In this case, also, there is but one real root; and when b is negative, the arcs v and u will be negative, as before.

a

v

v

pear, leaving = 2√√ cos. But this cos corresponds to the arcs

v, 360° +v, and 360° - v.

Dividing by 3, and putting 120° under the form of 180° 60°, we find the other two roots,

We are now prepared to recapitulate; it being recollected that 10 should be algebraically subtracted from the index of the logarithmic sines and tangents. First bring the given cubic to the general form x3 + a x = b.

I. When the coefficient a is positive. Find

2

II. When a is negative, and () less than 1. Find

b

III. When a is negative, and() greater than 1. Find

In the case where()* = 1, x = 2 (')*.

Thus all the real roots of any cubic equation may be found by logarithms. It is perhaps unnecessary to remark, that in these values of x, the coefficient a is to be taken as numerically positive, irrespective of its algebraic sign. The investigation of these solutions is new in part, and will be found convenient for reference.

If p2<4q the roots of (3) and (4) are imaginary.

We have taken the liberty to add trigonometrical solutions of equations of the second degree, and commend them, as well as Correspondent's cubics, to the attention of students. ED.

THE MOTIONS OF FLUIDS AND SOLIDS RELATIVE TO THE EARTH'S SURFACE.

[Continued from page 406, Vol. I.]

SECTION V.

ON THE MOTIONS OF THE ATMOSPHERE ARISING FROM LOCAL DISTURBANCES.

58. BESIDES the general disturbance of equilibrium arising from a difference of specific gravity between the equator and the poles, which causes the general motions of the atmosphere, treated in the last section, there are also more local disturbances, arising from a greater rarefaction of the atmosphere over limited portions of the earth's surface, which give rise to the various irregularities in its motions, including cyclones or revolving storms, tornadoes, and water-spouts. When, on account of greater heat, or a greater amount of aqueous vapor, the atmosphere at any place becomes more rare than the surrounding portions, it ascends, and the surrounding heavier atmosphere flows in below, to supply its place, while a counter current is consequently produced above. As the lower strata of atmosphere generally contain a certain quantity of aqueous vapor, which is condensed after arising to a certain height, and forms clouds and rain, the caloric given out in the condensation, in accordance with ESPY's theory, produces a still greater rarefaction, and doubtless adds very much to the disturbance of equilibrium, and to the motive power of storms. So long, then, as the ascending atmosphere over the area of greater rarefaction is supplied with aqueous vapor by the current flowing in from all sides below, the disturbance of equilibrium must continue, and consequently the local disturbances of the atmosphere to which it gives rise, whether those of an ordinary rain storm, or a cyclone, may continue many days, while the general motions of the atmosphere may carry this disturbed area several thousands of miles.

59. In the ordinary rain-storms of the United States, the area of greater rarefication seems to be, in general, very oblong in the direction of the meridians, as is shown by ESPY's charts. The atmosphere becoming more rare over the land, a current seems to set in from the Atlantic towards the Rocky Mountains, causing an ascent of the atmosphere in the west, and a line of greatest rarefaction in the direction of the meridians, arising from the condensation of the ascending vapor into clouds and rain, while the general motion of the atmosphere eastward, in those latitudes, carries this area of greater rarefaction, with its accompanying rain-storm, towards the east, at an average rate of about 30 miles per hour. As the velocity of the general eastward motion of the atmosphere is greater above, the rainy portion of the storm is for the most part on the east side of the line of greatest rarefaction, and as the currents below must be towards this line on both sides, when it passes over any place, the rain generally ceases and the wind changes.

60. When the area of rarefaction is such as to cause the atmosphere to flow in from all sides below towards a centre, and the reverse above, the disturbed portion of atmosphere, if it were not that its motions are resisted by the earth's surface, and the surrounding undisturbed part, would assume the outline and the gyratory motion in the case of no resistance, as represented in Fig. 3 and Fig. 4. But on account of the resistances, the motions of the atmosphere are very much modified, so that it has only a tendency to assume in some measure those motions, and instead of the atmosphere's receding entirely from the centre, on account of the rapidity of the gyrations near the centre, as represented in Fig. 3, it is only a little depressed in the middle, as represented in Fig. 6.

D1r,

Fig 6

61. Since the force which produces the gyrations depends upon that is, upon the velocity of the flow to and from the centre,

it is evident, that, at the centre and at the external part of the disturbed portion of atmosphere, where Dr must vanish, the resistances destroy all gyratory motion. Hence, instead of very rapid gyrations near the centre, as in the case of no resistances, there must be a calm there, and the most rapid gyrations be at some distance from the centre, in accordance with observation. The diameter of the comparatively calm portion, in the centre of the large cyclones, is sometimes about 30 miles. The velocity of gyration of the external part, which, in the case of no resistances, is small, is in a great measure destroyed by the resistances of the surrounding atmosphere, so that it is, for the most part, insensible to observation, and only the more rapid gyrations of the internal part are observed. The motion of gyration combined with the motion at the earth's surface towards the centre, gives rise to a spiral motion towards the centre, exactly in accordance with the observed motions of the atmosphere in great storms or hurricanes, as has been shown by REDFIELD, in a number of papers on the subject, published in the American Journal of Science.

62. According to (§ 29) the gyrations of the inner part of a cyclone must be from right to left in the northern hemisphere, and the contrary in the southern, which is the observed law of storms in all parts of the world, as shown by REDFIELD, and also by Reid, in his Law of Storms. It is also evident that at the equator, where cos vanishes, there cannot be a cyclone, and hence, of all those which REDFIELD has investigated, and given in his charts of their routes, none have been traced within 10° of the equator. The typhoons or cyclones, also, of the China sea, have never been observed within 9° of the equator.

63. That the atmosphere must run into a gyration, if it converge towards a centre, is evident from the principle demonstrated in (§ 32), by which, in flowing in from all sides towards the centre,