Page 190, to the alinea (beginning with): “the sole condition to be satisfied, is,” &c. (ending with): “… the versor of the expression may then become what it may,” add:

provided this versor does not differ from the versor of the product

made by successive multiplication (according to the definition of multiplication of more than two vectors), the quaternions w + ω, w−ω being likened respectively to the products

by a proper choice of the vectors p and .

The object sought to be gained by this tabulation may be best seen by considering a specific case. When we wish to produce, by help of toothed wheels, a certain ratio between the rotations of two axes, we must have that ratio represented by integer numbers, and these numbers must contain no prime factors exceeding the limit of the number of teeth which can conveniently be cut in any wheel. If, for example, we wish to represent the mean motions of the planets, we have to express the ratios of their periodic times by decomposable numbers; and as this, in general, can only be done approximately, we have to prescribe some limit of error on either side, and have to seek for all fractions between those limits, and then among these to search for those that are suitable.

In my treatise on wheel-teeth a solution of this problem is given, complete so far as the discovery of the fractions is concerned, but imperfect in this, that it does not place those fractions in the order of their magnitudes. I propose now to supply this deficiency by explaining an exceedingly simple process, which enables us to make a complete list, arranged in the order of their values, of all irreducible fractions whose denominators shall not exceed some specified number; and also to take up and interpolate any portion of the list. This process is founded on a general theorem flowing directly from the doctrine of continued fractions, but of which the following simple demonstration may be given.

Abundant as are the localities in which garnet is found in Scotland, there are but few which yield specimens such as can be analysed.

This is on account of an intermixture of quartz—for the most part in a granular form—the granules being promiscuously scattered throughout the mass of the crystals. In three localities the intermixture is not promiscuous, but has been governed by some intermittent crystalline action. These localities are Glen Skiag in Ross, where, around a central nucleus of leucitoidal crystals of garnet, translucent quartz is arranged in layers which alternate with those of the garnet, conformably to the figure of its crystal.

The two other localities are the first and third granitic veins to the east of Portsoy; in both of which garnet of a pale brown colour is laced with quartz, which is arranged in a graphic manner, as in the felspar from an adjacent vein. At no other locality do I know of such an occurrence. The appearance of a section of a rhombic dodecahedron is somewhat like the figure.

In a paper on “The Bifilar Magnetometer, its Errors and Corrections” &c., which appeared in the Transactions of the Royal Society of Edinburgh, vol. xxii. p. 467; it was shown that, in turning the apparatus which carries the upper extremities of the two wires through an angle exceeding by ν that through which the lower extremities and the magnet are moved, the upper end of each wire is also turned through an angle greater by ν than the lower end. The whole force acting on the magnet, to turn it from the meridian, is compounded of the bifilar and the unifilar torsion—only the former of which had been considered in the method issued by the Committee on Physics of the Royal Society of London for the instruction of directors of observatories.

It was sought in the paper cited to show the change that the neglected forces would introduce into the expression for the unit coefficient, which was found to be

instead of

where Δν is the arc value ot one scale division of the instrument, and β is the variation of ν due to the unifilar torsion. There were errors of copying, including omissions in the investigation given. I also remarked, after the paper was printed, that the value of k found was less than when the unifilar torsions were neglected (since cot (ν + β) < cot ν); whereas all the determinations by other methods, in which the unifilar torsions were included, had shown a greater coefficient.

Certain cinematical and physical questions lead to the problem: to determine the directions in which a given linear vector-function, ϕρ, assumes a direction parallel to that of the vector, ρ, on which it depends.

The condition of parallelism is expressed by

and it is translated into the equation

where g represents a certain scalar, on whose determination the whole problem depends.

Following the method traced out by Hamilton, we treat this equation successively by

α,β,γ, being any system of vectors not coplanar between each other; but we will state at once that throughout the whole of this paper we shall assume α,β,γ, to form a system of treble rectangular unit-vectors, of which hypothesis the justification is evident.

The determination by experiment of the thermo-electric relations of any one substance belonging to the electromotive series to all other such substances is sufficient to fix all mutual thermo-electric relations among these. The first endeavour of the experimenter is then to obtain as convenient a substance for this purpose as possible. In investigating charcoal and certain alloys, we have in almost all cases employed one or other of two alloys of platinum and iridium, which have been already used by Professor Tait for a like purpose. The wires we used were the same which he discusses in his “First Approximation to a Thermo-Electric Diagram,” under the names of M and N. Their complete freedom from oxidation, their elasticity, and the high temperatures of their fusing points, rendered them peculiarly suitable for thermo-electric investigations through long ranges of temperature.

Generally both the M and N wires were firmly bound, each by its one extremity to the end or ends of the wire or wires respectively which were under investigation, in a multiple junction. This triple or, as it was in some cases, quadruple junction constituted the “hot junction.” The free extremities of the wires thus united were each bound to a moderately thin copper wire by very thin wire of the same metal, and the copper wires were led from these “cold junctions” to a commutator, which was in connection with a galvanometer. The commutator consisted of an arrangement of small mercury pools, into which the galvanometer and circuit wires, carefully amalgamated to ensure contact, dipped. All the junctions were formed in the manner indicated above, namely, by tightly binding the extremities of the wires together by thin copper wire.

