270 
DR. T. R. MERTON AND PROF. J. W. NICHOLSON ON 
and the total breadth at any point is uniformly expanded from the normal breadth 2x. 
We may use a normal scale in which the interval d 0 (l + corresponds to the 
wave-length separation S A = e, and then the total breadth of the distorted parabolic 
fringe is the true normal breadth. In other words, if p and q are the observed intervals 
on the right and left of any fringe, the normal interval corresponding to e has a length 
d 0 + —, where d 0 = 2p — q, a = q— p, and the central fringe is on the right. This 
length becomes (p + q ), or the mean of the observed intervals. On this scale the 
apparent breadth of a distorted fringe is the true breadth of the undistorted one, if 
it is symmetrical about the central line when undistorted. 
In this normal system, the apparent breadth on the right, of a real breadth x, is 
x 
x - 
to a sufficient order, and on the left, 
x" — x - 
ax 
2d,: 
ax 
vL 
Conversely, the real breadths on this normal scale corresponding to observed breadths 
x' and x" are 
, cxx' 2 
X — X + - „ 
2d, 2 
X 
ax 
//2 
2d 2 
when insignificant terms are neglected. These results enable us in the next section 
to isolate the individual components of a broadened line, and to obtain their 
separation. 
The second manner in which the fringes are not normal is in regard to their 
heights. This effect has a much smaller influence on the contours of the individual 
fringes, although it appears as a striking phenomenon in itself on the photographs. 
A theoretical investigation, which need not be reproduced here, indicates that the 
summits of the maxima—P, Q, R, S, in the preceding figure—should lie almost 
exactly on a parabola when the light falls on the plate at nearly grazing incidence. 
This parabolic locus of the summits is indicated on the photographs, but in actual 
fact the parabola is very nearly a straight line. Indeed, a straight line can be passed 
with some precision through any three consecutive summits, and it has been found 
sufficient, for the degree of accuracy we desired to obtain, to consider only three 
summits, so that the straight line locus can be used. For any given fringe, say CR 
in the figure, the “ normal heights of points on the right must be slightly smaller 
than the observed values, and those of points on the left must be slightly larger. 
If axes are taken at C, the axis of y being along CR, and of x along CD towards 
the right, and if the equation to the “ normal ” parabola is 
h = CR, 
— hr 2 — y—h, 
