1. The integral occurs so frequently in various branches of research that, as far back as 1783, Laplace suggested that it would be useful to tabulate its values for successive ranges of integration. It is employed in investigations on the theories of refraction, conduction of heat, of errors of observation, of probabilities, etc. These are familiar to physicists and need not be dwelt upon.

1. The existence of relations between the coaxial minors of a determinant was first discovered by MacMahon in 1893. The whole literature of the subject is comprised in three papers, viz.:—

MacMahon, Phil. Trans., clxxxv. pp. 111–160.

Muir, Phil. Mag., 5th series, xli. pp. 537–541.

Nanson, Phil. Mag., 5th series, xliv. pp. 362–367.

My present object is to continue the investigation of the relations in question, and more particularly to draw attention to an explicit expression for a determinant of the 4th order in terms of its own coaxial minors. At the outset some fresh considerations regarding determinants in general will be found useful.

The earlier mineralogists laboured under two great disadvantages. They could not readily, on account of the small number of students of chemistry, call in the aid of that science: and at the time when mineralogy was becoming a distinct branch of science chemistry was in itself crude as well as cumbrous. They were thus forced to rely chiefly upon external properties; and, where crystalline form was absent, they were confined to what may be called physical properties alone.

Their knowledge of the composition of bodies being thus limited and uncertain, the old nomenclature was to a considerable extent founded upon external features alone.

The experiments described in this paper were commenced with the view, not only of determining the absolute thermal conductivity of nickel, but also of comparing the results found by Forbes's and Ängstrom's methods for the same specimen. Although some readings were taken for Ängstrom's method, that part of the investigation was not completed, because it was found that the experimental errors — unavoidable, on account of the necessity of measuring rapidly changing temperatures—would be too great for the results to be of any value. The thermal conductivity of a portion of the bar of nickel used for Forbes's method was determined by a direct method involving the determination only of steady temperatures, and the results so obtained are given in the latter portion of this paper.

As the result of two visits to Orkney, in which he was accompanied by Mr B. N. Peach, he pointed out that the yellow sandstones of Hoy did not pass down conformably into the flagstones which form the basis of that island, but were separated from them by a marked unconformity. At the base, of the upper sandstones lay a series of contemporaneous lavas and ash beds, which were in all probability erupted, from certain ‘necks’ in the low-lying district at the foot of the Hoy Hills. These rocks he regarded as belonging to the upper Old Red Sandstone. The lower Old Red Sandstone consisted principally of a great thickness of flagstones, with which were interstratified beds of yellow and red sandstone, and occasionally of conglomerate. The fossils belonged exclusively to this lower series; and a table is given, compiled by Mr C. W. Peach, showing the distribution 9f fossil fishes in the lower Old Red Sandstone of Lake Orcadie, including those of Orkney so far as known at that time. As Sir Archibald Geikie anticipated, subsequent revision has necessitated “considerable pruning of the fossil lists.” The conglomerates around the granite axis of Stromness formed merely a local base, “due to the uprise of an old ridge of rock from the surface of the sheet of water in which these strata were accumulated,” and were presumably not on the same horizon as the thick conglomerates on which, in Caithness, the lowest flagstones rest. The sandstones interbedded with the flagstones in South Ronaldshay were regarded as in all probability the northward continuation of the similar rocks at Gill's Bay, Huna, and John o'Groats, on the south side of the Pentland Firth. From a geological point of view, the brief notice of the Old Red Sandstone of the Orkneys contained in this paper forms by far the most important contribution to the knowledge of the subject published up to that time.

This paper is in continuation of two others, on the same subject, previously communicated to the Society. In the First Paper (Philosophical Magazine, July 1894) it was shown that the formula

where n, a, and b are constants in any one experiment, represents with accuracy the relation between y, the range of oscillation, and x, the number of oscillations which have taken place since torsion was first applied and the wire was left to itself, so that the oscillations gradually diminished. The apparatus employed, and the method of observation used, were identical with those described in the Second Paper above referred to. The wire which was experimented upon was the same as that used on the previous occasions. Its length, as given in the First and Second Papers, was 89·1 cm. A measurement made on the date 19.10.1897, in the course of the last series of experiments described in the present paper, showed that the length had become 89*3 cm. This increase was doubtless due to the fact that the heavy lead oscillator had been left attached to the wire during the whole of the intervening period. On the date given, it was also found that, with the same oscillator as was used in the experiments first described, ten oscillations were performed in 81 seconds, when the range was large, while 79 seconds were occupied when the range was small. This observation verified the result stated in the First Paper, that the period slightly increases as the range increases. It also showed that the wire was practically in the same condition as it was at first, in so far as elastic qualities are concerned; for the corresponding periods were only slightly less in earlier experiments, the difference being largely accounted for by the slight increase of length of the wire.

The remarkable changes produced by magnetization in the internal volumes of hollow cylinders of iron, steel, and nickel have been described in Part I. (see Trans. R.S.E., vol. xxxviii. pp. 531–555). As pointed out in the closing paragraph, a complete discussion of these changes had to be “deferred until direct measurements of elongation had been obtained with the various tubes under the same magnetic influences.” It was not possible, of course, to measure the elongations of all the tubes that had been experimented with; for of these, eighteen (Nos. I. to VI. of each inclusive) were no longer in existence, having been the successive stages through which No. VII. was brought from the condition of small bore and thick walls to that of wide bore and thin walls.

The first instalment of this paper was devoted in great part to the general subject involved in its title, but many of the illustrations were derived from the special case of the flight of a golf-ball. Since it was read I have endeavoured, alike by observation and by experiment, to improve my numerical data for this interesting application, particularly as regards the important question of the coefficient of resistance of the air. As will be seen, I now find a value intermediate to those derived (by taking average estimates of the mass and diameter of a golf-ball) from the results of Robins and of Bashforth. This has been obtained indirectly by means of a considerable improvement in the apparatus by which I had attempted to measure the initial speed of a golf-ball. I have, still, little doubt that the speed may, occasionally, amount to the 300, or perhaps even the 350, foot-seconds which I assumed provisionally in my former paper:—but even the first of these is a somewhat extravagant estimate; and I am now of opinion that, even with very good driving, an initial speed of about 240 is not often an underestimate, at least in careful play. From this, and the fact that six seconds at least are required for a long carry (say 180 yards), I reckon the “terminal velocity” at about 108, giving v2/360 as the resistance-acceleration.