feet from the surface, taken as a sample of dressed ore, and weighing about a quarter of a pound, gave me in the air-dry state, in summer, 93.83 per cent. binoxide of manganese, with barytes and a mere trace of iron. It is a very valuable property of this ore, as regards its use by glass-makers, that when cleaned it contains remarkably little iron. The first shipment sent to England, consisting of about seven tons and a half, gave, on analysis in Liverpool, 91.5 per cent. binoxide, and less than a half per cent. of iron.

South of Teny Cape, at a distance of some ten miles, large nodules of manganese ore are found resembling in appearance those described as occurring in the "soil" at the former place. One of these weighed 180 pounds; a fragment from another, weighing thirty-five pounds, was examined by Mr. H. Poole, a pupil of mine. The mass was black, of unequal hardness, portions scratching apatite, and therefore about 5.5, while the rest yielded easily to the knife. The powder of the harder parts was nearly as black as that of the softer. The water of composition was found by weighing in chloride of calcium; the binoxide of manganese by oxalic acid; the results were these:

which show that the mass consisted chiefly of pyrolusite. That the associated mineral was psilomelane follows from its appearance and hardness, the colour of its powder, and the amount of water contained, which is too little for manganite, and too much for any of the other manganese minerals.

The researches of MM. Deville and Debray (given in abstract in the 'Chemical News,' vol. i. p. 299) show that natural binoxide of manganese is a very complex substance, containing various soluble salts, among which are alkaline nitrates, and that nitric acid is one of its products of ignition. I have found the Teny Cape pyrolusite to give strongly acid fumes on ignition, no doubt from the presence of nitric acid. The nature of the soluble salts I have not inquired into further than regards iron and baryta. As a new illustration of the complexity of the mineral, it is an interesting fact that silver, to the amount of five ounces to the ton of ore, has been found in a specimen from Teny Cape, on assay by J. Taylor and Co., in London. I may here recall the fact that thallium to the extent of 1 per cent. has been detected by

Bischoff (Quart. Journ. of Science, October 1864, p. 688) in a specimen of pyrolusite from an unmentioned locality.

The majority of the localities affording pyrolusite in this province are almost certainly known to belong to the lower carboniferous beds; the country-rock of the ores has not in all cases been made known. I saw last summer, in a locality about five miles from the quartz and manganite conglomerate before mentioned, which may be of New Red Sandstone age, a hard highly siliceous rock, apparently quartzite (contiguous to slate), from which about a ton of ore, consisting of pyrolusite and psilomelane, had been recently taken.

Wad. This is found in various parts of the province, sometimes in abundance. One specimen, of black colour, from a considerable bed situated, I believe, to the east of Halifax, gave me, when dried at 212°, 56 per cent. binoxide of manganese, a great deal of iron, a little cobalt, and a large quantity of insoluble matter. In specimens of brown "paints" I have found from 11 to 20 per cent. binoxide of manganese, the greater part of the residue being water and peroxide of iron.

XXIII. On some Problems in Chances. By J. M. WILSON, M.A., Fellow of St. John's College, Cambridge, and Mathematical and Natural Science Master of Rugby School*.

THE

HE problem of determining the probability that, if four points be taken at random in an infinite plane, one of the four shall lie inside the triangle formed by the other three, has now acquired some degree of notoriety. Various solutions have been given;, 1, have all been obtained as results; and all the methods are considered fallacious from introducing a comparison between infinities, and from those infinities apparently having different relative values according to the mode of approaching them.

4,

I shall venture to offer a solution which depends on a different principle, and I shall first illustrate it by applying it to a simpler question.

To determine the probability that if three lines are drawn at random in an infinite plane, a fourth line drawn at random will intersect the triangle formed by the other three.

The peculiarity of this class of questions is that, if the triangle is supposed drawn, it must be finite in conception as compared with the infinities on all sides of it, and the chance required would appear to be indefinitely small. But since the first three

lines are drawn at random, as the fourth line is, it is equally likely that all of the triangles should be infinite.

Let the four lines be drawn in any manner; call the lines a, b, c, d, each of them in succession may be considered the fourth line; and it will of course happen that in two of the four cases the fourth line does intersect the triangle formed by the first three; and therefore the chance required is.

In this solution the italicized part contains the assumption, to which some attention should be given. It in fact substitutes simultaneous for successive drawings of the lines in the figure. Viewed in this light I do not think it can be objected to; for by hypothesis all the lines are at random and independent of one another; and this once granted, the rest of the solution presents no kind of difficulty.

It may be viewed as a compendious way of presenting the following solution. Let the lines be drawn in the order a, b, c, d. Among the infinite series of figures so obtained, it will necessarily follow that d and a, d and b, dand c will sooner or later change places, the others remaining where they were. Hence it is clear that the figures, if every possible figure could be conceived as drawn, could be arranged in groups of four, the figures being the same in each group, and d occupying each of the four positions of the lines in succession; that is, in each group d will twice intersect the triangle formed by a, b, c, and twice it will not; or the chance required is, as before,.

