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XL. On the Formula for the Dispersion of Light derived from 
M. Cauchy’s Theory. By the Rev. Baven Powet1, M.4A., 
F.R.S., Savilian Professor of Geometry, Oxford. 
(Continued from p. 24.) 
N a former Number of this Journal I stated in a general 
way the nature of some considerations of importance re- 
specting the formula of dispersion deduced from M. Cauchy’s 
theory, promising at the same time to give some further ac- 
count of the other researches of Sir W. R. Hamilton, with 
which it is connected, and of which the author has kindly 
allowed me to make use. In saying that that eminent mathe- 
matician had “taken up the subject,” I beg it to be under- 
stood (and I mention it at his request,) that I allude to no 
other investigations than what are here to be given. In the 
present paper, therefore, I shall proceed to explain his method 
of comparing, in the first place, the degree in which the ap- 
proximate and exact formulas accord with each other; and 
in the second, the analysis by which he deduces a process of 
computation by the exact formula: the numerical results, as 
derived from the two methods, will then be compared. Some 
illustrations of other points will form the subject of a future 
communication. 
Upon the principles explained in the last paper, if we take 
the development of the sine divided by the arc, and square 
the polynomial, using the approximate formula, 
] sin 6,\2 
a> js Bef i ') (1.) 
we shall have 
1 H 2 1 H,? 6,2 2 H,2 4 
Sg ey eg Ea fh a ae i? O4—, &e. (2.) 
2 
But if we use the exact formula, 
1 k > (sin 6y* 
ihn s{ (=) \ (3.) 
. - A 
and substitute for @ its value a 
we shall in like manner have, 
1 2 Diamine Z 2 4 
p= S(H)- Z (5) Sad) +e () Scag) 
=), Occ. (4.) 
ue 