Towards the end of last winter session, a Memoir by me, on the “Parallel Roads of Lochaber,” was read to this Society, and it has since been published in our Transactions. The subject was far from being exhausted. Nevertheless, I had no intention of continuing the inquiry, venturing to think, that enough had been adduced by me to support the conclusions at which I had arrived. But during the course of last summer, Dr Tyndall of London visited Lochaber. He went for the special purpose of studying the “Roads,” and of enabling him to give a public lecture regarding them in the Royal Institution, Albemarle Street, on 9th June.

In the course of his lecture, Dr Tyndall alluded to my recent Memoir, and also to researches described in a previous Memoir. But he dissented from my solution of the problem, and told his hearers (I quote his words) that they might “with safety dismiss it (the detrital barrier theory) as incompetent to account for the phenomena. The theory which ascribes the Parallel Roads to lakes clammed by barriers of ice, has, in my opinion, an amount of probability on its side, which amounts to a practical demonstration of its truth.”

These views having been rested on observations made in the district by Dr Tyndall himself, I felt that it was only due to a person of his scientific reputation, to reconsider my own opinions, and to weigh well his reasons for coming to a different conclusion.

In the early part of the present century, when stratigraphical geology, starting from the clear succession of Secondary rocks of England, was groping its way among the older formations in this country and abroad, the Old Red Sandstone occupied a somewhat indeterminate position. The series of deposits comprised under that name had been recognised chiefly in the British Islands, hardly at all on the opposite mainland of Europe. By most geologists they were classed as a subordinate and inconstant portion of the Carboniferous system, while by some they were placed rather at the top of the yet unexplored “Transition” or “Greywacke “ series. Murchison first claimed for them the dignity and importance of a distinct system. On the whole, they had yielded comparatively few organic remains; they consequently seemed to lie as a thick red barren zone between the richly fossiliferous Silurian deposits below them and the equally fossiliferous Carboniferous limestone above. By degrees, however, as they brought forth a rich harvest of new and strange ichthyolites, they indicated their own right to recognition, and when they were found covering a vast space in Russia with many of the same types of fish as they had yielded in Britain, their claim to rank as a distinct and independent system was no longer contested.

(I.) Let F(x) = 10 + b1x + … + bnxn = 0 be an equation whose roots are all real and different; then

where ϕ(x) = 1 + a1x + … + amxm and Σ indicates summation for all the roots of F(x) = 0: of which r is a root.

Let also

then

and equating coefficients of like power of x,

whence

the suffix number indicating the order of the determinant.

Malacolite was first observed in Scotland by Dr Macculloch, who included it with the second sub-species, under the name of Sahlite.

He, in his different works, noted its occurrence—“near the church of Rodel in Harris,”- “in Tiree,”-“Glen Elg,”-“Glen Tilt,”-and “in Rannoch.” With his usual power he describes the occurrence, varieties, and even the crystalline form of the mineral,—the clearness and precision of his descriptions being only equalled by the utter want of precision and frequent mystifications as to their localities.

The above sentence conveys all the direct information which Dr Macculloch vouchsafes as to the five localities indicated by him, though in a vague manner he may occasionally lead one to conjecture that certain particular spots are meant.

By the time that the mineralogist, led by Dr Macculloch to Glen Elg, Tiree, or Rannoch, has walked some twelve or twenty miles in an unsuccessful search, he begins to think that if he be successful, he has some claim to be himself regarded as an independent discoverer.

The first experiment which I have to describe relates to an attempt to employ the telephone as a means of measuring the hearing power of the ear in different individuals, or in the same individual at different times. For this purpose a constant source of sound, driven by clockwork, is made to act upon the disc of a telephone in a distant room. In the circuit of this telephone are included two others, so placed on a stand, that they can be brought close to both ears of the person to be experimented on. Soft india-rubber rings are attached to these telephones so as to enclose both ears, and Serve the double purpose of avoiding any disagreeable pressure on them, and preventing, as far as possible, all extraneous sounds from entering. In the circuit of the telephones there is also included a column of slightly acidulated water whose length can be varied at pleasure, so as to introduce a greater or less resistance to the electric current. This is managed in the following way:—A pretty wide glass tube, about 24 centimetres long, closed at the lower end, and graduated to millimetres, is mounted vertically on a stand. Projecting through the lower end of the tube is a platinum wire melted into the glass and having its point inside opposite the zero of the scale.

In Crelle's Journal for 1844, at the end of a paper on Cubic Forms, Eisenstein gives the following results:—

where m is any odd number and ρ a primitive root of the equation zm = 1;

where

No demonstrations are given of these identities; but they are said by the author to be only particular cases of a very general equation, the complete theory connected with which he hoped to give on another occasion.

At p. 193 of the same volume of Crelle he returns to the subject, not however for the purpose of giving the promised theory, but to add several other results similar to those before given.