to recur in a separate Memoir, and for that reason shall defer any further consideration of the subject until then.

1

Another observation that suggests itself is, that since p is a factor

of 1, and

€

q of E, the product fq is independent of e, and consequently would, were the action of the particles of æther alone influential, be the same for all substances. By a reference to the numerical values above given, it will appear that this only holds as a very rude approximation for the glasses, and altogether fails for the fluids. Thus much, however, may be gathered from an inspection of the tables, that there is a tendency to verify the result, and that we should not be induced to regard the effects of the material particles of such considerable magnitude, as to vitiate the general conclusions. When, however, our results are pursued into details, the action of the material particles (or whatever other actions we choose to consider, if these be rejected) produces a sensible effect. For it will be observed, that all the solids have negative; whilst, on the contrary, water and solution of potash make it positive. Now the distances between the particles cannot affect these signs, nor can the absolute forces, as long as these forces are supposed all of the same nature. This, then, is a difficulty in our way which it would be well to remove. I cannot, however, enter into this subject further than to observe, that if there were no extraneous forces, the quantity would undoubtedly be positive; and that, as the action of the particles on each other is attractive, an alteration in sign must arise from an addition of repulsive effects; and that since these effects are not particularly great in affecting p and q, the function to which they will give rise will be a series not so rapidly converging as that which expresses the velocity due to the actions of the particles of æther on each other.

I ought, however, to state that it is not impossible that this effect would have been explained, had we taken into account the terms in the expansion of (r+p). p (r+p), &c. to the third order. This expla

nation I have not been able to succeed in deducing, as the equation of motion then assumes the following form:

which would seem to indicate that the velocity is not altogether independent of the extent of vibration.

It would, however, lead us too far into speculations, which, after all, may have little grounds to rest upon, should we pursue what I have here barely alluded to.

We shall obtain a value of by dividing p by q, for supposing the wave transmitted parallel to the axis of y,

which will serve to determine e, if the above numerical quantities can be assigned.

The numerator can be determined without any considerable difficulty, but owing to the very slow convergence of the denominator, I have not been able to assign its value to any degree of accuracy, I shall consequently content myself with proving (what is essential to my remarks on the transversality of the vibrations) that each of the quantities is positive.

Let n be any values of 7 and , then the position of the numerator, omitting the known factor, for these particular values of the " and is

And for the next wave the expression becomes Σ.

which gives a result of the same form. Hence the sum of these two

terms is Σ taken only on one side, which is an essentially (§ 2 + (12 + m2 2) 3

2

positive quantity; and this is true for every particular value of " and X, and is therefore true for the sum of all the values; whence the numerator above is a positive quantity. Similar to the corresponding (nı2 + (12) (ni2 − (12)2, which is also essentially ({ 2 + n2 + ( 2 ) 2

term of the denominator is Σ

positive.

This result is necessary to the reasoning I adduced above, in order to shew that the forces which the particles exert on each other are attractive.

I wish it were in my power to offer any considerations relative to the phenomena of polarization by reflexion from the surface of glass, and so on. There appears to be little doubt of the truth of the results which have been deduced by M. Fresnel relatively to the coefficients of the intensity of reflected and transmitted light produced by the different vibrations. I cannot however think that the hypothesis of the æther within the glass being more dense than that without in the ratio of 1 is altogether satisfactory, but I forbear making any remarks on that subject further than to shew what is the corresponding relation of the densities deducible from our hypothesis.

and taking only a very approximate value, we obtain

but the density evidently varies as the cube of the reciprocal of €,

In conclusion, I would remark, that although what has here been treated of has been but roughly and approximately developed, there is good reason for supposing that the laws we have arrived at are the correct ones, not only as regards the action of the particles of æther, but as regards those of air also.

The law of the inverse square of the distance has always appeared to me a necessary law; necessary, I mean, as regards the actual state of the constitution of the Universe: and although I could allow that the particles of matter might have been impressed with any law at their creation, I cannot, in consistence with the simplicity of all known actions, conceive any other than Newton's law. It is true, the phenomena of Capillary Attraction seem to militate decidedly against it, but no person that I am aware of has proved that the phenomena could not arise from discontinuing the fluidity, and until that has been done, I think (I speak with deference to others far more capable of judging) we ought not to be too hasty in adopting a law of force, however simply it may account for the particular phenomena in question, which we have no reason to suppose is applicable to any others.

But I fear I am trespassing beyond the proper limits of my subject, and shall therefore proceed no further than merely to observe, that the farther we proceed in our investigations, the more simple do our conclusions become, and that from the apparent discrepancies, as, for instance, in the lateral spread of sound passing through an aperture, which is not the case for light, in general arise the strongest confirmations of the unity of the whole.

VII. Sketch of a Method of Introducing Discontinuous Constants into

the Arithmetical Expressions for Infinite Series, in cases where they admit of several Values. In a Letter to the Rev. George Peacock, &c. &c. By AUGUSTUS DE MORGAN, of Trinity College, Fellow of the Society, and Secretary of the Royal Astronomical Society.

[Read May 16, 1836.]

DEAR SIR,

Two years ago, I presented to the Society through yourself, the detail of some anomalies which I had observed to exist in certain series which I then produced. They arose out of investigations connected with Functions, and since published in my Treatise on that subject in the Encyclopædia Metropolitana. But on further consideration, I find that I have not distinctly expressed the method by which the anomalies of the series in question may be reconciled, or rather by which the series may be so obtained that the difficulties shall not appear.

I beg leave therefore, to request that you will lay the following view of the subject before the Society.

The assumption of a given form for a development amounted to an express exclusion of several considerations, which, so it happened, did not affect the results of ordinary operations, in cases where the form assumed was that of development in whole powers of a variable. Among the exclusions, was that of the possibility of a discontinuous constant, which was never considered, I believe, until the errors which the omission of it created in the inversion of periodic developments