since u+u2+ &c., and++&c., are geometrical series with

ratios u and

REMARK.

1

и

By dividing the terms of the first fraction, and multi

plying those of the second, by ut, we get

This form of the result was not given by any of the competitors.

PRIZE SOLUTION OF PROBLEM IV.

JOHN Q. HOLLISTON, Hamilton College, Clinton, N. Y.

Having given the Right Ascensions and Declinations of two stars, to find the formula for the distance between them. Also, find what the distance becomes, when for one star A. R. is 8h 12m 38.17, and Dec. 17° 23′ 49′′.8 north, and for the other A. R. is 13h 28m 19.92, and Dec. 21° 12′ 37′′.2 south.

Let and be the declinations of the two stars; then 90° — 8 and 90° will be their co-declinations; and since the right ascensions are measured on the equator, their difference will measure the angle at the pole made by the meridians passing through the two stars. Let H denote this angle, and the distance sought. We have then a spherical triangle in which the sides 90° 8 and 90° include the angle H, and 4, the third side, may be found from the formula

cos = cos (90°—8) cos (90°—d)+ sin (90°—8) sin (90° — d′) cos H,

(1) = sin 8 sin & + cos & cos & cos H,

Formulas (3) and (4) need only tables of logarithmic sines, cosines, and we find 4 86° 24′ 12′′.2.

&c.;

PRIZE SOLUTION OF PROBLEM V.

By ASHER B. EVANS, Madison University, Hamilton, N. Y.

In a frustum of any pyramid or cone, the area of a section, parallel to the two bases and equidistant from them, is the arithmetical mean of the arithmetical and geometrical means of the areas of the two bases.

The three sections are evidently similar figures; hence their areas will be as the squares of any homologous lines. Let x, xy, and xy, represent any homologous lines in the upper base, middle section, and lower base, respectively. Then their areas may be represented by m (x − y)2, m x2, m (x + y)2, respectively. The arithmetical mean of the two bases is

and the arithmetical mean of m (x2 + y2) and m (22 — y2) is m x2, as was to be shown.

SIMON NEWCOMB.

W. P. G. BARTLETT.

TRUMAN HENRY SAFFORD.

NOTES AND QUERIES.

1. What fraction is that, to the numerator of which if 1 be added, its value will be ; but if 1 be added to the denominator, its value will be ?

2 3

4

The successive multiples of are %, 8, 12, 15, 1, &c.; hence, taking unity from the numerator of each of these, we have the series 4, 3, 12, 15, 18, 21, &c.,

1 2
69

3

4

5

3

4

5

and the required fraction must necessarily be one of this series. Again, the successive multiples of are, 12, 16, 20, 24, &c.; whence taking unity from the denominator of each of these, we get 4, 11, 15, 19, 23, 27, &c.,

3

4

5

a series of fractions in which the required fraction must necessarily be found.

Examining these two series, we find that the third fraction in the second series is the same as the fourth fraction in the first series, and therefore, as the sought fraction must be found in both series, we conclude with the utmost certainty that is the fraction fulfilling the conditions of the question.

2. What fraction is that, whose numerator being doubled, and denominator increased by 7, the value becomes; but the denominator being doubled, and the numerator increased by 2, the value becomes ?

Reasoning as in the preceding example, we must first take the successive multiples of ; divide the numerator of each multiple by 2, and decrease each denominator by 7; then we have the following series, viz. :—

Again, take the consecutive multiples of, decrease the numera

tor of each by 2, and divide the denominator by 2; then we have

the series

26

号 8, 8, 18, 18, 18, 18, &c.

The required fraction must exist in both series, in order that the two conditions in the enunciation may be complied with, and as is the only fraction common to both, we conclude that will satisfy the conditions stated in the question. Prof. W. RUTHERFORD, Woolwich. The Northumbrian Mirror, 1838.

1. Isochronous Motions.-The principle of isochronous motions, as exhibited in the small oscillations of the pendulum, in the oscillations of the hair-spring balance, and in the descent of weights on the cycloid, is not usually set forth in the analytical treatment of these examples with sufficient prominence to be distinctly generalized and abstractly comprehended, though the analysis necessarily involves it.

In the following theorem and demonstration this principle is distinctly presented, and from it may be deduced all the cases that have been separately investigated.

Theorem.

If a material point, constrained to move in any given path, tend to approach a given fixed point in the path, urged along by a force proportional to the length of the path between it and the given point; it will pass from a state of rest at any point whatever to the given point always in the same time; or it will pass from the given point with any velocity whatever to a state of rest in the same time.

Demonstration. If two material points start at rest in two positions on the path, and if we suppose that the two lengths of the path included between the given fixed point and these positions be divided into the same number of very small parts, the ratio of the corresponding parts, and the ratio of their distances from the given fixed point along the path, will be equal to the ratio of the lengths themselves; hence, the forces acting in corresponding parts will be in the ratio of the parts themselves; hence, the sum of all the

actions of the forces through any number of parts in one length will be to the sum of the actions of the forces in the same number of corresponding parts in the other length in the same ratio; hence, the two velocities which the two material points will have in passing through corresponding parts will be in the ratio of the parts themselves, and these corresponding parts will therefore be passed over in the same time, and any number of them in one length will be passed over in the same time as the same number of the corresponding ones in the other length; hence, the whole lengths will be described in the same time; that is, the two material points will arrive at the given fixed point at the same instant, and with velocities proportional to the lengths of the path they have described. If a material point set out from the given fixed point with any velocity, it is obvious that it will come to rest at that point from which it would attain this velocity in moving back to the given fixed point; and that it will have the same velocities in every part as in moving from rest to the given fixed point, though in a contrary direction; hence the time will be the same.

Since a body, descending upon any path under the action of gravity, is impelled along the path by the force g sint, in which g is the direct force of gravity, and the angle made by the direction of the curve at any point with the horizon, it follows, that the times of descent will be the same from all parts of the curve to the point for which = 0, or the lowest point of the curve, if the arc s, reckoned from this lowest point, be proportional to the force g sin T. The equations 4 R sint is an equation of the cycloid, a curve which is therefore called the Tautochrone. In the common pendulum, the force g sint, which impels the ball along its circular path, is nearly proportional to the path itself when the amplitudes of the vibrations are small. The small motions of the pendulum are therefore nearly isochronous. The small motions of