XLII. Notices respecting New Books.

Elements of Quaternions. By the late Sir WILLIAM ROWAN HAMILTON. Second Edition, edited by C. J. JOLY. (Longmans, Green & Co. Vols. I & II.: 1900, 1901.)

PROFESSOR JOLY is to be congratulated on having accomplished his great task of issuing a new edition of Hamilton's classic work. By virtue of its larger type and broader page, the new edition, with its two handsome volumes beautifully printed at the Dublin University Press, is an improvement upon the criginal book, dear though that was in its very compactness to students of quaternions. The distinction which Hamilton himself made between the large and small type portions is sacrificed; but in other respects, even to the characteristic use of italics and capitals, the new edition seems to be a faithful reproduction of the earlier.

The position to be assigned to quaternions in the mathematical developments of the last century is still matter of dispute. The pure mathematician looks askance at it, and will not take the pains to master its transformations, Cayley, who himself in the pages of this Journal nearly sixty years ago gave the subject some attention, seems never to have considered the geometrical significance of the Quaternion. After Tait had shown how to extract the square root of the linear vector function, that is, of a particular matrix, Cayley investigated the matter by his analytical methods and expressed surprise that his result agreed with Tait's! More recently he has contrasted coordinates and quaternions as geometrical methods and has given his verdict in favour of the former. Nevertheless, is there any mathematical treatise that can be compared to Hamilton's Elements for its wealth of geometrical applications, or is there any analytical method that is so incisive as the quaternionic? As with a search-light Hamilton directs his symbols into every region of tridimensional geometry, illuminating wherever he touches. There is, we believe, no other single treatise from which a student, familiar with ordinary mathematical processes in their elementary applications, could gain so wide or so deep a knowledge of geometry as from Hamilton's great work. Even familiarity with ordinary mathematical processes might in special cases be dispensed with. Tait, the great example of a man who got, as Maxwell expressed it, "the Quaternion mind directly from Hamilton," used to speculate on the possibility of training a mathematically gifted mind wholly along quaternionic lines. To such a mind all artificial systems of coordinates would appear in their true light, as valuable aids for mathematical counting but not as fundamental necessities for mathematical thinking-these being according to Maxwell the two kinds of work of mathematicians. There is no doubt as to the necessity of thinking on the part of the man who would use quaternions. But surely the thinking is well bestowed when two or three lines of appropriate transformations are the equivalent of a page of Cartesian eliminations.

One chief interest peculiar to the new edition is the Appendix of Thirteen Notes covering 111 pages of the Second Volume. These are the work of Professor Joly, and are intended to show the various directions along which Hamilton's powerful calculus has been or may be suitably extended. They include such subjects. as the theory of screws, finite rotations, the kinematical treatment of curves and surfaces, systems of rays, and in particular important sections on the linear vector function and on the differential operator .

The further development of the theory of the linear vector function and its application to strains and stresses constitute part of Tait's important additions to Hamilton's own labours; and the fact is duly acknowledged by Professor Joly. But by far the most important of Tait's quaternion investigations have to do with the operator. Hamilton discovered it; but it was left to Tait to disclose its full potency and show how to use it. It is not a little surprising, therefore, that Professor Joly should not give the remotest hint that this whole department of quaternions was virtually created by Tait, whom Maxwell named "the chief Musician upon Nabla"

When, in 1870, Tait was awarded the Keith Prize by the Royal Society of Edinburgh, Maxwell, referring to the papers for which the award was given, wrote in November of that year:-"That on Rotation is very powerful, but the last one on Green's and other allied theorems is really great "+.

In the same communication Maxwell says further: "No one can tell whether he (Tait) may not yet be able to cause the Quaternion ideas to overflow all their mathematical symbols and to become embodied in ordinary language so as to give their form to the thoughts of all mankind. I look forward to the time when the idea of the relation of two vectors will be as familiar to the popular mind as the rule of three, and when the fact that ij=-ji will be introduced into hustings speeches as a telling illustration." It was, no doubt, with this prophetic thought in his mind that Maxwell introduced the expressive and compact notation of quaternions into his Electricity and Magnetism.' In these days Maxwell's example is followed by many workers in electromagnetic theory. Most of them are no doubt content to use merely as a notation the particular form of vector analysis which they adopt; and with the rectangular space scaffolding ever before their eyes they spoil all hope of real quaternionic work by discarding the associative principle which makes quaternions workable, and without which a vector analysis speedily becomes too complicated. "I can work Heaviside's methods but I don't understand

* See Maxwell's Life, p. 634. The name Nabla, which Professor Joly introduces almost as if Hamilton himself had given it, was suggested by W. Robertson Smith from the resemblance of the symbol to an Assyrian harp. See Tait, "On the Importance of Quaternions in Physics," Phil. Mag. vol. xxix. p. 91; Scientific Papers, vol. ii. P. 303.

+ In his Electricity and Magnetism,' the same paper is characterized as "very valuable."

quaternions," was the somewhat curious remark once made by a vector analyst. He might as well have claimed full knowledge of Modern Greek and absolute ignorance of Ancient Greek.

