320 Dr. S. P. Thompson on 



lenses A and B. The sum of these components would be 

 A cos 2 -f B cos 2 (6 — (f)). Similarly there might be taken, at 

 right-angles to c, the two other cylindrical components, 

 whose sum would be A sin 2 + B sin 2 (0 — 0) . Now there 

 will be in every case one particular value of which will 

 make the former sum a maximum and the latter a minimum. 

 If we can find this value of 0, the problem is solved. 



Differentiating with respect to the expression above 

 obtained for the sum of the two components along c, and 

 equating to zero, we find that when this sum is a maximum 

 the angle will be such as to give the relation 



A_ Bin 2(0-0) m 



B " sin 20 ' l } 



to which may be given the alternative form 



^ -fcos 20 



cot 20 = . _ (2) 



T sin 20 ' 



From this latter may be reckoned by the aid of trigono- 

 metrical tables, A, B, and being all given. Angle being 

 thus found, it can be used to calculate the maximum and 

 minimum values, namely the two sums previously expressed. 

 The cylindrical lens representing the maximum sum being 

 set at angle with the original direction of A, and the 

 cylindrical lens representing the minimum sum being set at 

 90° — 0, they will, thus crossed at right-angles to one another, 

 together act as the optical equivalent of the two obliquely- 

 crossed cylinders. 



These two rectangularly- crossed cylindrical lenses may 

 again be resolved into the combination of (1) a cylindrical 

 lens, w T hose axis is along the axis just found for the maximum 

 lens, and of power C dioptries equal to the difference between 

 the maximum and minimum cylindrical powers, and (2) a 

 spherical lens whose power Dis that of the minimum. Hence 

 we shall have 



G = A cos 2 + B cos 2 (0-0) -A sin 2 0-B sin 2 (0-0) ; 



C = Acos20 + Bcos2(0-0), (3) 



and 



D = Asin 2 + Bsiu 2 (0-0) (4) 



4. The solution thus found is capable of further sim- 

 plification. 



Dividing equation (3) by A, we have : 



B 



-^=cos20 + £-cos2(0— 0). 



