REFRACTION BY CONVEX AND CONCAVE LENSES. 69 



different curvatures ; the principal focus of such a lens is found 

 by multiplying the radius of one surface by the radius of the 

 other, and dividing this product by half the sum of the same 

 radii. The rules by which the foci of convex lenses may be 

 found, for rays of different degrees of convergence and diver- 

 gence, will be found in works on Optics. 



6. The refracting influence of concave lenses will evidently be 

 precisely the opposite of that of convex. Kays which fall upon 

 them in a parallel direction, will be made to diverge as if from 

 the principal focus, which is here called the negative focus. This 

 will be, for a plano-concave lens, at the distance of the diameter 

 of the sphere of curvature ; and for a double-concave, in the 

 centre of that sphere. In the same manner, rays which are con- 

 verging to such a degree, that, if uninterrupted, they would have 

 met in the principal focus, will be rendered parallel ; if con- 

 verging more, they will still meet, but at a greater distance ; and 

 if converging less, they will diverge as from a negative focus at 

 a greater distance than that for parallel rays. If already diverg- 

 ing, they will diverge still more, as from a negative focus nearer 

 than the principal focus ; but this will approach the principal 

 focus, in proportion as the distance of the point of divergence 

 is such, that the direction of the rays approaches the parallel. 



7. If a lens be convex on one side and concave on the other, 

 forming what is called a meniscus, its effect will depend upon the 

 proportion between the two curvatures. If they are equal, as 

 in a watch glass, no perceptible effect will be produced ; if the 

 convex curvature be the greater, the effect will be that of a less 

 powerful convex lens : and if the concave curvature be the more 

 considerable, it will be that of a less powerful concave lens. The 

 focus of convergence for parallel rays in the first case, and of 

 divergence in the second, may be found by dividing the product 

 of the two radii by half their difference. 



8. Hitherto we have considered only the effects of lenses upon 

 a " pencil" of rays issuing from a single luminous point, and 

 that point situated in the line of its axis. If the point be situated 

 above the line of its axis, the focus will be below it, and vice versd. 

 The surface of every luminous body may be regarded as compre- 

 hending an infinite number of such points, from -every one of 

 which a pencil of rays proceeds, and is refracted according to the 

 laws already specified ; so that a perfect but inverted image or 

 picture of the object is formed upon any surface placed in the 

 focus, and adapted to receive the rays. It will be evident from 

 what has gone before, that if the object be placed at twice the 

 distance of the principal focus, the image, being formed at an 

 equal distance on the other side of the lens ( 5), will be of the 

 same dimensions with the object: whilst, on the other hand, if 

 the object (Fig. 5, a b) be nearer the lens, the image A B will be 

 farther from it, and of larger dimensions ; but if the object A B 

 be farther from the lens, the image a b will be nearer to it, and 



