1904.] 



Some Uses of Cylindrical Lens- Systems. 



281 



" Some Uses of Cylindrical Lens-Systems, including Eotation of 

 Images." By George J. Bukch, M.A., D.Sc. Oxon., F.R.S., 

 Lecturer in Physics, University College, Reading. Eeceived 

 February 29,— Read March 10, 1904. 



It was, I believe, first shown by the late Sir George Stokes that if 

 two similar cylindrical lenses are placed in contact, with their axes of 

 curvature at right angles, the combination acts as a spherical lens, and 

 that if the axes are not at right angles the system is equivalent to a 

 cylindrical and a spherical lens combined, the spherical element dis- 

 appearing when the axes are parallel. 



I have recently had occasion to employ a modification of this 

 arrangement which has proved of some practical utility as well as 

 theoretical interest. 



When, as in Sir George Stokes's experiment, the two lenses are 

 placed in contact, the combination acts as a spherical lens in every 



respect, that is to say, the formula - + — = — holds in regard to it. 



u v f 



But when the two lenses are not in contact there is only one pair of 

 conjugate points at which a real object will give a real image. 

 Reference to the ordinary focometer formula will show that this must 

 be the case. Let I be the distance between an object and a screen, and let 

 a convex lens of focal length / < \l be moved along the line joining 

 them. Then there must be two, and only two, positions of the lens 

 at which it will form images on the screen, one at a distance u from the 

 object and v from the screen, and the other at a distance v from the 

 object and u from the screen where 



I 2 - a 2 



u + v = /, u-v = a, and — ^ — = . 



In either of these two positions a cylindrical lens of equal power 

 will, if its axis be vertical, form sharp images of vertical lines, or of 

 horizontal lines if its axis be horizontal. If therefore we place one 

 cylindrical lens with its axis vertical at a distance u from the object 

 and a second of equal power with its axis of curvature horizontal at 

 a distance u from the screen, there will be produced on the screen a 

 sharp image magnified vertically v/u times, and horizontally u/v times. 



This method may be employed for comparing by photography curves 

 plotted to different scales, or for increasing or diminishing the ordinates 

 of a curve or record the scale of which is unsuitable. Figs. 1 and 2 

 are an example of this latter use. 



Fig. 1 is a record taken with the capillary electrometer of the dis- 

 charge of the electrical organ of Malapterurus, the two sides of the 



