322 Dr. S. P. Thompson on 



It will then be evident that the value of C is at once 

 calculable directly from A, B, and 6 by the equation 



C 2 = A 2 + B 2 + 2ABcos2<9 ( 6 ) 



Haying obtained C, the angle cf> can be most easilv calcu- 

 lated by the relation 



D 



sin 2<j>=^ sin 26 (7) 



It only remains to find the corresponding expression for 

 the power D of the spherical part of the equivalent com- 

 bination. 



Dividing equation (4) by A we have 



z = sin 2 <£+! sin 2 (#-(/>). 



_ 1 f sin 2 [6 -(f)) + sin 2<f> -sin 26 } 

 2 X sin 2(0-0) J 



-i{ 



+ A A J 



whence 



1 A+B-C 



D = ^±^ (8) 



This last result might also have been obtained by remem- 

 bering that the maximum power being C + D and the minimum 

 power being D (at right angles), ihe sum of these, namely 

 C + 2D, will be equal to A + B, whatever the angle between 

 the latter. 



An example of the use of the three working formulae (6), 

 (7), and (8) will suffice. 



To find the equivalent sphero-cylindrical combination for 

 the following obliquely-crossed cylindrical lenses : — 



+ 3-5 cyl. ax. 20° Q + 2'5 cyl. ax. 35°. 

 Here A = 3*5 dioptries, 



B = 2*5 dioptries, 



= 15°, 

 cos 26 = 0-866, 

 sin 20 = 0-5, 



