and, instead of it, I have maintained the simple conclusion of two kinds of heat simultaneously emanating or originating from luminous hot bodies.

As to contending that this distinction of two kinds of heat "suffices to explain all the facts relative to transmission," I should have been glad if M. Melloni had pointed out any passage in which I have contended for anything of the kind. The facts of transmission (and for all these most curious and important facts we are entirely indebted to the experimental skill of M. Melloni) are of a kind as yet appearing so little reducible to fixed laws that I should imagine any theory entirely premature; certainly I have offered none.

The experiment described by M. Melloni (p. 477 at the bottom) is undoubtedly a most curious and interesting one, but how it applies to the question relative to mine I fail in perceiving. It proves, the author conceives, that there are in this case "several different kinds of dark heat." It is true we have hitherto known of but one, and I have referred only to one in my researches, but I have never denied that there may be two, three, or a hundred kinds. I have merely maintained that there are characteristic and well-marked distinctions in the properties of any or all non-luminous heat (to adopt for brevity the barbarously incorrect language which is becoming current) from those of the luminous kind; but there may still be many more such characteristic distinctions, and some such M. Melloni seems to have established in this experiment.

I look with great interest to the further extension of this curious inquiry on a point requiring the most careful examination; while I acknowledge that the thermomultiplier of M. Melloni has opened to us an entirely new field of investigation, and in the hands of its inventor and of Prof. Forbes has done more for the advance of our knowledge in this department within a very short time past, than the most sanguine would have ventured to anticipate.

VIII. Further Observations on M. Cauchy's Theory of the Dispersion of Light. By the Rev. BADEN POWELL, M.A., F.R.S., Savilian Professor of Geometry, Oxford.*

Na

IN

a former paper, which is inserted in successive portions of this Journal, No. 31 et seq. (vol. vi.), I have given an abstract of M. Cauchy's highly important researches on the undulatory theory, so far as they bear on the great question of the dispersion of light, and in conclusion have deduced a simple for

• Communicated by the Author: on this subject see also a preceding article by Mr. Tovey.

mula expressing the relation between the length of a wave and the velocity of its propagation. In a paper in the Philosophical Transactions for 1835, Part I., 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 accordance is as close as can be reasonably expected.

Since that paper was printed I have been indebted to Professor Sir W. R. Hamilton for bringing to my notice the circumstance 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 motions of an entire system, considered as produced by the combination 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 introduce 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.

is obtained by collecting together into one constant (H') the sum of a number of terms of analogous forms which compose 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 perceive 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

k2

= S{ (F (m, r, a) + F (m, r, ß) +, &c.
c.). 4 (k, r, cosε) } .

Hence by the same process as that employed before, we may obtain a corresponding abridged expression

To perceive more clearly the difference between the exact and approximate expressions, we may first observe, that since we have from equations (19.) and (45.)

the arc which is involved in the formula becomes A..

Now, if we take the simplified formula, develope the sine in terms of the arc, and divide by the arc, we shall have

whereas the exact formula, in the same way, would give

;

S (H2)

supposing the series to converge rapidly enough.

Now, this would manifestly be the same as the last if we were at liberty to suppose

$ [(H)2 (^; : *)'] = [S (Ho)] ·[^;· *]',

and similarly in the other terms: in which case we should

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) (Ag2)] = (S (H2) (A g2)

for the same value of a; or, since Ag is the difference in perpendicular distance from a given plane, of the molecule at the point ry z at the end of the time t, it will be evident, on a little consideration, that to disregard the sign of summation altogether corresponds to taking into account only the action of two adjacent molecules. If again we apply it only to H2 (as in our simplified formula), without regarding Ag as variable, this is equivalent to considering only the action of two adjacent parallel strata of molecules, for all of

which Ag is the same. But if (~) be small, and the series consequently converge rapidly, (4) being stili 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 on 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 specificially pointed out by the Irish Astronomer Royal in his third Supplement to the Theory of Systems of Rays in the Transactions 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 dispersionformula, 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 important researches.

μ

in a series of powers

The development of the value of (1) of λ, in a form available for the actual comparison of theory with observation, by the use of a peculiar method for determining the coefficients, appears also to have been lately investigated by M. Cauchy. His "Exercices de Mathématique," which, as I stated in a former paper, were broken off abruptly in 1830, have now been resumed, and are in the course of publication

at Prague, under the title of "Nouveaux Exercices," &c. They will contain the continuation of the theory of dispersion, and the development in a form adapted to calculation. The distinguished author also has recently produced a memoir on interpolation, by a new method, which in conclusion he briefly applies (but without sufficient explanation) to the calculation of the refractive indices, in one instance of flint glass from Fraunhofer.

One thing, however, is clear, viz. that from the close accordance between all the results which I have calculated (by the approximate formula) and those of observation, viz. the ten sets of indices obtained by Fraunhofer, and since that, ten other sets determined by M. Rudberg (very recently communicated to the Royal Society*), it is sufficiently evident that at least for all these cases the approximate supposition is as near the truth as, perhaps, will be thought sufficient, when all circumstances are considered.

It is, however, still quite conceivable that the differences, minute as they are, may be accounted for by a more accurate prosecution of the analysis. Again, it remains to be seen whether in other cases, especially those of more highly dispersive media, the same method will still apply, or whether we must have recourse to a more complex investigation, which shall yet include, as a simplified case, the formula which holds good for media of low dispersive power.

IX. A Sketch of the Geology of West Norfolk. By C. B. ROSE, Fellow of the Royal Medical and Chirurgical Society of Lon

don.

[Continued from vol. vii. p. 376, and concluded.]

Diluvium.-CLAY, sand, or gravel of varying thickness, and 'frequently alternating beds of these substances, are found immediately incumbent on the chalk, and obscure in many places its outcrop, as they also do that of the gault, lower greensand, and clays of Marshland+. These irregular beds, alternating with each other, without any order of superposition, have received the name of diluvium; but it is so difficult to determine what has been deposited by diluvial agency, in other

So long ago as 1827 we received and inserted in Phil. Mag. and Annals, N.S., vol. ii. p. 401, a paper on the undulatory theory of dispersion from M. Rudberg. Has this been lost sight of in the recent investigations of the subject? Some of the calculated numerical results obtained by M. Rudberg, we observe, are identical with those obtained by Professor Powell, as given in Phil. Trans. 1835, pp. 252, 254.--EDIT.

The diluvium in Marshland is covered by a considerable thickness of alluvial deposits.