DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
349 
We have 
e Vdp ( ve h T 
JO 
odv 
+ ^ e-^dp J 
ve 
jp \ 2 fvf_ y 
XV J + [2-jtVJ 
~ k t 
\ 
xv 
dv 
2ttV 
— 00 
CO 00 
-p 
e 
0 'J V 
-hv 2 
v dp d v 
(- - £)'* (f,) 
V \ 2 + 
P \ ( v .t 
x + XV) + \ 27tV 
or, changing the order of integration, 
d o dp . ve /,i:2 
(w o 
p \ 2 / vf 
X ~ xv) + (brV 
• /' \ 2 + 
■ * £)'* (&; 
• (vL) 
Visibility Curve. 
§ 43. Professor Michelson* has shown that, although the breadths of elementary 
spectrum lines cannot in general be examined directly, yet the application of his 
interference method enables one to obtain much more detailed information. The 
light is made to interfere with itself, at a relative retardation u of the two half 
streams. Interference bands are produced and their “ visibility ” estimated for 
different values of u, the path-difference. From the visibility-curve thus constructed 
we can work backwards to the breadth of the spectrum line, and find out something 
about the distribution of light in this breadth. 
Michelson has shown that, if <f>(x) represent the intensity of light for position x 
in the spectrum, and 
+ co 
C = | COS 27 TUX 
dx 
s 
= j <£(*) si 
sin 27 tux dx 
Q = I <f)(x)dx 
and 
V- = ° s + s 
Q* ’ 
then V is the visibility-function, in terms of u, the path-difference. 
Michelson, ‘ Thil. Mag.,’ vols. 31 and 34. 
