432 Transactions of the Society. 



constant throughout the system for a given ray. But the absolute axial 



distances are different, and different in the proportion -. — ^ . It is evi- 



sm 



dent that when 6 is large this discrepancy becomes enormous. 



But the tangent law does not apply only to unfocussed beams. If 

 we were to place a narrow-angled lens at C and cause it to rotate through 

 the angle 6, adjusting its focal length to the sec 6 as it rotated, we should 

 have focussed beams falling in the tangent positions. And it is equally 

 plain that we should in that way obtain a flat field by an optical appliance 

 which could not yield an image of a plane wave-front. Such systems 

 are in use. For the photographic camera they are indispensable. In 

 panoramic cameras the lens actually rotates. But the rotation is not 

 necessary to the embodiment of the principle. All that is indispensable 

 is that the focal length should vary with the position angle 6, and in the 

 ratio of sec 0. A fixed lens that satisfies this condition yields images on 

 tangent scale. 



The great advantage of a lens of that description lies in its breadth 

 of field. The centred system, to use Helmholtz' term for the system 

 first discussed, cannot cover a field any larger than its aperture. But 

 this uncentred system, in which the optical axis wanders to all parts of 

 the field, is limited only by the extravagant dimensions to which the 

 tangent attains at very wide angles. Indeed, a panoramic camera, with 

 a circular field, will take in an angle of 360°, and more too if you will 

 let it. But then, of course, it substitutes the angle for its tangent 

 as the modulus of the image scale. This case, therefore, does not fall 

 within our present scope, since we are only concerned with the formation 

 of images in flat fields. 



It may be useful, however, to point out that there are three and only 

 three practicable image scales for aplanatic systems and that these three 

 scales are related to one another as the sine, arc and tangent. The 

 sine scale applies to centred systems with flat fields, the arc scale to 

 systems centred or uncentred with spherical fields and the tangent scale 

 to uncentred systems with flat fields. It is not possible to deal with the 

 one system fully apart from the others for we may have a spherical 

 image of a flat object or vice versa and again we may have a centred 

 system with its fixed optical axis on one side of the aperture, and an 

 uncentred system with its wandering optical axis on the other side of 

 the aperture. We are at present, however, only concerned with the sine 

 and tangent scales ; — but even so it is simplest to begin by considering- 

 all three scales together. 



Fig. 105 exhibits these relations at a glance. Here we have three 

 fields which have a common point upon the common axis and a common 

 position angle 6 at the common optical centre C. The scales are 

 obviously proportional to the lengths which subtend this common angle, 

 that is to say, they are proportional to -q . . . rj a , v ... r) $ , and v . . . Vt 

 respectively. If we take the length of the arc as the standard magni- 

 tude and denote it by A we shall have for the sine scale S = ^- A, 



