magnesium, and probably of iron. The presence of sodium was also indicated in Procyon and a Cygni.

In no single instance have the authors ever observed a star-spectrum in which lines were not discernible, if the light were sufficiently intense, and the atmosphere favourable. Rigel, for instance, which some authors state to be free from lines, is filled with a multitude of fine lines.

Photographs of the spectra of Sirius and Capella were taken upon collodion; but though tolerably sharp, the apparatus employed was not sufficiently perfect to afford any indication of lines in the photograph.

In the concluding portion of their paper, the authors apply the facts observed to an explanation of the colours of the stars. They consider that the difference of colour is to be sought in the difference of the constitution of the investing stellar atmospheres, which act by absorbing particular portions of the light emitted by the incandescent solid or liquid photosphere, the light of which in each case they suppose to be the same in quality originally, as it seems to be independent of the chemical nature of its constituents, so far as observation of the various solid and liquid elementary bodies, when rendered incandescent by terrestrial means, appears to indicate.

III. "A Second Memoir on Skew Surfaces, otherwise Scrolls." By A. CAYLEY, Esq., F.R.S. Received April 29, 1864.

(Abstract.)

The principal object of the present memoir is to establish the different kinds of skew surfaces of the fourth order, or Quartic Scrolls; but, as preliminary thereto, there are some general researches connected with those in my former memoir "On Skew Surfaces, otherwise Scrolls" (Phil. Trans. vol. 153. 1863, pp. 453, 483), and I also reproduce the theory (which may be considered as a known one) of cubic scrolls; there are also some concluding remarks which relate to the general theory. As regards quartic scrolls, I remark that M. Chasles, in a footnote to his paper, "Description des Courbes de tous les ordres situées sur les surfaces réglées du troisième et du quatrième ordres," Comptes Rendus, t. liii. (1861), see p. 888, states, "les surfaces réglées du quatrième ordre. . . . admettent quatorze espèces." This does not agree with my results, since I find only eight species of quartic scrolls; the developable surface or "torse" is perhaps included as a "surface réglée;" but as there is only one species of quartic torse, the deficiency is not to be thus accounted for. My enumeration appears to me complete, but it is possible that there are subforms which M. Chasles has reckoned as distinct species.

IV. "On the Differential Equations which determine the form of the Roots of Algebraic Equations." By GEORGE BOOLE, F.R.S., Professor of Mathematics in Queen's College, Cork, Received April 27, 1864.

(Abstract.)

Mr. Harley* has recently shown that any root of the equation

y"—xy+ (n-1) x=0

satisfies the differential equation

(D-2-1) (D_3-2). (D_2+1)

n

..

n

D(D—1) .. (D−n+1)

d 'do'

n

in which ex, and D= provided that n be a positive integer greater

than 2. This result, demonstrated for particular values of n, and raised by induction into a general theorem, was subsequently established rigorously by Mr. Cayley by means of Lagrange's theorem.

For the case of n=2, the differential equation was found by Mr. Harley to be

Solving these differential equations for the particular cases of n=2 and n=3, Mr. Harley arrived at the actual expressions of the roots of the given algebraic equation for these cases. That all algebraic equations up to the fifth degree can be reduced to the above trinomial form, is well known.

A solution of (1) by means of definite triple integrals in the case of n=4 has been published by Mr. W. H. L. Russell; and I am informed that a general solution of the equation by means of a definite single integral has been obtained by the same analyst.

While the subject seems to be more important with relation to differential than with reference to algebraic equations, the connexion into which the two subjects are brought must itself be considered as a very interesting fact. As respects the former of these subjects, it may be observed that it is a matter of quite fundamental importance to ascertain for what forms of the function (D), equations of the type

u+¢ (D) enou=0 . .

(3) admit of finite solution. We possess theorems which enable us to deduce from each known integrable form, an infinite number of others. Yet there is every reason to think that the number of really primary forms-of forms the knowledge of which, in combination with such known theorems, would enable us to solve all equations of the above type that are finitely solvable

* Memoirs of the Literary and Philosophical Society of Manchester.

is extremely small. It will, indeed, be a most remarkable conclusion, should it ultimately prove that the primary solvable forms in question stand in some absolute connexion with a certain class of algebraic equations.

The following paper is a contribution to the general theory under the aspect last mentioned. In endeavouring to solve Mr. Harley's equation by definite integrals, I was led to perceive its relation to a more general equation, and to make this the subject of investigation. The results will be presented in the following order :

First, I shall show that if u stand for the mth power of any root of the algebraic equation

n-1

y" — xy" 1-1=0,

then u, considered as a function of x, will satisfy the differential equation

Secondly, I shall show that for particular values of m, the above equation admits of an immediate first integral, constituting a differential equation of the n-1th order, and that the results obtained by Mr. Harley are particular cases of this depressed equation, their difference of form arising from difference of determination of the arbitrary constant.

Thirdly, I shall solve the general differential equation by definite integrals.

Fourthly, I shall determine the arbitrary constants of the solution so as to express the mth power of that real root of the proposed algebraic equation which reduces to 1 when x=0.

