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SUMMARY OF CURRENT RESEARCHES RELATING TO 
can construct afterwards or can calculate with great rapidity the image 
of any point placed at any distance whatever from the system. 
The greater simplicity of the new method arises from considering 
those rays which undergo neither deviation nor displacement either at 
the entrance into or exit from the different media, so that the faces of 
the lens or the external surfaces of the system perform the function of 
the principal planes of Gauss, the centres of curvature of these surfaces 
that of the nodal points of Listing, and their images or centric points 
that of the principal foci of the optic system. 
Without now entering into minute details of the new method, it wall 
be sufficient to show how, by having recourse to it, we can easily find 
the centric points of a given lens, and how, once these points are found, 
we can easily construct the image of any object seen through the lens. 
We shall thus see whether the proposed method deserves or not to be 
preferred to others. 
In order to find practically the position of the centric points of a 
given lens, we measure its thickness y, and with the spherometer, or by 
reflection or otherwise, the radii of curvature r and r x of its first and 
second surfaces. Having obtained these quantities we place normally 
to the axis of the lens an object of a known size o g, at a determinate 
distance a g from one of the faces, and we find the image g l either 
real or virtual of the object, seen through the lens, measuring this 
image, and determining its distance b g from the other surface. 
Then by drawing a straight line from the extremity o of the object to 
the centre c of curvature of the first face of the lens, this straight line -will 
cut the last face in a certain point ; by drawing a straight line from 
the extremity o 2 of the image to the centre of curvature c x of the last 
face, we mark by m the point in which this straight line cuts the first 
face of the lens. Join o x to wq, the point q, in which the straight line 
Ox mx cuts the axis of the lens, will be the first centric point, that is, the 
place of the image of the centre c of the first face seen through the 
second. Let o be similarly joined to m, the point g u in which the line 
o m cuts the axis, will be the second centric point, that is, the image of 
the centre c x of the second face seen through the first. Having thus 
obtained the points q and q l9 the construction of the principal or conju- 
gate foci of the system and that of all the images which it can give, can 
be made exceedingly rapidly, and we can then deduce very easily the 
places of the principal planes, the nodal points, the optic centre, &c., 
if we wish to treat the problems relating to the given lens by the 
methods of Gauss, Listing, or other mathematicians. 
The preceding diagram shows at once how we may obtain the image 
of a point o placed outside the axis of the lens. (If the point given were 
on the axis, we might raise from it a perpendicular to the axis, and 
determine the image of any point on this perpendicular, drawing from 
the image obtained a normal to the axis itself. The meeting point of 
this normal and the axis would be the place of the image of the given 
point.) Let a straight line be drawn from the point o to the centre c of 
the face through which it is intended the light should pass ; such a 
straight line will represent a luminous ray, which starting from o will 
pass, neither deviating nor displaced through the lens, until it meets in 
m, the second face. The ray having reached m, will deviate towards the 
