place in a horizontal direction. In order to get quit, as much as possible, of the effects of friction on the sides and bottom of the vessel, the experiment was repeated in a pond, with a like result.

From these experiments we may conclude that, save under very exceptional circumstances, the wind can only give rise to a horizontal circulation of the oceanic waters,-the exceptional cases being, when the wind-driven current is deflected by the irregularities in the outline of the bed of the ocean, or strikes against a deep and nearly vertical coast; or when the north wind drives the waters of the Antarctic Sea against the great barrier of ice-cliffs which surround the south pole. The depth to which the vertical currents will descend in these cases will depend much on the relative densities of the water in the currents and of the water surrounding them.

It may possibly be objected that we are not entitled to come to any conclusion, from experiments made on so small a scale, as to what takes place in the ocean. Such objections would be perfectly valid, if there was not some evidence, in the conditions we find in the ocean, to support these conclusions. If the return currents in the ocean flowed underneath the wind-driven surface currents, then we should be perfectly justified in expecting some evidence of their presence. For instance, we would expect that the water near the bottom, underneath the wind-driven currents near the equator, would be hotter than the water at corresponding depths at other parts of the ocean. An examination, however, of the temperature sections of the Atlantic Ocean, taken by the "Challenger," shows no evidence whatever of the return current by this route; we are therefore compelled to conclude that the water must return by the surface, and that the wind does only produce horizontal currents, and therefore cannot account for the presence of the cold water which we find all over the bottom of the ocean, from the poles to the equator.

I

The second point 1 wish to refer to, is the effect of these winddriven surface currents on the cold water underneath them. have said that almost the whole of the motion of the wind-driven currents takes place in a horizontal direction. Such was the general result given by the experiment. But, in addition to this, there is another point to which I must refer. The wind driven

horizontal currents have an influence on the water underneath them which, for the sake of clearness, I have reserved for separate consideration here. Let us draw an imaginary section across a wind-driven current, at a point near its source, that is, near where the wind begins to act on it. And let us imagine another section of this same current, at a point some distance farther "down" the stream. As the current is acted upon by the wind between these two sections, it will be much deeper at the second section than at the first, and will also be going at a greater velocity. There will, therefore, be much more water passing the second section than passed the first, and the water necessary to supply this growing stream must be supplied to it between the two imaginary sections. The result is, part of the necessary supply rushes in at the sides, but part of it rises from the still water underneath the surface stream. This lifting of the deeper water by the surface current was very evident in the experiment already referred to, so long as the surface current was shallow, and gradually became less, as might be expected, when the current deepened.

From these considerations, we are naturally led to expect that the hot surface water at those parts of the ocean over which winds are constantly blowing, will be much reduced in depth, and that the cold bottom water will be found at a less depth underneath these surface currents than at any other part of the ocean. Part of this cold water will, in all probability, get mixed up with the bottom water of the surface current. And further, we would expect that this wind-driven hot surface water, after it passes beyond the windy regions, will gradually lose its motion and increase in depth.

These expectations are in a remarkable manner supported by the evidence of the temperature sections of the Atlantic taken by the Challenger," and given in Dr Carpenter's paper in the "Proceedings of the Royal Geographical Society," vol. xviii. No. iv., 1874. Take, for instance, the section between Tenerife and St Thomas. In the first part of the journey there is not much alteration in the relative position of the isotherms, but after crossing the Tropic of Cancer and getting into the region of the north-east trade winds, the hot surface isotherms gradually approach each other, and the isotherm of 40°, which off the coast of Tenerife was

at a depth of nearly 1000 fathoms, rises to about 700 fathoms before it arrives at St Thomas. Again, in the passage north from St Thomas to Bermuda, and on to Halifax and New York, the temperature sections show that after getting out of the region of the trade winds, the drifted hot surface water has gradully lost its motion and increased in depth. This refers to the great mass of ocean water, and not to the comparatively shallow Gulf-Stream. For instance, the isotherm of 60°, which at St Thomas was found at a depth of only 200 fathoms, was found at a depth of 330 fathoms for hundreds of miles all round Bermuda, notwithstanding a considerable reduction in the temperature of the surface water.

