and this is the relation existing between the mean anomaly and the true anomaly.

SOLUTION OF A QUESTION INVOLVING A MINIMUM.

THE following problem is found in the Mathematical Monthly (Runkle's). "Find a point, 0, within a triangle, such that (OA)"+(1B)"+(OC)"= a minimum."

[The solution there given is a good one; but a friend of mine, in the United States Coast Survey, dreamed out the solution below, and conveyed it to paper in the morning. I give it for publication; but he disallows the use of his name.]

that is, the point is the center of gravity of the triangle.

THEO. L. DE LAND, TREAS. DEP'T, WASHINGTON, D. C.

REPETENDS.

BY PROF. M. C. STEVENS, SALEM, OHIO.

IN the February number of the Michigan Teacher, for 1869, there appeared an article contributed by Mr. William Wiley of Detroit, entitled, "New Theory of Repetends", in which it is shown that the figures of a repetend are easily deduced from the common fraction successively from right to left, instead of from left to right as is done by the ordinary method.

Thinking it deserving of more notice than it received from its publication in that journal, I propose in this article to reproduce, in substance, Mr. Wiley's method, and also demonstrate some of the curious properties of repetends.

First Principle.-In every common fraction which reduces to a pure repetend, the unit's figure of the denominator must be either 1, 3, 7 or 9. Second Principle.—If the numerator of the fraction be unity, the last figure of the repetend is either 9, 3, 7 or 1, respectively, according as the units of the denominator is 1, 3, 7 or 9.

Third Principle.-If 1÷d be the fraction that reduces to a repetend, q1 the last figure before it commences to repeat, r1 the last and r2 the next to the last remainder, then evidently

Fourth Principle.—If In, In−1, ...... 92 and q1 be the digits of the repetend, then is

in which we have the same sequence of digits, the last taking the first place and the remaining figures being each removed one place to the right. For an example, take ✈ .1428574; then = .714285. This is ev.14285, and the value of the fraction at the end is five times the part found, § = .714284; whence .14285714284 = .1428

ident, since

=

=

57142855. We thus see the necessity of the sequence above stated.

When we multiply (2) by r2, if we represent the tens of r291, 292, &c., by m1, m2, &c., we evidently have

(1′)

r293+m2 = 10mg +94, &c.

(3')

We have here the key to Mr. Wiley's method; for by the second principle 91 is known at a glance, and by the third principle r, is found by eq'n (1), whence q, becomes known from (1'), and 93, 94, &c., are successively found from equations (2′), (3′), &c.

(To be continued.)

PROBLEMS.

1. Find the value of x and y in the following equations:

a2x1 + b2y* = a2b3(x + y)2,

a2x2 + b2y2 = a2b3.

-Communicated by U. JESSE KNISELY, Pres't and Prof. of Mathematics in Luther College, Newcomerstown, Ohio.

2. Let a regular polygon of 14 sides be described, each of whose equal sides shall be one. Then will the radius of its circumscribing circle, which put =r, be more than two and less than three. Put r = 2+x; then is x a positive quantity less than one. Let another regular polygon of half the number of sides (7) be inscribed in a circle whose radius is one, and determine one of its equal sides in functions of x expressed in its simplest form.

3. If a line make an angle of 40° with a fixed plane, and a plane embracing this line be perpendicular to the fixed plane, how many degrees from its first position must the plane embracing the line revolve in order that it may make an angle of 45° with the fixed plane?-Communicated by PROF. A. SCHUYLER, Berea, Ohio.

4. A cask containing a gallons of wine stands on another containing a gallons of water; they are connected by a pipe through which, when open, the wine can escape into the lower cask at the rate of c gallons per minute, and through a pipe in the lower cask the mixture can escape at the same rate; also, water can be let in through a pipe on the top of the upper cask at a like rate. If all the pipes be opened at the same instant, how much wine will be in the lower cask at the end of t minutes, supposing the fluids to mingle perfectly?-Communicated by ARTEMAS MARTIN, Mathematical Editor of Schoolday Magazine, Erie, Pa.

NOTE.-To those who use "Nystrom's Mechanics": Nystrom prints "29.869650000+."

