Theory of the Dispersion of Light. . 25 
mula expressing the relation between the length of a wave 
and the velocity of its propagation. In a paper in the Phi- 
losophical Transactions for 1835, Part L, I have exhibited the 
results of calculation by means of this formula, by which theory 
is compared with observation for all the cases determined by 
M. Fraunhofer, and it will, I believe, be admitted that the ac- 
cordance is as close as can be reasonably expected. 
Since that paper was printed I have been indebted to Pro- 
fessor Sir W. R. Hamilton for bringing to my notice the cir- 
cumstance that the formula as there deduced, owing to certain 
assumptions made in the course of the investigation, is not 
absolutely rigorous, although under conditions which may be 
easily admitted as likely to subsist it is reduced to the form 
which I have used. ‘The state of the case will be rendered 
evident from the following considerations. 
In order to simplify the investigation M. Cauchy adopts 
the method of supposing an expression, which really consists 
of the sum of a series of analogous terms, reduced to a single 
term; upon this he pursues his inferences with respect to it, 
and then in the conclusion recurs to the summation again. 
The complete resulting expression would represent the mo- 
tions of an entire system, considered as produced by the com- 
bination of many, or even an infinity of, similar motions, each 
represented by the simplified equations obtained with the 
omission of the sign of summation. This will be understood 
on a comparison of those parts of my abstract which intro- 
duce equations (21.) and (56.). On the same supposition 
I have proceeded to that deduction which leads to the formula 
expressing the relation between the length of a wave and the 
refractive index. (See p. 265, Lond. and Edin. Phil. Mag., 
April 1835.) ; 
The formula thus deduced, in its simplified shape, viz. 
\- (2 
at Lala sin 
(9) 
= ~ 
(eau 
~ 
is obtained by collecting together into one constant (H/’) the 
sum of a number of terms of analogous forms which com- 
pose the values of the coefficients L, M, &c. Now if we recur 
to the expressions from which these values were originally 
derived, the equations (22.) and (12.), (or in the original 
memoir, more explicitly, equation (20.),) we shall readily per- 
ceive that the values of these coefficients in their exact form 
(that is, retaining the sign of summation,) are such as these: 
Third Series. Vol. 8. No. 43. Jan. 1836. E 
