110 
SUMMARY OF CURRENT RESEARCHES RELATING TO 
patch, and spread it as nearly equally as possible over the area covered 
by the transparent slide, so that the whole is equally illuminated, and 
so that the light so transmitted shall be on the whole slightly con- 
vergent. To measure the aberrations or exact focal powers of such 
lenses would be a useless work. 
However, it will be convenient at the outset to enumerate all the 
things which might be made the subject of measurement with respect 
to a lens or combination of lenses. These are no fewer than eighteen in 
number: — (1) Diameter, or linear aperture. (2) Thickness, or length 
from pole to pole. (3) Focal power, or its reciprocal the focal length. 
(4) Position of principal focal planes. (5) Position of optical centres 
(“ principal points ” of Gauss). (6) Angular aperture. (7 ) Chromatic 
aberration. (8) Spherical aberrations, lateral and longitudinal. 
(9) Chromatic difference of the spherical aberration. (10) Loss of 
light by reflexion from surfaces. (11) Absorption of light in trans- 
mission. (12) Illumination of field, central and marginal. (13) 
Complaneity of focus (included in 7 and 8). (14) Degree of 
distortion of image (rectilinearity). (15) Cvlindricity, or degree of 
astigmatism, including angle of axis of cylindricity. (16) Accuracy 
of centering. (17) Definition in margin of field (involved in 7, 8, and 
16). (18) Refractive indices of materials. 
Now, of all these varied matters, there are but three with which the 
present paper will deal : namely, the focal power of lenses, and the 
position of their focal planes and principal points. 
By focal power I mean, of course, that property on which their 
convergivity (positive or negative) depends, and on which in turn their 
magnifying (or minifying) action is dependent. It must be borne in 
mind, as a fundamental principle of elementary optics, that all that any 
lens or mirror (or combination of mirrors or lenses) can effect is to 
imprint a curvature on the wave-front of the light that enters it. If 
the wave is plane — i. e. consists of parallel rays, to use the old lan- 
guage — then the lens prints a curvature, positive or negative, upon it by 
virtue of which its march is changed, and made convergent or divergent. 
If the wave before impinging on the lens is initially non-plane, but 
either convergent or divergent, then the lens will alter the curvature of 
the surface, the resultant curvature on emerging being simply the 
algebraic sum of the initial curvature and the impressed curvature.* 
The focal power is the curvature imprinted by the lens on a plane 
wave, and is the reciprocal of the true focal length. It is appropriately 
expressed in terms of the proper unit of focal curvature, the dioptrie . f 
* All the ordinary formulae of the text-books for lenses are more or less particular 
statements in symbols of this general rule ; for example, the well-known formula 
1 1 , 1 
- = - T Ti 
V U J 
in which / is the focal length, and u and v the respective distances of two conjugate 
points serving as object and image. The reciprocal of a length is a curvature ; so 
that this formula merely states that one curvature is the result of adding two other 
curvatures together. I have pointed this out in a paper in the ‘ Philosophical Maga- 
zine,’ in Oct. 1889. 
f The dioptrie , originally proposed by Monoyer as the unit of focal power of a 
lens, is now in international use, having been formally adopted by the International 
Medical Congress of Brussels. It is a unit of curvature, and as such may be used 
