SPHERICAL AND CHROMATIC ABERRATION. 71 



certain proportions ; but the difficulties in the way of the mecha- 

 nical execution of lenses of this description are such, that, for 

 practical purposes, this plan of construction is altogether un- 

 available. 



9 a. Various means have been devised for reducing the Aber- 

 ration of lenses of spherical curvature. It may be considerably 

 diminished, by making the most advantageous use of ordinary 

 lenses. Thus, the aberration of a plano-convex lens, whose con- 

 vex side is turned towards parallel rays, is only T Vbths of its 

 thickness ; whilst, if its plane side be turned towards them, the 

 aberration is 4J times the thickness of the lens. Hence, in the 

 employment of a plano-convex lens, its convex surface should be 

 turned towards a distant object, when it is used to form an image 

 by bringing to a focus parallel or slightly diverging rays ; but it 

 should be turned towards the eye, when it is used to render 

 parallel the rays which are diverging from a very near object. 

 The single lens having the least spherical aberration, is a double- 

 convex whose radii are as one to six : when its flattest face is 

 turned towards parallel rays, the aberration is nearly 3J times 

 its thickness; but when its most convex side receives or trans- 

 mits them, the aberration is only T J- w tlis of its thickness. The 

 aberration is further diminished, by reducing the aperture or 

 working-surface of the lens, so as to employ only the rays that 

 pass through the central part, which, if sufficiently small in pro- 

 portion to the whole sphere, will bring them all to nearly the 

 same focus. The use of this may be particularly noticed in the 

 object-glasses of common (non-achromatic) Microscopes ; in 

 which, whatever be the size of the lens itself, the greater portion 

 of its surface is rendered inoperative by a stop, which is a plate 

 with a circular aperture interposed between the lens and the rest 

 of the instrument. If this aperture be gradually enlarged, it 

 will be seen that, although the image becomes more and more 

 illuminated, it is at the same time becoming more and more in- 

 distinct; and that, in order to gain defining power, the aperture 

 must be reduced again. Now this reduction is attended with 

 two great inconveniences ; in the first place, the loss of intensity 

 of light, the degree of which will depend upon the quantity 

 transmitted by the lens, and will vary, therefore, with its aper- 

 ture; and, secondly, the diminution of the "angle of aperture," 

 that is, of the angle (a b c, Fig. 8) made by the most diverging of 

 the rays of the pencil issuing from any point of an object, which 

 can enter the lens ; on the extent of which angle depend some 

 of the most important qualities of a Microscope ( 100). 



10. The Spherical Aberration may be got rid of altogether, 

 however, by making use of combinations of lenses, so disposed 

 that their opposite aberrations shall correct each other, whilst 

 magnifying power is still gained. For it is easily seen that, as 

 the aberration of a concave lens is just the opposite of that of a 

 convex lens, the aberration of a convex lens placed in its most 



