ASTRONOMICAL PHOTOGRAPHY 749 



If a represents the distance between the two openings in the cardboard screen and 

 hi and hi represent the corresponding distances between the images formed on the 

 plate taken inside and outside the focal plane at distance di and d^ from the focal 

 plane, then the following proportion holds: 



61 di a 



h^^d^F (4) 



where F is the distance of the desired focus from the lens or cardboard screen. 



In general it is inconvenient to determine by measurement di and d^ since the 

 position of the focus is itself unknown. If, however, the attached scale reads continu- 

 ously with increasing numbers from a point inside the focus to points outside the 

 focus, if Si and S2, respectively, represent the scale readings of the index for the posi- 

 tions of the plateholder to which the two exposures are made, and if /o equals the scale 

 setting for the focal point, we observe that 



S2 — Si = di + dg (5) 



From the geometry of the situation 



bi _ /o - si 



~a F~ (6^ 



and similarly 



hi ^ Si -/o 



a F ^ ' 



Eliminating both a and F from these equations and solving for /o, we find 



. S261 + S162 -OS 



^' ~ 61+62 ^^^ 



This gives very simply by calculation the scale setting of the plate for the focal point /o. 

 This method has the advantage, in that less labor is entailed in making the series of 

 determinations of the focus for changes in temperatures. It will be observed that 

 neither a, the distance between the apertures in the cardboard screen, nor F, the exact 

 distance to the focal point from the lens, enters into the final result. If the precise 

 focal length of the lens or mirror is desired, e.g., to determine the scale of the plate, this 

 may be found from the expression 



F =^{j, -si) =£(S2 -/o) (9) 



where a is known from measurements. 



It is often desirable, instead of using cardboard diaphragms with a single pair of 

 apertures connected by a diameter, to employ a screen containing two pairs of aper- 

 tures arranged along diameters at right angles to each other. If the focal setting is 

 different as determined from these separate pairs of apertures, some astigmatism is 

 present in the optical system. A cardboard screen containing multiple holes arranged 

 at different radii from the geometrical center of the lens and in different position angles 

 is frequently employed in observatories for determining the optical quality of the lens 

 or mirror by actually calculating the focus for the objective for different zones and in 

 different planes distributed around the optical axis in position angle. 



If a prism is placed to intercept two converging pencils of light from the two- 

 aperture screen, curved spectra of a star will be photographed. By measuring the 

 distances 61 and 62 between the components of each pair of spectra at the positions 



