of Edinburgh, Session 1881-82. 
419 
represents the general form of a central section of such a bull’s-eye. 
It is evident that the focal length of successive annuli of such a piece 
of glass, treated as a lens, increases from the central portion to the 
circumference, where it becomes infinite. For an approximate study 
of its behaviour we may assume that the focal length of an annulus 
of radius r is b 2 /(a - r), where a is the extreme radius, at which the 
sides of the pane become parallel. Suppose sunlight, passing 
through a narrow slit, to fall on such a lens at a distance e from its 
centre, and to be received on a screen at a distance c from the lens. 
It is easy to see that the polar equation of the illuminated curve on 
the screen is (the pole being in the axis of the lens) 
b 2 - ce sec 
This curve can be readily traced by points for various values of the 
constants. In fact, if r be the radius vector of a straight line, the 
vector of any one of these curves (drawn in the same direction) is 
proportional to r (A — rj, and the curve can therefore be con- 
structed from a straight line and a circle. Here the value of A is 
( ac - b 2 )/c; i.e., it is a fourth proportional to c, a, and the distance 
of the screen from the focus of the central portion of the lens. 
When A is small compared with the least value of the curve has 
a point resembling a cusp, but as A increases the kink appears. 
This is easily observed by gradually increasing the distance of the 
screen from the lens ; and the traced curves present forms which 
are precisely of the general character of those observed. 
2. On the Nature of the Vibrations in Common Light. 
One of the few really unsatisfactory passages in Airy’s well- 
known “ Tract ” on the Undulatory Theory of Optics is that which 
discusses the nature of common light To explain the production 
of Newton’s rings in homogeneous light to the number of several 
thousands, it is necessary that at least several thousand successive 
waves should be almost exactly similar to one another. On the 
other hand, we cannot suppose the vibrations (which will in general 
be elliptic) to be similar to one another for more than a small 
fraction of a second ; if they were so, we should see colour pheno- 
vol. xi. 3 H 
e sec 0 
(ac — 
