of Chromatic Dispersion. 167 
a view to determine the constants e and a. Seeing that, in the 
absence of 2, the difference between the wave-lengths correspond- 
ing to any two fixed lines before refraction, when divided by the 
difference between those same two wave-lengths within the 
refracting medium, would always be equal to e, it follows that, 
by taking the whole of the fractions which may be thus formed, 
we shall obtain as many values of ¢ as there are fractions ; and by 
taking the average of all these, the effects produced on the value 
of € by the variable quantity « will be mutually neutralized. 
Hence, by taking the sum of the 21 differences between the 
wave-lengths corresponding to the seven fixed lines before re- 
fraction (or as they exist in the unrefracted spectrum), and 
dividing them by the sum of their 21 differences within the 
refracting medium, the quotient will be «. The wave-length 
within the medium is found by dividing each normal wave- 
length by its own refractive index; so that, « bemg the index, 
we have —=b, —=c, &c. In this manner the positive and 
negative extrusions are made to neutralize each other. The pro- 
cess, however, may be shortened thus : 
(3B+4+2C+ D)—(F+2G+3H) ie 
(8b+4+2c+d) — (f+2g+3h) ~~ 
The numerator of this fraction is of course constant for all media, 
and its denominator varies with the refractive indices. 
The constant ¢ having been thus found, a may be easily de- 
termined as follows. Call B+C+D+E+F+G+H=sS, which 
is of course constant. Call b+ce+d+e+f+g+h=s, which will 
S 
=——s 
7é 7 
The two constants e and a having been thus determined, it is 
easy to find a second series 6, Co, d,, &c., showing what the 
wave-lengths within the medium would be, apart from their ex- 
trusions; that is, if the fixed lines retained the same relative 
mutual distances within the medium as they present in the 
unrefracted spectrum. This series is obtained by the formula 
C D 
——a=b),, ——a=c ——@g=d,, &e. 
2 BI | Pe a 
vary with the indices. Then is ze = u,, OF 
=a. 
In this series, b,, cs, do, &c., the following relations constantly 
subsist: €(b.—¢s) =B—C, e(cy— d,) =C —D, and so of the others. 
Then the differences between } and b,, between ¢ and cy, between 
d and d,, &c. are the extrusions. In this manner the extrusion 
due to each of the fixed lines may be ascertained in all those media 
whose seven indices have been observed with sufficient accuracy ; 