Looking back over half a century of scientific progress, we cannot but be struck with the remarkable way in which Hamilton prepared a mathematical method peculiarly suitable to the physical theories that find their source in Maxwell's great work. It is from this point of view that the utility of quaternions is at present to be judged. Whatever may be the merits of other similar systems, one thing is incontrovertible. Hamilton, in his quaternions, gave to the world the first complete symmetric and workable system of analysis directly applicable to vector quantities. O'Brien, one of Hamilton's brilliant contemporaries, constructed a vector analysis which has been reproduce with modified notations by Professor Willard Gibbs and Dr. Heaviside. It has a certain superficial similarity to Hamilton's method, and, in so far, is effective enough, but it lacks the homogeneity and solidarity so characteristic of the latter. In this connection it is well to bear in mind the verdict both of Tait and of Professor Joly, that any endeavour to improve or modify the calculus of quaternions 'should be made with extreme caution.

Surely we are justified in hoping that, now that Hamilton's Elements has been made accessible to all students, there will be a real endeavour on the part both of mathematicians and physicists to "get the quaternion mind." For certain special types of academic problems the quaternion method has no special fitnessit "degenerates" simply into ordinary scalar coordinates; but for taking a direct hold of the essential nature of a general problem, whether of tridimensional geometry or of dynamics in the widest sense of the term, the quaternion method is facile princeps. Take as an example Hamilton's own treatment of the curvature of surfaces, to which in one of his notes Professor Joly has given a most interesting extension.

We have already referred to the linear vector function, which comes to the frout at the very outset when the simplest quaternion equation is considered. Hamilton developed its theory with great power, and Tait discovered other curious and important properties of it. One of Tait's latest utterances was that there was a vast deal more in ቀ than had yet been made out-its depths were not yet sounded. Professor Joly devotes five of his notes to various aspects of linear vector functions, and materially extends our knowledge, especially in regard to their invariants.

By far the longest of the notes in the Appendix is that on VProfessor Joly's discussion is very instructive, covering a good deal of the ground already familiar to readers of Tait's and McAulay's books and papers, but developing along different lines. A comparison of the modes in which these three quaternionists attack the same problem shows well the extraordinary variety in detail of the quaternion method. It is a variety which, like that of organisms, springs naturally from the fundamental vital principle of the whole.

THE

LONDON, EDINBURGH, AND DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[SIXTH SERIES.]

NOVEMBER 1901.

XLIII. The Mechanism of Radiation. By J. H. JEANS, B.A., Scholar of Trinity College, and Isaac Newton Student in the University of Cambridge*.

1.

Introduction.

§ THIS class of phenomena by referring them

HIS paper attempts to give a consistent interpretation

all to the same hypothetical view as to the structure of matter. As the result of some work on the kinetic theory of gases the author was led, rightly or wrongly, to the conclusion that the kinetic-theory phenomena of matter compel us to attribute certain definite properties to the molecules of which matter is composed. The range of view as to the structure of matter could, it was found, be still further narrowed by the help of certain optical phenomena. It was then of interest to examine to what extent the view arrived at in this way was capable of giving an account of the remaining phenomena, and how it compared in this respect with other views as to the structure of matter.

To this end, several hypotheses as to the structure of matter were examined by the present writer, and it seemed to him that the one which is put forward in this paper was much more capable of giving an account of the phenomena in question than were any of the others tested by him. For the sake of brevity, it has been thought advisable to remove the scaffolding, by the help of which the theory of the present paper has been formed, and simply to submit for

* Communicated by the Physical Society: read June 14, 1901. Phil. Mag. S. 6. Vol. 2. No. 11. Nov. 1901.

2 F

judgment the view ultimately arrived at, together with the account of the phenomena of matter to which it leads, in so far as it has been found possible to examine these phenomena.

The theory will not, it is hoped, be judged as an attempt to attain to ultimate truth. At most the author hopes that by attempting a definite and consistent hypothetical interpretation of certain phenomena, some kind of clue be suggested as to the real significance of these phenomena, and perhaps something of the nature of a foreshadowing of the real truth arrived at.

may

Analytical Expression for the Radiation from a Gas.

§ 2. It will be best to begin by a consideration of the general question of spectroscopy.

We shall suppose the radiation emitted by a gas to be the aggregate of contributions from a great number of similar vibrators. Each of these vibrators will be supposed to be capable of vibrating with certain definite frequencies of vibration, and the disturbances of the æther which are set up by these vibrations constitute the radiation.

Let us, in the first place, consider only the radiation propagated along a certain line of sight, say the axis of a, and polarized in a certain plane, say that of xy. If the vibrator were at rest in space, this radiation might be represented, so long as the vibrator was undisturbed by collisions, by

A cos (pt+€),

in which A and e would be constants, which would, in general, depend upon the orientation of the vibrator.

If, however, the vibrator is rotating with an angular velocity w about some axis fixed in space, the above expression will no longer represent the radiation in question. This radiation may, however, be represented by

A cos pt+B sin pt,

where A, B are themselves periodic functions of the time, of period 2π/w.

If we expand A and B in Fourier-series, the foregoing expression can be at once expressed as a series of simply harmonic terms of frequencies

P1 p±w, p±2w, ... &c.

Suppose, to take a particular case, that the vibrator is an electrical doublet of strength A cos pt, rotating about a fixed axis with angular velocity w, the axis of the doublet always