PHENOMENA RELATING TO THE SPECTRA OF HYDROGEN AND HELIUM. 269 
n+ 1 th fringe should be very accurately given by the formula a + ftn where a and ft 
are constant. In other words, these distances are in arithmetical progression. 
Measurement of the plates shows that this law is almost exactly fulfilled in our 
photographs, the term in n 2 , which would follow, having so small a coefficient as to be 
negligible for fringes not too far from the central one—signifying, in practice, the first 
five or six. This inequality of dispersion of the plate could be corrected graphically, 
but a simple interpolation formula deduced as follows is more convenient, for it 
elucidates the principal effects in a clear manner. 
Let d 0 be the “ normal ” interval between two maxima, for a certain uniform 
dispersion, and d 0 + a 3 , d 0 + 2a, ... , the intervals actually found. Thus d 0 is the 
interval corresponding to the wave-length separation e. The intervals d 0 + a, d [: + 2a 
are DC, CB, in the figure, and may be called p and q. Then 
d 0 = 2p-q 
gives the normal scale to which the fringe of axis BC is being reduced. The whole 
visible fringe for this axis is shaded in the figure. 
The law of the intervals between the fringes can be expressed by the formula 
t a \ , a z 2 
2dj2d} 
where a “ normal ” interval 2 reckoned from any specified starting point becomes z' on 
the photograph. Thus if 2 = d t) , z' — d 0 + a, and if 2 = 2 d 0 , 2 ' = (d 0 + a.) + (d 0 + '2a,). A 
line of breadth x on the normal scale, on one side of the point d Q , becomes situated at 
the point d\ +xJ on the photograph, where 
d' 0 + x' 
and as d' 0 = + a, we find 
There is a difference between the right and left sides of the central line in the 
figure of the fringe. For on the left x' is positive, and on the right it is negative, 
while the last term in this formula is always positive. Thus the fringes are dis¬ 
torted—shortened on one side and expanded on the other. If x' is the breadth on 
the right, and x" on the left, corresponding to a normal breadth x on either side 
whence 
/, ,oa \ 
= x [ 1+ u) 
3a \ ax 2 
+ 
0 / 2d 0 
2 > 
x —-xi I + 
a \ ax 
2 dj 2 dft 
x’+x"= 2x11 
\ 2 d 0 / 
2 p 
(do + x ) ( 1 + TftY ) + (d 0 + x )\ 
-J _ ^ . 3 
Xa . X a 
X + f-^r + 
da 2d ft 
VOL. OCXVII. A. 
