ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 123 
of the real or virtual images which may be produced by the system 
itself. 
The determination a, priori of these points (like the determination of 
the points and planes of Gauss and Listing), demands the knowledge 
of the length and sign of the radii of curvature of the two surfaces of 
the lens, that of the thickness of the lens, or the axial distance of the 
two refracting surfaces, and finally that of the relative velocity of light 
in the three successive media— that is to say, of their relative indices of 
refraction. We can with these data alone construct or calculate the 
Fig. 9. 
o 
place of the centric points q and q u without first determining the 
principal foci and the principal distances or anterior foci of the two 
surfaces of the lens ; we can also, if we wish, determine these quantities 
which, introduced in successive calculations or in ulterior constructions, 
abbreviate or simplify the work. 
In any case, having obtained the two centric points, we have no 
further need either of the optic centre, or of its two images, or the 
nodal points of Listing, or Gauss’ principal planes, or the principal 
foci of the whole lens, to construct or calculate the places, positions, 
and size of the images. And as such constructions are made very 
quickly, we may use them in order to find the final effect of any series 
whatever of surfaces and of different refracting media, centered on the 
same axis. 
It is not, therefore, necessary in the case of optic systems to have 
recourse to the laborious process of construction or calculation by means 
of successive images, for there can always be determined in every 
optic system (however complex) the images of the centres of curvature 
of its first and of its last surface, seen successively through the whole of 
the rest of the system, observing the image of the centre of the first 
surface through the second, then the image of this image through the 
third, and so on to the image of all the preceding images, seen through 
the last surface, and repeating the same operation in the opposite 
direction for the centre of the last surface and for its successive images 
up to the last, which is seen through the first surface. In this way the 
centric points of the whole system are obtained, by means of which we 
