44 PHOTOGRAPHIC FACTS AND FORMULAS 
together, and divide by their sum less the distance of separa- 
tion. Let f’ and f’’ be the foci and d the separation of the 
two lenses; then the final focal length will be (f’ X f”) + 
(f’ + f’’ —d). Example: a 6-inch and an 8-inch lens are 
to be combined with a distance of separation of 1 inch; the 
focus will be (6 X 8) + (6+ 8—1) =48~13=—3¥Y. To 
find the focal length of a supplementary lens or magnifier to 
reduce or increase the focal length of a given lens, multiply 
the focal length F to be altered by the final focal length 
desired, and divide the product by the original focal length 
less the final focus. Example: it is desired to reduce the 
focal length of a 10-inch lens to 7 inches, then 10 XK 7 = 70, 
10—7=3, and 70~—3—231/3. To reduce the focal 
length, positive or convex lenses must be used; to increase 
the focus, negative or concave lenses must be used; and, in 
this case, the focus of the lens to be added is prefixed by the 
minus sign in the formula. Example: to lengthen a 7-inch 
lens to 10 inches, 7 X 1070, and 7— (+10) =—3, 
therefore 70 ——3 = —23 1/3. A very simple rule to re- 
member as to the focus of a magnifier for reducing the focus, 
or, in other words, to take near objects close to the camera, 
is that the focus of the supplementary lens must be equal to 
the distance of the object. In this calculation the separation 
of the lenses has been neglected; to take this into considera- 
tion is a refinement that merely complicates calculations. 
Ratio APERTURE OF D1aPHRAGMS.—The ratio or effective 
aperture of the stops or diaphragms does not coincide with 
their actual diameters in consequence of the condensation of 
the light by the front lens component in compound lenses. 
To determine the correct ratio aperture, focus the lens for 
parallel rays, that is, for a very distant object; then replace 
the focussing screen with an opaque card, in the center of 
which is a pinhole. In a dark room place a light behind the 
