The root of it, which is found by the summations of the above series, is .2016, and the others may be found from the reduced quadratic, they are 2.1284 and -2.3300. In both the above examples, it will be seen that or v2 a2 is a is small fraction, and so two terms at most of the series will amply suffice. It might, however, have happened that was nearly a2 v2 equal to unity. The method, however, would still be applicable, only we should have to take a greater number of terms. But in practice this may be avoided; and it will always suffice even in the most unfavourable cases to take two terms at most, and then apply a correction, as will presently be explained. For even if we were to omit all the terms in the latter series after the unit, the error in the value of the root would never exceed a fifth or sixth of its entire value, and the error would be much further lessened if we were to take one or two of the terms which follow the unit. Suppose, then, that by doing so, we find a value c for a root of the equation, but which, on substituting it for x, does not satisfy the equation so nearly as we could wish. Suppose also, that c+h is the true value of the root; then we should have (c+h)3-q(c+h)—r=0 or c2+3e2h+3ch2+h3—qc—qh—r=0 As h is supposed to be small, we may neglect its square; and this will give us for its value and so we might proceed to a still nearer reduction. Take as an example but by taking two terms of the second series, and applying the correction, one of the roots will be found to be .6566. The others are both possible, and may be found in the usual way. It appears, then, that the formula of Cardan is equally capable of reduction whether the roots be all possible or not, and with precisely the same degree of exactness; the only difference being that when they are all possible, the operation is somewhat more troublesome than when two of them are impossible. Moreover, the formula is capable of being reduced algebraically, and without the use of tables of cosines. Nor is there much difference between the two methods as regards simplicity; perhaps the algebraical method will have the advantage when it is only required to calculate the root to four or five places of figures; but beyond this we might, perhaps, have to refer to the tables oftener than we should in reducing the trigonometrical formula, but not otherwise; and at all events, it will have the advantage of treating in a purely algebraical manner, and without the introduction of other branches of mathematics, what is a purely algebraical problem. ART. VI. On the Vertical Currents of the Atmosphere. By HENRY HENNESSY, F.R.S. §. 1. T has been long recognized that, although currents of wind in a direction nearly parallel to the horizon are those which usually prevail, the atmosphere is frequently subjected to vertical and oblique motions among its particles. Under favourable conditions these motions may acquire such a development as to force themselves upon the attention of observers, and thus become objects for meteorological inquiry. The interesting researches of M. Fournet upon the vertical currents of mountains, appear to have arisen from the opportunities enjoyed by that physicist of studying such phenomena among the Alps. Among the deep ravines and valleys, as well as along the elevated slopes and escarpments of the Alps, a regular periodicity in the action of vertical winds has been frequently observed during the course of twenty-four hours, which has led to the conclusion that their development depends upon changes of temperature resulting from the presence and absence of the sun. As it is now well established that the distribution and changes of temperature in these islands are dependent upon other influential causes besides the direct action of the sun,' we cannot, in general, expect to find 1 See ATLANTIS, vol. I. p. 396, also a letter from the author to Major-General Sabine, on the influence of the Gulf-stream on the winters of the British Islands. Proceedings of the Royal Society, vol. IX. p. 324. in our climate, a similar diurnal periodicity so distinctly defined as that observed in the centre and south of Europe. Here, as well as on the continent, mountains are favourable to the production of inequalities of temperature, moisture, and density among the aerial strata, which thus become liable to a multitude of disturbances, and especially to the action of vertical currents. It seems to follow that in mountainous countries vertical currents have well marked relations with the changes of the weather. If, as usually happens, lakes exist among the mountains, the mysterious occurrence called the "bore" is also thus explained. The circumstance that the suddenly-formed wave thus designated always proceeds from a side of the lake bordered by steep mountains, immediately suggests such an explanation. Although a similar idea has occurred to other inquirers, I may be permitted to refer to an instance where a demonstration was presented by me of the efficiency of vertical currents in producing the "bore" on the surface of one of our Irish lakes. The fact that such a sudden wave usually preceded a change of the weather in the district surrounding the lake, led me to think that the study of the effective cause of the bore itself might become of importance in meteorology. But to do this, we should possess means for observing the actual direction, and, if possible, the force of the atmospheric currents. §. 