ON (CERTAIN NEW METHODS AND KESn/rs IN ol'TICS. 



OiiAiJLEs S. Hastings. 



Tlie usual formulas for the calculations of lens systems of auy but those of the greatest sim- 

 plicity are extremely complicated. Should such accura(!y be desired as is absolutely requisite far 

 the designing of optical apparatus, iu which case the ordinary approximations are insufficient, the 

 fornuilas become almost unmanageable. 



The tamous i)apcr by Gauss, which was iniblished in 1840, and which first introduced the 

 conception of cardinal points in an optical system, was a very remarkable advance on thv. okler 

 theories. Alter its appearance it was possible to take into account the thicknesses and separations 

 of the lenses with rigid accuracy without introducing greater (complexity in the rtnale<iuiitions than 

 already secured by a method which assumed that these elements might bo neglected. But even 

 more important than this was his i)erfect definition of terms which had been used for ideal sys- 

 tems, but which conld not have any exact meaning wheu applied to any physical a])iiaratus, such 

 as focal length, magnification, optical center. This paper, with Listing's addition of tlie two nodal 

 jioints, left nothing to be desired iu giving a complete and definite geometrical notion of ano]>tical 

 system whose cardinal points were known. IJntthis is not the primary end in eitlier designing an 

 optical system nor in determining its optical efficiency when completed. The absolute focal length 

 and magnification of any optical instrument is of the least possible consequence, as becomes at 

 once obvious when we consider how few working astronomers or microscopists can give these data 

 with precision for their own instruments. 



What is necessary in designing an oi)tical instrument is a knowledge of its aperture, in the 

 more general sense, of the variation of its focal plane with varying wave length of light, of its 

 spherical aberration, of the variation of the spherical aberration with varjnng wave length, and, if 

 the character of the image remote from the axis is of consequence, the variation of the magnifica- 

 tion with varying wave length, and, finally, its astigmatism. Neither the older methods nor that 

 of Gauss define the first of these important (piantities, except, of course, in the simple case of the 

 telescope, nor are they convenient for calculating the others. 



From the character of the quantities named above, it is obvious that their mathematical 

 expressions nuistbe for the most part of the nature of derivations from tlie fundamental cipiations. 

 But all of the fundaiticiital ciiuations for lenses of sufiicient exactness to be of the least value are 

 s«i complicated tli;it Mieir derivations, at least, are (piite valueless for practical purposes. 



It occurred to the, writer a good many years ago that greater simplicity might be attained iu 

 mathematical expressions for optical systems if the conceptions of rays, radii, and indices of re- 

 fraction, the first of which at any rate is most artificial, were replaced by the more natural notions 

 of wave and lens surfaces and light velocities, the surfaces being characterized by their curvatures 

 rather than by their radii. This jn-oved so highly satisfactory in practice that he has used no 

 other method for a long time, but a serious attempt to find how far it would go in replacing the 

 older method in formal optics; that is to say, how far it would be of interest indci)endent of more 

 practical advantages, was left until about a year ago. To show the residts obtained, so far as the 

 fundamental equations are concerned, is the object of the present paper, leaving the discussion of 

 the derivative equations and their applications for another occasion. 



