Obliquely '-crossed Cylindrical Lenses. 319 



curvature at of the trace of this plane, where it cuts the curved 



surface along P P', will be - cos 2 <£. This follows from the 



circumstance that P P' is part of an ellipse whose major and 

 minor axes are respectively rj cos (f> and r. We may further 

 consider the intersection Q Q' of another oblique planeat 

 right-angles to P P'. The curvature at along the line 



Q Q' will be - sin 2 </>. The sum of these two curvatures will 



obviously be equal to the original maximum curvature along 

 NN', since the minimum curvature along A A' is zero. 

 Hence we may regard the curvature along N N' as contri- 

 buting two components of cylindricity of the respective values 

 named in the two directions P P' and Q Q'. If light were 

 admitted through narrow parallel slits set respectively along 

 P P' and Q Q', the convergivity of the two beams respectively 

 impressed by the lens would be (/*■— 1) cos 2 <j>/r and (/*—!) 

 sin 2 <j>/r. If r is expressed in metres, these two convergivities 

 will be expressed in dioptries according to the practice now 

 internationally adopted by ophthalmists. It is obvious that 

 the angle (/> may be measured either between N N' and P P', 

 or between A A' and Q Q'. 



3. Returning to the problem enun- Pig- 3. 



ciated w T ith reference to fig. 1, we may 

 now find a solution by resolving each of 



the two cylindrical lenses into compo- 

 nents and then recombining these com- 

 ponents in the manner presently to be 



considered. 



Let the line a (fig. 3) represent the 



direction of the axis of one of the given 



cylindrical lenses, having a power of A 



dioptries, and the line b the direction of 



the axis of the other given cylindrical lens 



of power B dioptries. The angle 6 between 



a and O b is also given. It is required 



to find the respective number of dioptries 



C and D of the cylindrical and spherical 



lenses which shall together constitute a 



combination whose optical effect is the 



equivalent of that of A and B. It is also 



required to find the angular position of 



the axis of the equivalent cylindrical lens. 



It is clear that we might take any line c through O 



making an angle (j> with a, and take the cylindrical com- 

 ponents, along that direction, of the two given cylindrical 



