554 Lord Rayleigh : Further 
not exactly opposite the contacts with the distance-pieces, 
but are displaced somewhat inwards or outwards in the 
radial direction (fig. 2). I£ the plates are too convex, the 
points of pressure must be displaced outwards. „. r 
In this form I have tried the experiment with a 
< 
K 
certain degree of success, but the displacements 
that I could command (1 mm. only) were too 
small in relation to the thickness of the plate. If 
it were intended to give this plan a proper trial, 
which I think it would be worth in order to 
render a larger aperture than usual available, 
the plates, or at least one of them, should be 
prepared of extra diameter, so that the bending 
forces could act with a longer leverage and at a 
greater distance from the parts to be employed optically. 
Such a construction need not involve a much enhanced cost, 
inasmuch as the outer parts would not need to be optically 
accurate. 
It may be worth while to consider the question here raised 
more generally. The problem is so to deform one surface, by 
forces and couples applied at the boundary, as to compensate 
the joint errors of the two surfaces and render the distance 
between them constant. If we take rectangular coordinates 
x, y in the plane of the surface with origin at the centre, the 
deformation obtainable in this way is expressed by terms in 
the value of £ (the other coordinate of points on the surface) 
proportional to x, y, a? 2 , xy, y 2 . x s , x 2 y. xy 2 , y 3 . For such 
terms are arbitrary in the solution of the general equation of 
equilibrium of a plate, viz. 
\dx 2 dy- 
4V)?=o. 
Of these terms those in x and y correspond of course merely 
to the adjustment for parallelism, and those of the second 
degree to curvature at the centre. The conclusion is that 
we may always, by suitable forces applied to the edge, render 
the distance between the plates constant, so far as terms of 
third order inclusive. 
Another inference from the same argument is that, in aut- 
optical apparatus, approximately plane waves of light may be 
freed from curvature and from unsymmetrical aberration 
(expressed by terms of the third order) by means of reflexion 
at a plate to the boundary of which suitable forces are applied. 
And the surface of the plate need not itself be more than 
approximately flat. 
