470 
PROF. J. W. NICHOLSON AND DR. T. R. MERTON ON THE 
where n is a definite number, \fs ( y ) cc y n } and f(x) = e = e <lx \ where q is constant. 
The law of intensity in the original line is therefore 
I = I Q e ~ qx \ 
being the probability curve when the final curve is a parabola. A case of obvious 
interest is that of an ordinary exponential distribution of intensity 
I = \e~ qx , 
where q is constant, in the original line. The equation to the photograph would 
then be 
qy = px tan a, 
so that it becomes a straight line, from which the value of q could be measured 
at once when the optical properties of the wedge, defining p and a, are known. This 
equation, of course, like those preceding, applies to one side of the photograph only, 
the other side being the optical image of the first in the axis of the photograph. 
For example, the present case would present, as the complete boundary, two straight 
lines intersecting at the apex of the curve, and inclined at an angle closely given 
by 2 pa/q, where a. is the small angle of the wedge. 
If the law were partly exponential and partly of the probability type, or 
I = I 0 exp (— k 2 x 2 —qx), where k 2 and q are positive constants, the graph on the plate 
would become 
k 2 y 2 +qy = px tan a, 
and it is easily demonstrated that this is a parabola exactly similar—and, in fact, 
equal—to the parabola obtained when q = 0, but shifted on the plate so that its axis 
has moved parallel to itself. The curve is therefore still symmetrical about its axis 
and curved towards it, so that this mixed law is incapable of explaining the 
characteristics of the photographs even on general grounds. In fact, the constant q 
has no influence on the radius of curvature. It would, however, have an influence 
if the law with which the simple exponential is combined were anything other than 
the law of probability. 
This lack of influence of q , however, only applies to corresponding points, and 
in order to avoid misconception, a more complete account is necessary, for the curve 
as seen on the plate would actually appear flatter and be of smaller extent. At the 
same time it undergoes a sudden change of curvature at the vertex. In the annexed 
figure (fig. 3 ) the dotted curve is the parabola which would be obtained when q = 0 . 
The other parabola ABODE is the shifted parabola obtained when q is not zero, 
as explained already. ' But only the portion ABC appears on the plate, for the 
axis OX cannot be disturbed by the presence of the new exponential factor, which 
must also lessen the height of the curve to the value CX. What is actually seen 
is ABC and its image in OX, or the shaded area in the figure bounded by two arcs 
