Theory of the Dispersion of Light. 27 
only have a common multiplier for the whole series, which 
would be represented by H'. Now this supposition would be 
the same as that of 
S [(H®) (Ag)] = (S (HP) (Ag) 
for the same value of A; or, since A g is the difference in per- 
pendicular distance from a given plane, of the molecule at the 
point vy = at the end of the time ¢, it will be evident, on a 
little consideration, that to disregard the sign of summa- 
tion altogether corresponds to taking into account only the 
action of two adjacent molecules. If again we apply it 
only to H* (as in owr simplified formula), without regarding 
Ae as variable, this is equivalent to considering only the ac- 
tion of two adjacent parallel strata of molecules, for all of 
ae 
which Ag isthe same. But if i) be small, and the series 
A : : : 
consequently converge rapidly, (=") being still of sensible 
magnitude, we may suppose that this is not far from the truth. 
I will not, however, say more with regard to the analysis 
of the theory at present, as the subject has been taken up by 
Sir W. R. Hamilton, with whose researches Gn systems of 
rays, in fact, the other parts of M. Cauchy’s investigations 
are closely connected. My abstract has been restricted to so 
much of those investigations as refers directly to the subject 
of the dispersion; but the entire theory, of which it forms a 
part, embraces the curious and beautiful discussion of wave 
surfaces: and the connexion and analogy of some of the most 
important of these results with his own researches are speci- 
ficially pointed out by the Irish Astronomer Royal in his third 
Supplement to the Theory of Systems of Rays in the Trans- 
actions of the Royal Irish Academy, vol. xvii. p. 125 and 141. 
Since this paper went to the press, that eminent mathematician 
has kindly given me permission to make what use I please of 
some further investigations on the subject of the dispersion- 
formula, including its numerical applications, which he had 
communicated to me. I hope, therefore, in a subsequent 
Number of this Journal to give some account of these im- 
portant researches, 
The development of the value of fe) in a series of powers 
of A, in a form available for the actual comparison of theory 
with observation, vy the use of a peculiar method for determin- 
ing the coefficients, appears also to have been lately investigated 
by M. Cauchy. His “ Eercices de Mathématique,” which, 
as | stated in a former paper, were broken off abruptly in 1830, 
have now been resumed, and are in the course of publication 
EK 2 