2. Hitherto, all instruments which had been employed for observing the wind were devised exclusively with reference to its horizontal direction and intensity, from the simple wind-vane to the most finished anemometer. I have attempted to modify the ordinary vane so as to make it an indicator of the actual direction of the current, both in altitude and azimuth. Instead of the fixed surface against which the wind impinges in ordinary vanes, I had a disk suspended at the tail of the vane, capable of rotating on an axis perpendicular to the line of direction of the instru A pair of flanges were attached to this disk in such a manner that, when the whole was at rest and the air free from motion, the flanges would be horizontal. With perfectly horizontal currents, the flanges would still continue in the same position, although the head of the vane would as usual move about ment. * In a letter to the Rev. T. R. Robinson, D.D., of Armagh. See Proceedings of the Royal Irish Academy, vol. vi. p. 279. * Some time after the anemoscope had been devised, my attention was called by my friend, the Rev. Dr. Robinson, to a passage among the notes to Dr. Darwin's poem of the "Botanic Garden", wherein the writer indicates such an instrument; but he seems never to have realized this idea, and the apparatus which he proposed was essentially different from mine. in azimuth. But if a current happened to be inclined to the horizon, the flanges would be pressed upwards or downwards, showing the direction and amount of the inclination, precisely as the position of the head or tail of the ordinary vane shows the direction and inclination of a current with reference to the meridian. When we know the inclination of a given current to the horizon, we can readily estimate its absolute force from its horizontal force, as can be easily shown. §. 3. Let the origin of co-ordinates be at the centre of the axis of the vertical disk; y dx will represent an element of the area of the flange. Let represent the angle of inclination of the flange, H the pressure exercised by the wind in a horizontal direction. upon a square unit of surface, and V the vertical pressure exercised upon a similar unit. The entire moment of the horizontal forces acting on the entire flange will be and the moment of the vertical forces will be HСsino.xydæ, Both of these moments tend to cause a rotation of the disk, but in contrary directions: hence when the disk is in equilibrium they must be equal, and therefore, because is independent of and y, we shall have Hsin0=Vcos0, V=Htang0 (1) and if we write F for the absolute force of the wind, we shall have F=Hsec0 (2). Hence it follows, that if we can observe the absolute direction of the wind, we can estimate its vertical force as well as its absolute intensity without any special instrument, using the results obtained by the existing anemometers which give the horizontal intensity. §. 4. A wind-vane or anemoscope, capable of showing the absolute direction of an atmospherical current, having been constructed in accordance with my directions, I proceeded to make some observations during the months of June, July, and August, 1857. It was placed on the top of a strong mast, about twentysix feet in height. The mast was fixed near the end of a large garden, far from buildings. As my first series of observations were intended to be merely provisional, I did not make them at specific fixed hours, but at such times as presented disturbances in the atmosphere, or which afforded sufficient leisure for continued attention. A journal was kept, from which I make the following extracts. Before doing so, it is proper to remark that by the term "vertical currents" in these extracts, as well as in the title of this paper, I do not mean currents actually perpendicular to the horizon, but rather oblique currents with an upward or downward tendency. June 28, 7h. A.M.-Air perfectly still, flanges horizontal, head of vane towards the east. 7h. 30m. A.M.-Breeze with slight vertical currents until after 8. The currents were upward from the ground. The flanges were often perfectly horizontal, and their mean angle of inclination was small. About 10 A.M, a few fine scattered clouds (cirro-cumuli) were observed to move in a direction contrary to the wind as observed near the earth. From 3h. P.M. to 3h. 45m.-Wind extremely gentle from E.S.E., upward current, angle of inclination estimated at about 5°. The upward currents often continued for several minutes together. The angle was sometimes almost imperceptible. The sky became gradually overcast towards evening. June 30, 10 A.M.-Sky completely overcast, strong wind from E.S.E., rapid oscillations of the disk during the greater part of the day. About 6 P.M., the wind blew in violent gusts from the east, and the disk showed alternations of upward and downward currents with occasional short intervals. These observations led me to conclude that rapid currents of air cannot generally advance with the same steadiness as currents of water, the greater mobility and elasticity of the former fluid probably allow its movements to easily acquire a species of undulation. Thus we may account for the motions of the branches of trees, which generally swing backwards and forwards, showing rapid variations in the intensity of the wind. During breezes composed of a succession of strong sudden gusts, it was difficult to estimate the inclination of the flanges, as each fresh impulse drove the flange beyond the angle due to the pressure, and before it had been sufficiently long oscillating about its true position to allow a correct observation, a fresh gust would perhaps drive it in a different direction. July 1, 9 A.M.-Wind N.E., strong breeze with vertical currents. The position of the flanges was sometimes steady for many minutes, with a very small inclination, upward currents appeared to predominate in duration. July 2, before 9 A.M.-Air still and warm, head of vane directed to S.E. After 9 a gentle breeze from E. and E.S.E., with an upward tendency. The disk remained steady at a small |