91. It is extremely interesting to find that, although the lunar and solar semidiurnal tides are very small in value. the series of means from which they were obtained were extremely regular and good, and the consequent determination of the phase of spring-tides from their respective epochs is probably correct within a few minutes. The proportion between the amplitudes of the lunar and solar semidiurnal tides is the nearest to equality yet obtained, being in the ratio of 11 to 6. The comparatively large value of R, of Series S is undoubtedly a genuine tide, but the smallness of the corresponding value of Series M must forbid the conclusion of its being purely astronomical. It is perhaps produced by temperature or wind, its time of maximum being about 40 minutes after noon. There are also indications of a similar and large annual tide of 0.3 foot amplitude, and maximum about July, which is also probably meteorological in its origin. The proportion between the lunar and solar diurnal (Declinational) tides (R, of Series O and P) will be, on the assumption of the variation of R, of Series O being as the square of the sine of the declination, about 4 to 1.

92. The following are the values of the long-period tides which have been obtained since the Edinburgh Meeting:

The above epoch for the solar annual tide would place the maximum about

August 16.

Address by Professor P. G. TAIT, M.A., F.R.S.E., President of the Section. IN opening the proceedings of this Section my immediate predecessors have exercised their ingenuity in presenting its widely differing component subjects from their several points of view, and in endeavouring to coordinate them. What they were obliged to leave unfinished, it would be absurd in me to attempt to complete. It would be impossible, also, in the limits of a brief address to give a detailed account of the recent progress of physical and mathematical knowledge. Such a work can only be produced by separate instalments, each written by a specialist, such as the admirable "Reports" which form from time to time the most valuable portions of our annual volume.

I shall therefore confine my remarks in the main to those two subjects, one in the mathematical, the other in the purely physical division of our work, which are comparatively familiar to myself. I wish, if possible, to induce, ere it be too late, native mathematicians to pay much more attention than they have yet paid to Hamilton's magnificent Calculus of Quaternions, and to call the particular notice of physicists to our President's grand Principle of Dissipation of Energy. I think that these are, at this moment, the most important because the most promising parts of our field.

If nothing more could be said for Quaternions than that they enable us to exhibit in a singularly compact and elegant form, whose meaning is obvious at a glance on account of the utter inartificiality of the method, results which in the ordinary Cartesian coordinates are of the utmost complexity, a very powerful argument for their use would be furnished. But it would be unjust to Quaternions to be content with such a statement; for we are fully entitled to say that in all cases, even in those to which the Cartesian methods seem specially adapted, they give as simple an expression as any other method; while in the great majority of cases they give a vastly simpler one. In the common methods a judicious choice of coordinates is often of immense importance in simplifying an investigation; in Quaternions there is usually no choice, for (except when they degrade to mere scalars) they are in general utterly independent of any particular directions in space, and select of themselves the most natural reference lines for each particular problem. This is easily illustrated by the most elementary instances, such as the following:The general equation of Cones involves merely the direction of the vector of a point, while that of Surfaces of Revolution is a relation between the lengths of that vector and of its resolved part parallel to the axis; and Quaternions enable us by a mero 1871.

1

mark to separate the ideas of length and direction without introducing the cumbrous and clumsy square roots of sums of squares which are otherwise necessary.

But, as it seems to me that mathematical methods should be specially valued in this Section as regards their fitness for physical applications, what can possibly from that point of view be more important than Hamilton's ? Physical analogies have often been invoked to make intelligible various mathematical processes. Witness the case of Statical Electricity, wherein Thomson has, by the analogy of Heat-conduction, explained the meaning of various important theorems due to Green, Gauss, and others; and wherein Clerk-Maxwell has employed the properties of an imaginary incompressible liquid (devoid of inertia) to illustrate not merely these theorems, but even Thomson's Electrical Images. [In fact he has gone much further, having applied his analogy to the puzzling combinations presented by Electrodynamics.] There can be little doubt that these comparisons owe their birth to the small intelligibility, per se, of what has been called Laplace's Operator, x2+ dy2+ dz2 which appears alike in all theories of attraction at a distance, in the steady flow of heat in a conductor, and in the steady motion of incompressible fluids. But when we are taught to understand the operator itself we are able to dispense with these analogies, which, however valuable and beautiful, have certainly to be used with extreme caution, as tending very often to confuse and mislead. Now Laplace's operator is merely the negative of the square of Hamilton's V, which is perfectly intelligible in itself and in all its combinations; and can be defined as giving the vector-rate of most rapid increase of any scalar function to which it is applied-giving, for instance, the vector-force from a potential, the heat-flux from a distribution of temperature, &c. Very simple functions of the same operator give the rate of increase of a quantity in any assigned direction, the condensation and elementary rotation produced by given displacements of the parts of a system, &c. For instance, a very elementary application of to the theory of attraction enables us to put one of its fundamental principles in the following extremely suggestive form :-If the displacement or velocity of each particle of a medium represent in magnitude and direction the electric force at that particle, the corresponding statical distribution of electricity is proportional everywhere to the condensation thus produced. Again, Green's celebrated theorem is at once seen to be merely the well-known equation of continuity expressed for a heterogeneous fluid, whose density at every point is proportional to one electric potential, and its displacement or velocity proportional to and in the direction of the electric force due to another potential. But this is not the time to pursue such an inquiry, for it would lead me at once to discussions as to the possible nature of electric phenomena and of gravitation. I believe myself to be fully justified in saying that, were the theory of this operator thoroughly developed and generally known, the whole mathematical treatment of such physical questions as those just mentioned would undergo an immediate and enormous simplification; and this, in its turn, would be at once followed by a proportionately large extension of our knowledge".