It must be observed that wherever the first method of solution is applicable, it may be looked on as a succinct way of stating this kind of proof, if this latter is thought more satisfactory. The next problem is the "four-point problem.'

If four points be taken at random in a plane, what is the chance that one of them lies inside the triangle formed by the other three.

(1) A point at random may be viewed as the intersection o two lines at random.

(2) Four random lines determine three sets of four random points.

(3) In one only of these three sets is one of the four within the triangle formed by the other three.

(4) And therefore the chance required is.

With respect to (2), it may be remarked that if the lines are a, b, c, d, and ab represent the point of intersection of a and b, the three sets of four points are

ab, ac, db, dc;

ba, bc, da, dc;

ca, cb, da, db.

The steps by which this reasoning may be justified may be presented as follows:

A blindfolded man is told that four points are marked on a large sheet, and is asked the chance that one of them is enclosed by the other three. He requests that the points may be joined by four lines, no three passing through a point. There are now six points; and if he is allowed now to inspect the figure, there are three original groups of four points from which it is equally probable that the figure originated, and in one only, and one always, of these three groups one point is enclosed by the other three; so that the probability is.

The reasoning may also be justified in the way by which the result of the first question was arrived at.

This method, which might be called instantaneous, is easily applied to similar problems; and I have given these remarks in full, in order to remove, if possible, doubt as to its validity.

XXIV. On the Level of the Sea during the Glacial Epoch in the Northern Hemisphere. By Archdeacon PRATT.

To the Editors of the Philosophical Magazine and Journal. Baitool, Central Provinces, India, January 3, 1866.

GENTLEMEN, URING a journey in these out-of-the-way parts I have lighted upon an interesting letter in a late Number of the Reader' (September 2, 1865) by Mr. Croll of Glasgow, suggesting that the greater depth of the ocean in the upper parts of the northern hemisphere, during the Glacial Epoch, may have been caused by the change of the centre of gravity of the earth by the presence in the northern regions of an enormous accumulation of solid matter in an "Ice-Sheet," which has since disappeared. This idea appears to me so ingenious and so probable, that it deserves a more careful examination. The hypothesis appears to be, that in time past a grand cosmical change has been going on, according to which the northern and the southern hemispheres (at any rate the higher portions of them) have been alternately bound up in ice, and have alternately yielded to milder influences, when the ice-sheet has become broken up, moving off in huge fragments which have caused the phenomena of the drift, and has finally disappeared. The centre of gravity of the earth has therefore slightly shifted during these enormous periods, first north and then south, and produced a corresponding effect upon the depth of the ocean. The question is, whether the matter deposited from the air in snow and hail and held fast

rise in the sea-level. The problem is, in fact, one of Attractions. I have of late years given much attention to this subject in attempting to estimate the influence of the vast superficial mountain-mass north of India, and also of the deficiency of matter in the extensive ocean south of India down to the south pole, upon the plumbline used in the Trigonometrical Survey of Hindostan. The present problem is one of precisely the same character, as it is to ascertain the attracting force of the supposed Ice-Sheet upon the waters of the ocean and its effect in drawing them up. In the absence of all knowledge of the form and extent of the ice-mass, which has now disappeared, I will adopt Mr. Croll's hypothesis, as a means of making a calculation from which a general idea of the operation of the causes can be obtained. He takes the Ice-Sheet to be a hemispherical meniscus of a certain thickness at the pole, and gradually getting thinner towards the equator, where it is supposed to be zero. The specific gravity of ice is 0-92, and of superficial rock 2.75. ratio of that of ice to that of rock = 1 to 3.

Hence the

2. Following the same course by which I have estimated the attraction of the Himalayan mass and of the deficiency of matter in the ocean, I have calculated that the Horizontal Attraction of a Spherical Meniscus of rock (of thickness h miles at the pole and radius a) at a place in the opposite hemisphere the polar distance of which is 0, measured from the pole of the attracting meniscus,

h

= (0·14-46 sin @ + 0·0958 sin 20 +0-024-4 sin 30) —g, . (1)

where g is gravity*.

From this formula it is not difficult to find the attraction of the meniscus upon a place on its own surface. For the tangential attraction of an oblate spheroid of small ellipticity e, at a place on its surface of which the distance from the nearer pole is p, 06 sin 24. e.g, and acts from the nearer pole. From this it is easily seen, by taking the difference of effects of two oblate spheroids having the same equator and differing in their compressions by h, that the combined attraction of the northern and southern hemispherical meniscuses at the place in the north

* See my treatise On the Figure of the Earth,' 3rd edit. p. 58. I avail myself of this opportunity to state that in two formulæ, (2) p. 59, and (6) p. 60, derived from the above formula, there is a misprint. It is not of much importance, as no use is made of those two formulæ. They should stand as follows:

(0-1446 sin +0-5042 sin 2p+0.0244 sin 3)

h

• a

(2)