A. A. MtcheUon — Measurement by Light-waves. 117 



il = smallest distance between lines which can be clearly ''resolved. 



?. = wave-length of the light employed. 



h =s breadth of the diffraction fringes with this kind of light. 



M = Magnification. 



R = Resolution. 

 D = Pelinition. 

 A = Accuracy. 

 Theu if M is the ratio of size of image to object, 



Bin ;5 



The resolution is measured by the closeness of two lines which ean be clearly 

 distinguished or "resolved.'' Let us therefore put 



-5 



m two lines are clearly distinguishable when the central fringes of their 

 images are separated by the width of one fringe. The actual limit at which the 



resolution disappears may be anywhere between b and -. (Sec " Wave Theory," 



1 



Lord Ravleigh, Enc, Brit.) But it can readilv be shown that b = and 



2 sin (3 

 2 

 d=6/M. hence, R=^ sin /3. 



A 



The definition of an objective is measured by the ease with which the forms of 

 minute objects may be recognized. Thus, were it not for diffraction. D would be 

 simply proportional to M. But for a given magnification the form of the image is 

 clearer, or the definition greater as the fringes are narrower; hence, we may put 



M 



D=-=R 



b 



Definition is not capable of being so precisely formulated as Resolution, and 

 would undoubtedly vary with the form of the object, its nearness to other objects, 

 etc. In view of these uncertainties it would scarcely be worth while to introduce a 

 constant coefficient in the last equation. 



The error of setting of the cross hairs of an eye-piece on the middle of a diffrac- 

 tion band of sufficient width will be b/e where e is a constant not far from 100, and 

 the corresponding error in distance would be &/e-f-M. This smallest measurable 

 distance is therefore e times as small as the smallest resole able distance; hence, 



A=eR. 



These formula? may be applied to the microscope (in which case the maximum 

 values correspond to a=90 c ). or to the telescope (in which a is nearly zero, and 

 angular measurements alone are of importance). Accordingly we obtain the 

 following : 



Microscope. Telescope. 



M = l/sin/3 F 



R = 2/a B/A 



D = 2/A B/a 



A = el p. eBp. 



The formulas for microscopes apply when the object is very near the lens. 

 They show, first, that with a microscope of given length the magnifying power 

 depends on the smallness of the objective and on nothing else ; second, that the 

 resolution, definition and accuracy (upon winch the usefulness of a microscope 

 chiefly depends) are the same for all microscopes, no matter how large the objec- 

 tive may bo. or how great the magnifying power, provided the latter be sufficient 

 to show diffraction fringes: third, that these qualities vary inversely with the 

 wave-length of the light employed. 



There seems to be a prevailing impression that a microscope may have a high 

 resolving power with but moderate definition and \a. This may be due 



to the difficulty in giving an exact signification to the terms. If those here em- 

 ployed be admitted it is evident that the two qualities must go together. 



