678 Lord Rayleigh on Hamilton's Principle and 
in another, but rather that the nature of the errors is different. 
In earlier writings the term aberration was, I think, limited 
to imperfect focussing of rays which, issuing from one point, 
converge upon another. Three of the five aberrations are of 
this character ; but the remaining two relate, not to imper- 
fections of focussing, but to the position of the focus. It is, 
in truth, something of an accident that, e. g. in photography, 
we desire to focus distant objects upon a plane. The second 
thing to which I wish to refer is that, although Seidel did 
much, four out of the five aberrations were pretty fully dis- 
cussed by Airy and Coddington before his time. To these 
authors is due the rule relating to the curvature of images, 
generally named after Petzval, so far, at any rate, as it 
refers to combinations of thin lenses. 
Some remarks are appended having reference to systems 
of less highly developed symmetry. 
According to Hamilton's original definition of the charac- 
teristic function Y, it represents the time taken by light to 
pass from an initial point (V, y', z') to a final point (.r, y, z), 
and it may be taken to be j \x ds, where ja is the refractive 
index and the integration is along the course of the ray which 
connects the two points. If the path be varied, the integral 
is a minimum for the actual ray : and from this it readily 
follows that 
I = dY/dx, m = dY/dy, n = dY/dz, . (1) 
- V = dY/dx 1 , -m' = dY/dy', - ri = dV/dz', . (2) 
where 2,m,w, l',m',n' are the direction-cosines of the ray 
at the end and beginning of its course, the terminal points 
being situated in a part of the system where the refractive 
index is unity. 
In his communication to the British Association (B. A. 
Report, Cambridge 1833, p. 360) Hamilton transforms these 
equations. As his work is so little known, it may be of 
interest to quote in full the principal paragraph, with a 
slight difference of notation : — " When we wish to study the 
properties of any object-glass, or eye-glass, or other instru- 
ment in vacuo, symmetric in all respects, about one axis of 
revolution, we may take this for the axis of z, and we shall 
have the equations (1), (2), the characteristic function Y being 
now a function of the five quantities, x 2 +y 2 , xx'+yy 1 , 
a' 2 -\-y' 2 , z, z', involving also, in general, the colour, and 
having its form determined by the properties of the instru- 
ment of revolution. Reciprocally, these properties of the 
