the same length of time, 18 weeks, and of 4317

the answer.

acres,

NOTE. We have appended the Prize Solution, to give our readers an opportunity to judge for themselves. It does not seem to us that there is any ambiguity in the statement of the problem, nor much difficulty in its solution; and we add the following for those who may wish to study it. Let unity denote the amount of grass at first on each acre, g the growth on each acre per week, and x the required number of oxen. Then, from the first condition, 1+4 g is the amount of grass consumed from one acre in 4 weeks, (1+4 g) 3 the amount from 31 (1 + 4 g) 3 (1 + 4 g) 3 the amount consumed by one ox in 4 weeks, the amount con12 X 4 sumed by one ox in one week. From the second condition, 1 + 9 g is the amount of grass con(1 + 9 g) 10 sumed from one acre in 9 weeks, (1 + 9 g) 10 the amount from 10 acres, the (1 + 9 g) 10 amount consumed by one ox in 9 weeks, the amount consumed by one ox in 1 week. 21 X 9 From the third condition, (1+18 g) 24 is the amount consumed by one ox in 1 week. Therefore

12

2 X 18

21

By solving these equations we get x = 37, which is the only number of oxen which will satisfy all the conditions of the problem.

SOLUTION OF CUBIC EQUATIONS BY THE COMMON LOGARITHMIC TABLES.*

BY A CORRESPONDENT.

By removing the second term, in the usual mode, every cubic takes the form 23+ ax = b. Assume xyz, and 3 y z=—a. By substituting these, the given equation will become y3+23 From the last two equations, the value of y3 and of 3 can be found by a quadratic. Since xyz, we thus obtain CARDAN's formula,

=

= b.

* For the trigonometrical solution of equations of the second, third, and fourth degrees, the student may consult CAGNOLI's Trigonométrie, Chap. XIV.; or CHAUVENET'S Trigonometry, pp. 95-100, for the solution of equations of the second and third degrees.-ED.

1

3

= (')' { ( 1 + √ √ 1 + ( ) ( )')'+ (1 − √('(')'}·

In adapting this expression to the logarithmic tables three cases are

x=

=

(†)* {(1 + √1 + tan2 v)* + (1 − √ 1 + tan3 v)*},

= ( )* { (1 + 1 )' + (1 − 1)'},

COS

This equation may be further simplified by making tan u=

(tan);

In this case there is but one real root. When b is negative it will only change the signs of v and u.

2

II. When a is negative and of such value that () is less than 1.

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.

III. "The irreducible case." When a is negative and of such value that

By applying DEMOIVRE's theorem, the imaginary quantities disappear, leaving = 2√ cos. But this cos corresponds to the arcs

a

v

v

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 23 + ax = b.

I. When the coefficient a is positive. Find

2

tan v =

;

tan u = (tan); x = 2√√ cot 2 u.

x=

2

II. When a is negative, and (3)* less than 1. Find

2

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

b

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 sur rounding 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.