OPTICAL INSTRUMENTS. 



the point/, the imaginary focus of the 

 lens; the perpendicular cd shews that 

 the rays obey the same law of refraction 

 as in a convex lens. It should be here 

 observed, that when the lens is concave, 



Fig. 9. 



the focal point will be on the same side 

 as the object, and it is termed negative, 

 as objects are diminished by concave 

 lenses. 



When converging rays from a convex 

 lens are transmitted by a concave, they 

 are rendered parallel, as shewn in fig. 

 10, where the converging rays at c 



Fig. 10. 



(from" a convex lens) are brought pa- 

 rallel L at p, after passing through the 

 lens. 



(1 0.) The manner in which the foci 

 of lenses of different curves are calcu- 

 lated, and how the foci of combined 

 lenses may be obtained, are as follows. 

 When the lenses are made of plate 

 glass, the focal distance is nearly the 

 diameter of the sphere from which we 

 may suppose a plano-convex lens to be 

 cut, or it is equal to twice the radius of 

 the circle that forms the convex surface 

 Of the lens. For example, if the globe of 

 glass is one inch in diameter, and a por- 

 tion is cut off to form a plano-convex 

 lens, the focus will be one inch, or twice 

 the radius of the circle. If the lens is 

 double convex, the focus will be equal to 

 the radius, or half the diameter. When 

 the lens is crossed or unequally convex, 

 the focal length will be twice the product 

 of the two radii, divided by the sum of 

 the radii. For example, let the radius on 

 one side be 2 inches, and on the other 



side 6 inches ; the focus of this will be 

 2 x 2 x 6 = 24, divided by 2 + 6 = 8 

 or 3 inches. The focus of the miniscus 

 lens is found by dividing twice the pro- 

 duct of the two radii by their difference. 

 Example ; let the radius on the convex 

 side be 2 inches, and on the concave 

 side 4, the focus is 2 x 2 x 4 = 16 di- 

 vided by 4 2 = 2, or 8 inches, the focus 

 of the lens*. 



If two lenses are placed in contact, 

 the compound focus, when each lens 

 has the same power, will be half the 

 focus of the single lens. When two 

 convex lenses are in contact, having dif- 

 ferent focal lengths, then, as the sum of 

 the two foci is to one of them, so is the 

 other to the compound focus required. 

 For example, let the foci of the lenses 

 be 2 and 6 ; then, as 2 + 6 = 8 : 2.'. 6 : 

 1^, the compound focus. Lastly, if two 

 lenses are not in contact, the compound 

 focus is found by dividing the product 

 of the two lenses by the sum lessened 

 by their distance. Example: let the 

 foci of the lenses be 2 and 4, their 

 distance 2 ; then 2x4 = 8 divided by 

 (2 + 4) 2=4 gives 2 as the compound 

 focus. 



(11.) If lenses be made of different 

 substances, although the curves may be 

 the same, the focal lengths will vary ; 

 while, in like mediums, the action will 

 always be equal. Let a b (Jig. 11.) be 

 a ray of light, and let it enter the me- 

 dium c c? at the point b ; instead of con- 

 tinuing in a right line to e it will pass on 

 in the direction b f, should the medium 

 cdbe denser than the first a b ; now if 

 on the point b a circle be drawn, and a 

 line si t parallel to the surface of the 

 medium, touching the incident ray a b 

 be produced to e, this line will be the sine 

 of incidence ; and if another line pr be 

 drawn in the same manner to the re- 

 fracted ray, it will be the sine of refrac- 

 tion. Now if the angle a b c be varied 

 to any degree, the sine s i will always 

 be in the same proportion to the sine of 

 refraction, p r. If the dense medium is 

 water, the sine p r will be f of 5 i. When 



* In many cases.it is found advisable to ascer- 

 tain the radii of the two surfaces of a convex lens, 

 as well as its focus, by a more accurate manner. 

 This may be effected by forming a reflected image 

 by the posterior surface, which distance will be half 

 of the radius of curvature (or one quarter the focus of 

 a plano-convex lens) ; then, by exposing the other 

 side, we obtain the radii of the opposite surface. 

 This method was adopted by Professor Robinson, to 

 measure the different radii of double and triple 

 achromatic object-glasses. 