The differential equation which forms the chief subject of these investigations certainly occupies an important place, if not one of exclusive importance, in the theory of that large class of differential equations of which the type is expressed in (3). At present, I am not aware of the existence of any differential equations of that particular type which admit of finite solution at all otherwise than by an ultimate reduction to the form in question, or by a resolution into linear equations of the first order. It constitutes, in fact, a generalization of the form

u+

a (D-2)2±n2 20 u=0
D(D-1)

given in my memoir "On a General Method in Analysis" (Philosophical Transactions for 1844, part 2).

V. "A Comparison of the most notable Disturbances of the Magnetic Declination in 1858 and 1859 at Kew and Nertschinsk, preceded by a brief Retrospective View of the Progress of the Investigation into the Laws and Causes of the Magnetic Disturbances." By Major-General EDWARD SABINE, R.A., President of the Royal Society. Received April 28, 1864.

(Abstract.)

The author commences this paper by taking a retrospective view of the principal facts which have been established regarding the magnetic disturbances, considered as a distinct branch of the magnetic phenomena of the globe, from the time when they were first made the objects of systematic investigation by associations formed for that express purpose, at Berlin in 1828 and at Göttingen in 1834, and dwelling more particularly on the results subsequently obtained by the more complete and extended researches instituted in 1840 by the British Government on the joint recommendation of the Royal Society and of the British Association for the Advancement of Science.

The Berlin Association, formed under the auspices of Baron Alexander von Humboldt, consisted of observers in very distant parts of the European continent, by whom the precise direction of the declination-magnet was recorded simultaneously at hourly intervals of absolute time, at forty-four successive hours at eight concerted periods of the year, which thence obtained the name of "Magnetic Terms." By the comparison of these hourly observations it became known that the declination was subject to very considerable fluctuations, happening on days which seemed to be casual and irregular, but were the same at all the stations, consequently over the continent of Europe generally. This conclusion was confirmed by the Göttingen Association, established at the instance and under the superintendence of MM. Gauss and Weber, by whom the "Term-observations" were extended to six periods in the year, each of twenty-four hours' duration, the records being made at intervals of five minutes. The number of the stations at which these observations were made was about twenty, distributed generally over the continent of Europe, but not extending beyond it. They were continued from 1834 to 1841. The observations themselves, as well as the conclusions drawn from them by MM. Gauss and Weber, were published in the well-known periodical entitled 'Resultate aus der Beobachtungen des magnetischen Vereins.' The synchronous character of the disturbances, over the whole area comprehended by the Association, was thoroughly confirmed: the disturbing action was found to be so considerable as to occasion frequently a partial, and sometimes even a total obliteration of the regular diurnal movements, and to be of such general prevalence over the greater part of Europe, not only in the larger, but in most of the smaller oscillations, as to make it in a very high degree improbable that they could

have either a local or an atmospherical origin. No connexion or correspondence was traceable between the indications of the magnetical and meteorological instruments; nor had the state of the weather any perceptible influence. It happened very frequently that either an extremely quiescent state of the needle or a very regular and uniform progress was preserved during the prevalence of the most violent storm; and as with wind-storms, so with thunder-storms, as even when close at hand they appeared to exercise no perceptible influence on the magnet. At some of the most active of the Göttingen stations the fluctuations of the horizontal force were observed contemporaneously with those of the declination-magnet, by means of the bifilar magnetometer devised by M. Gauss: both elements were generally disturbed on the same days and at the same hours. The magnitude of the disturbances appeared to diminish as their action was traced from north to south, giving rise to the conclusion that the focus whence the most powerful disturbances in the northern hemisphere emanated might perhaps be successfully sought in parts of the globe to the north or north-west of the area comprehended by the stations. The intercomparison of the records obtained at the different stations showed moreover that the same element was very differently affected at the same hours at different stations; and that occasionally the same disturbance showed itself in different elements at different stations. The general conclusion was therefore thus drawn by M. Gauss, that "we are compelled to admit that on the same day and at the same hour various forces are contemporaneously in action, which are probably quite independent of one another and have very different sources, and that the effects of these various forces are intermixed in very dissimilar proportions at various places of observation relatively to the position and distance of these latter; or these effects may pass one into the other, one beginning to act before the other has ceased. The disentanglement of the complications which thus occur in the phenomena at every individual station will undoubtedly prove very difficult. Nevertheless we may confidently hope that these difficulties will not always remain insuperable, when the simultaneous observations shall be much more widely extended. It will be a triumph of science should we at some future time succeed in arranging the manifold intricacies of the phenomena, in separating the individual forces of which they are the compound result, and in assigning the source and measure of each."

In the British investigations, which commenced in 1840, the field of research was extended so as to include the most widely separated localities in both hemispheres, selected chiefly with reference to diversity of geographical circumstances, or to magnetic relations of prominent interest. Suitable instruments were provided for the observation of each of the three magnetic elements; the scheme of research comprehended not alone the casual and irregular fluctuations which had occupied the chief attention of the German associations, but also "the actual distribution of the magnetic influence over the globe at the present epoch in its mean or average state,