80 THEORY OF THE MICROSCOPE. 



Microscope, while the two others are displaced to the right 

 and left, in a parallel direction ; and let a I be the object, 

 whose separate points give equally perfect images with accurate 

 centering. If, with the help of the rays of direction, we con- 

 struct the path of the pencils which proceed from the marginal 

 points a and b to the objective, and compare it with the path 

 as it would be in the accurately concentric system, the errors, 

 due to the displacement can readily be comprehended. In the- 

 Fig. the ordinary lines refer to the excentric and the dotted 

 lines to the concentric system. The virtual or real images, which 

 the separate double-lenses form, are denoted by a V, a" b", and 

 a" V" in the first, and with similarly-accented Greek letters in 

 the second. The position of these images relatively to the 

 axis is, of course, previously determined by the direction of the- 

 rays, if necessary produced backwards ; their distances are regu- 

 lated by the focal lengths, and are, of course, in both cases the. 

 same ; for greater clearness they are, however, represented in 

 the Fig. as somewhat different. It is evident that the virtual image 

 a V of the first lens suffers no alteration by the displacement. 

 Considered as an object to the second lens, its right-hand margin,, 

 V, comes into a somewhat unfavourable position, since it is in 

 this case further from the optic axis o. 2 by the amount of dis- 

 placement. The lens therefore forms a less perfect image of 

 this portion of the margin, as may reasonably be supposed. The. 

 entire remaining portion (by far the greater) fulfils, on the other- 

 hand, afterwards, as before, the condition that none of its surface- 

 elements, for which the second lens is as far as possible aplanatic, 

 fall outside of the field of view. If we make b' x equal to the 

 amount of displacement, then in the image a" l>" a corresponding 

 portion I" x will be unsatisfactory, while a" x retains its original 

 distinctness. v Now, however, the whole image, as the Fig. shows,, 

 lies somewhat further to the right than before the displacement,, 

 and at a distance which is in the same ratio to b" x as a" l>" to. 

 a" b" a b': To verify this point, it is only necessary to draw 

 through I) and the upper principal point of the second lens, in its 

 two different positions, two lines, produce them downwards to /3" 

 and b", and compare the triangles meeting in b'. 



The virtual image a" l>" assumes, therefore, in general, a different, 

 position to the axis o 3 . The relative position, of course, remains, 

 the same only when it moves exactly as far as the axis itself. 