But π2 ==

9.86960440108+.-U. JESSE KNISELY.

QUERY.-What is the explanation of the phenomena described below? If a ball of cork or other light substance be placed in a vertical jet of water of sufficient force to elevate the ball, it will rise to a point where the force of the ascending jet, or so much of it as is efficient in elevating the ball, is just equal to the weight of the ball, and will there revolve; and its equilibrium will continually be restored, notwithstanding the ball may be disturbed by slight horizontal forces.

BOOK NOTICES.

Comets and Meteors. By DANIEL KIRKWOOD, LL.D., Professor of Mathematics in Indiana University, and author of "Meteoric Astronomy." J. B. Lippincott & Co., Philadelphia.

To those who have not yet seen this very interesting book by Prof. Kirkwood, the following quotation from the Preface will serve to indicate its character:

"The origin of meteoric astronomy, as a science, dates from the memorable star-shower of 1833. Soon after that brilliant display it was found that similar phenomena had been witnessed, at nearly equal intervals, in former times. This discovery led at once to another no less important, viz.: that the nebulous masses from which such showers are derived revolve around the sun in paths intersecting the earth's orbit. The theory that these meteor-clouds are but the scattered fragments of disintegrated comets was announced by several astronomers in 1867- -a theory confirmed in a remarkable manner by the shower of meteors from the debris of Biela's comet on the 27th of November, 1872. To gratify the interest awakened in the public mind by the discoveries here named, is the design of the following work. Among the subjects considered are cometary astronomy; aerolites, with the phenomena attending their fall; the most brilliant star-showers of all ages, and the origin of comets, aerolites and falling stars."

Surveying and Navigation, with a Preliminary Treatise on Trigonometry and Mensuration. By A. SCHUYLER, A. M., Professor of Applied Mathematics and Logic, in Baldwin Univer sity, author of "Higher Arithmetic", "Principles of Logic" and "Complete Algebra." Wilson, Hinkle & Co., Cincinnati and New York.

We would be pleased to give an extended notice of this book did our space permit. We must be content to say, however, that as a text-book for the student, and as a manual for the surveyor, we think it admirable, both in plan and execution. The subjects discussed are thoroughly and yet concisely dealt with; and the paper, wood cuts and typography are perfect.

Yates County Chronicle. Persons who are fond of solving mathematical problems and who want something new upon that subject every week and a good newspaper besides, will do well to obtain the Yates County Chronicle; published at Penn Yan, New York. DR. S. H. WRIGHT, Mathematical Editor.

In bringing forward the nebular hypothesis to explain the cause of the primitive movements of our solar system Laplace states the following five phenomena which must be considered. 1. The motions of all the planets in the same direction and very nearly in the same plane: 2. The motions of the satellites in the same direction as that of the planets: 3. The motions of rotation of these different bodies and of the sun in the same direction as their projected motions, and in nearly the same planes: 4. The small excentricities of the orbits of the planets and their satellites: 5, and finally the great excentricities of the orbits of comets, while the inclinations of these orbits seem to occur wholly at random.

The only previous hypothesis to which Laplace refers is that of Buffon, the naturalist. Buffon assumed that a comet falling upon the sun had thrown out a torrent of matter, which, uniting at different distances into various globes, had become opaque and solid by cooling, and thus formed the planets and their satellites. This hypothesis would explain the first of the five preceding phenomena stated by Laplace, for all the bodies thus formed would move very nearly in the plane passing through the center of the sun and the direction of the torrent; but the four other phenomena cannot be explained by it. In fact, the smallness of the excentricities of the planetary orbits is directly opposed to this hypothesis; for if a body moving in an ellipse touches the sun it will do so at each of its revolutions, and, although this condition of things might be somewhat modified in Buffon's hypothesis, the chance that the excentricities of the orbits would be small is very slight. Finally, this hypothesis does not account for the comets at all. Among those who preceded Laplace in speculations on the constitution of the universe, after the discovery of the law of gravitation, may be mentioned Kant, the metaphysician, Lambert and Sir William Herschel. In a work published in 1755 under the title