reckoned outwards from 

 the origin, as shown by the 

 arrow-heads. For negative 

 (i. e. divergent) cylindrical 

 lenses the directions must be 

 reckoned inwards. Compound- 

 ing A and B in the ordinary 

 way, we obtain the resultant 

 (J C which represents in magni- 

 tude (and in sign) the resultant 

 cylindrical part of the desired 

 equivalent combination: but 

 the angle A will be double 

 of the angle that the axis of 

 the resultant cylindrical lens will make with the axis of A. 