"

*The following extracts from letters of Sir W. R. Hamilton have a perfectly general application, so that I do not hesitate to publish them:- 'De Morgan was the very first person to notice the Quaternions in print; namely in a Paper on Triple Algebra, in the "Camb. Phil. Trans. of 1844. It was, I think, about that time, or not very long after"wards, that he wrote to me, nearly as follows:-I suspect, Hamilton, that you have "caught the right sow by the ear!' Between us, dear Mr. Tait, I think that we shall begin "the SHEARING of it!!" 66 'You might without offence to me, consider that I abused the "license of hope, which may be indulged to an inventor, if I were to confess that I expect "the Quaternions to supply, hereafter, not merely mathematical methods, but also phy"sical suggestions. And, in particular, you are quite welcome to smile if I say that it "does not seem extravagant to me to suppose that a full possession of those à priori prin"ciples of mine, about the multiplication of vectors (including the Law of the Four Scales "and the conception of the Extra-spatial Unit), which have as yet been not much more than hinted to the public, MIGHT have led (I do not at all mean that in my hands they ever would have done so) to an ANTICIPATION of the great discovery of OERSTED."

"

"

"6 It appears to me that one, and not the least, of the services which quaternions may be expected to do to mathematical analysis generally, is that their introduction will compel

And this is but one of the claims of Quaternions to the attention of physicists. When we come to the important questions of stress and strain in an elastic solid, we find again that all the elaborate and puzzling machinery of coordinates commonly employed can be at once comprehended and kept out of sight in a mere single symbol-a linear and vector function, which is self-conjugate if the strain be pure. This is simply, it appears to me, a proof either that the elaborate machinery ought never to have been introduced, or that its use was an indication of a comparatively savage state of mathematical civilization. In the motion of a rigid solid about a fixed point, a quaternion, represented by a single symbol which is a function of the time, gives us the operator which could bring the body by a single rotation from its initial position to its position at any assigned instant. In short, whenever with our usual means a result can be obtained in, or after much labour reduced to, a simple form, Quaternions will give it at once in that form; so that nothing is ever lost in point of simplicity. On the other hand, in numberless cases the Quaternion result is immeasurably simpler and more intelligible than any which can be obtained or even expressed by the usual methods. And it is not to be supposed that the modern Higher Algebra, which has done so much to simplify and extend the ordinary Cartesian methods, would be ignored by the general employment of Quaternions; on the contrary, Determinants, Invariants, &c. present themselves in almost every Quaternion solution, and in forms which have received the full benefit of that simplification which Quaternions generally produce. Comparing a Quaternion investigation, no matter in what department, with the equivalent Cartesian one, even when the latter has availed itself to the utmost of the improvements suggested by Higher Algebra, one can hardly help making the remark that they contrast even more strongly than the decimal notation with the binary scale or with the old Greek Arithmetic, or than the well-ordered subdivisions of the metrical system with the preposterous no-systems of Great Britain, a mere fragment of which (in the form of Tables of Weights and Measures) forms perhaps the most effective, if not the most ingenious, of the many instruments of torture employed in our elementary teaching.

It is true that, in the eyes of the pure mathematician, Quaternions have one grand and fatal defect. They cannot be applied to space of n dimensions, they are contented to deal with those poor three dimensions in which mere mortals are doomed to dwell, but which cannot bound the limitless aspirations of a Cayley or a Sylvester. From the physical point of view this, instead of a defect, is to be regarded as the greatest possible recommendation. It shows, in fact, Quaternions to be a special instrument so constructed for application to the Actual as to have thrown overboard everything which is not absolutely necessary, without the slightest consideration whether or no it was thereby being rendered useless for applications to the Inconceivable.

The late Sir John Herschel was one of the first to perceive the value of Quaternions; and there may be present some who remember him, at a British Association Meeting not long after their invention, characterizing them as a "Cornucopia from which, turn it how you will, something valuable is sure to fall." Is it not strange, to use no harsher word, that such a harvest has hitherto been left almost entirely to Hamilton himself? If but half a dozen tolerably good mathematicians, such as exist in scores in this country, were seriously to work at it, instead of spending (or rather wasting) their time, as so many who have the requisite leisure now do, in going over again what has been already done, or in working out mere details where a grand theory has been sketched, a very great immediate advance would be certain. From the majority of the papers in our few mathematical journals one would almost be led to fancy that British mathematicians have too much pride to use a simple method while an unnecessarily complex one can be

"those who adopt them (or even who admit that they may be reasonably adopted by other "persons) to consider, or to admit that others may usefully inquire, what common grounds "can be established for conclusions common to quaternions and to older branches of ma"thematics."

"Could any thing be simpler or more satisfactory? Don't you feel, as well as think, "that we are on a right track, and shall be thanked hereafter? Never mind when."