October 8, 1891] 



NATURE 



553 



whence, substituting for ^ in (i) we have 



^^ /./ 2D-J 

 V 2 Vd - F 



(2) 



Let the nearest object be at n times the focal length of the 

 lens. Then, putting «F for D, 



_ /(jY in - I 



- V — --- - 



V« - I 



(3) 



This gives the value of A as a linear quantity; it is usual, 

 however, to reckon the diameter of stops as fractions of the focal 

 length. 



Dividing, therefore, (3) by F, 



A^ A 2«- I 



F V 2F ^n - I •••••• ^^' 



From (4) the accompanying table has been computed, giving 



=^ for various values of F and 

 graphically. ) 



(Fig. I gives the same 



Table shnving ratio of aperture to focal length which gives the 

 best average definition when the nearest object to be photo- 

 graphed is at "m" times the focal length of the lens, and 

 distant objects are a 'so in vieiu. 



•C244 

 ■0.99 



•0172 



■0154 

 •0I4I 



0130 

 ■0122 



■0II5 



■0109 



I have not before seen it pointed out that the ratio --, which 



gives the best average definition, alters with the value of F. 



If a is the least angular distance between two points (as seen 

 from the centre of the lens) which are shown as separate points 



on the photograph, a must at any rate not be less than -|-, or 



showing that, if the foreground is kept at a distance proportional 

 to the focal length of the lens, the definition improves with an 

 increase of the focal length. 



On the other hand, if the nearest object is at some fixed dis- 

 tance, D, from the lens, we have as the limit for o, 



\^2(J 



sId - F 

 2D - F' 



an expression which increases with F, so that for a given picture 

 taken from a fixed position, definition will be gained by the use 

 of a short focus. 



The gain, however, in this respect is not great, for in practice 

 D is always a considerable multiple of F, and writing 



^/ 



F- 

 it will be seen that when D is many times F, 7^ „ 



neglected in comparison with 4D. 



Thus, in ordinary cases the limit for a is »/ ^ 



dependent of the focal length of the lens employed 



may be 



and 



if we inquire how close the nearest object may be to the lens 

 NO. I 145. VOL. 44] 



when a view containing also distant objects has to be photo- 

 graphed with a definition reaching a certain standard, we have, 

 on the above iuppotilion, 



D= ^ ; 

 2a- 



and if we put a = i', which is often taken as the least angle 

 separable by the unaided eye, and \ as zTsiirs inch, 



D = 150 inches, 



showing that if the picture is to appear as well defined as the 

 natural objects themselves, to the eye placed at the position of 

 the lens, no object in the view must be nearer the latter than 

 about 13 feet.^ 

 ' Though, as above stated, the focal length does not affect the 

 definition, when the right-sized stop is used, it does the rapidity 

 with which a picture may be taken, for the intensity of the light 



on the plate is measured by — or '''- - ^ ~ ' . 

 ^ ^ F^ 2F « - I 



That is, in these circumstances, the exposure is inversely as 

 the focal length. 



All that has been hitherto said refers to the definition in the 

 central parts of the plate. 



The definition for the oblique pencils is necessarily worse. For 

 even if it were assumed that the lens was perfect for oblique 

 pencils, the points out of focus would be no longer represented 

 by circular areas, but by the elliptic projections of these circles 

 on the plane of the plate. 



The assumption, however, that a lens is perfect for oblique 

 pencils is too far removed from actual fact to make it worth 

 while to consider the results to which such a supposition would 

 lead. 



The definition for the marginal parts of the photograph 

 depends on the various aberrations which all combinations of 

 lenses suffer from in some degree, but which in well-made 

 examples are completely, or almost completely, corrected for 

 direct pencils. 



These aberrations are (i) spherical, (2) chromatic, (3) astig- 

 matism, (4) curvature of field. 



The effects of the two last are much the most important, and 

 will be considered first. 



Let O (Fig. 2) be the optic centre of the lens, OF the axis 

 of the lens, and F the principal focus. 



Let Fx be the plane of the plate, FP and FS the curves on 

 which the primary and secondary foci respectively He. 



Let Op be the axis of a pencil inclined to OF at an angle 6, 

 and meeting FP and FS in / and s. Then sp measures the 

 astigmatism of the lens for a pencil of obliquity 0. 



Putting 7;» and j»'j for the ordinates of the curves FP and FS 

 at p and s, it will be seen that a point distant from the axis 

 of the lens, will be represented on the plane of the plate by an 



oval patch of light whose axes are A -^^ and A^-^-, in direc 



^ ^ cos 9 cose 



tions parallel and perpendicular to Fx ; A, as before, being the 

 aperture of the lens. 



Any formula depending on the actual data of real combina- 

 tions of lenses, and giving the values of V/, and j>s in terms of 

 radii of curvature and refi active Indices, &c., of the lenses 

 composing ihem, would be a very unmanageable thing for the 



' I have verife J ih's with a lens o' lo-Inch focus* 



