6 THEORY OF THE MICROSCOPE. 



which we shall retain in the following examination of s3 T stems of 

 lenses, so that the aberration due to spherical form will be dis- 

 regarded. 



When the refracting surfaces, which bring about the union of 

 the rays (or of their prolongations) to form the image, are situated 

 so far from each other that a neglect of these distances in com- 

 parison with the other constants is inadmissible, the rules by 

 which the paths of the refracted rays are determined undergo an 

 essential modification. This case may occur as well with single 

 lenses of considerable thickness as with systems of lenses as 

 employed in the Microscope, and in the treatise of Gauss, the 

 most general case is supposed of there being no limitation of the 

 distances between the refracting surfaces. 



Such a system of refracting surfaces, with their centres of 

 curvature lying in a straight line, has, in the first place, the 

 property, in common with an infinitely-thin lens, of reuniting in 

 a point rays emanating from a point, or, in other words, of re- 

 fracting homocentric pencils of light in such a manner that after 

 the refraction they still remain homocentric. Secondly, it may be 

 shown that under all circumstances there exist two points on the 

 common axis, F and F*, which exactly coincide with the foci of a 

 single lens, since the planes drawn through these points at right 

 angles to the axis possess all the properties of focal planes. As, 

 however, the latter may, under certain circumstances, lie within 

 the limiting surfaces of the system, they can be characterized only 

 by the definition already given above, which is universally applic- 

 able, viz. : that to every incident homocentric pencil of rays 

 whose centre lies in the anterior 1 focal plane, there corresponds 

 an emergent parallel pencil ; and, conversely, to each parallel 

 pencil an emergent homocentric one whose centre lies in the 

 posterior focal plane. 



The distinguishing feature of a compound refracting system 

 consists in this, that two points, E and *, take the place of the 

 optical centre, the former in reference to the incident rays, and 

 the latter to the emergent rays. These points are called the 

 principal points of the system, and the planes drawn through them 

 at right angles to the axis are called the principal planes. Their 

 importance will be at once evident, if we try to determine by con- 

 struction the image of any object a b (Fig. 4), as before in the case 

 1 With regard to the direction of transmission. 