The temperature sections of the South Atlantic do not illustrate these points so well as the temperature sections of the North Atlantic, partly because a very large part of the hot surface water of the south equatorial current does not return to the South Atlantic, but is driven into the North Atlantic, and partly on account of the great amount of cold surface water of the Antarctic drift, which gets mixed np with the return current; and further, the temperature sections are not taken at the best places for our present purpose. The only two available sections, however, point to the same conclusion as the North Atlantic sections. There is no section of the south equatorial current, but we may suppose it to be somewhat similar to the section taken between St Paul's Rocks and Pernambuco, which gives the isotherms of the branch of the south equatorial current which passes into the North Atlantic. If we compare this section with the part of the stream which has flowed southwards, as given in the section taken between Abrolhos Island and Tristan d'Acunha, we find that the hot surface water, in flowing southwards beyond the region of the south-east trade winds, has deepened, notwithstanding a considerable fall in the temperature of the surface water. The isotherm of 40° which was found at a depth of 300 fathoms off Pernambuco, sunk to a depth of between 400 and 500 fathoms between Abrolhos Island and Tristan d'Acunha. We might have expected that this hot surface water would have kept its depth all the way to the Cape of Good Hope. It, however, does not do so, probably on account of the cold surface water of the Antarctic drift.

I am aware Dr Carpenter has offered a different explanation of the

rising of the glacial water under the equator. He considers that the rising of the glacial water under the Line is due to the meeting of the Arctic and Antarctic underflows. That part of the effect is due to this cause is very probable, but when we consider the very great area of section of the under glacial currents, and the small amount of water that can be carried by them, it is evident that their rate of motion must be very slow, and it is very doubtful how far the whole phenomena can be explained in this way. Our doubts on this subject are somewhat confirmed by the consideration of the fact that the Arctic underflow is warmer than the Antarctic underflow, and would therefore-other things being equal-tend simply to overflow the Antarctic underflow, and not to rise vertically as Dr Carpenter supposes. That the wind-driven currents, so long as they are increasing in volume, have the power of drawing the bottom water upwards cannot be doubted, and I have already said was most marked in the experimental illustration. We should not, however, expect to find this lifting power so marked in the ocean, as the ocean currents are so much deeper and do not so rapidly increase in volume as in the experiment. And further, the bottom water in the ocean is denser than the surface water, and does not rise so easily as the bottom water in the experiment.

If we take the evidence of the temperature sections of the Atlantic on the subject, we shall find that they also point to the wind as one of the causes of the rising of the cold bottom water. If we take the section between Madeira and lat. 3° N. and long. 15° W, we find that the isotherm of 40°, which at Madeira lies at a depth of about 900 fathoms, rises to a depth of only 300 fathoms at the equatorial position, and further, neither this section nor the section between lat. 3° N. and long. 15° W. and Pernambuco, show the least evidence of the presence of Antarctic water so far north as the equator, in the eastern basin of the Atlantic. This rising, then, of the glacial water on the eastern side of the Atlantic cannot therefore be due to the meeting of the glacial streams, and in the absence of further evidence we may suppose it to be due to the wind-driven currents. We thus see, that though the great bulk of the wind-driven circulation is a horizontal one, yet there is also produced in a comparatively very small degree a modified vertical circulation.

VOL. IX.

3 G

It may here be asked, Is this lifting power of the surface currents sufficient to account for the vertical circulation which we find in the ocean? In all probability it is not. There seems to be no reason why this vertically rising current under the equator should draw its supplies from the furthest limits of the ocean, which it would require to do to explain the conditions we find existing. Yet there can be no doubt but that these horizontal surface currents really do assist in producing a vertical circulation.

3. On a New Investigation of the Series for the Sine and Cosine of an Arc. By Edward Sang.

The sines of the successive equidifferent arcs form a progression having for its general character the relation

and the properties of sines may be deduced from this general formula. Viewed in this light, the angular functions become cases only of more general ones.

If we suppose A, B, C to be three consecutive terms of such a progression we must have

from which, when three of the four quantities, A, B, C, v are given, the fourth may be found. Let then A and B, the first and second terms of the progression, and v the common coefficient, be given; the succeeding terms may be computed thus: