The Mathematical Monthly

Since, by § 19, the plane of a given scalar may be considered as coinciding with any plane we please in space, all the terms in the last member of (47) except the last, satisfy one or the other of the conditions of § 35; whence, by (43), (46), and (47),

p.qr SpSq.Sr+Sp Sq. Vr+Sp Vq.Sr+ Sp V q . V r (48) +VpSq.Sr+Vp Sq. Vr+VpVq. Sr+VpVq. Vr = (Sp+Vp) (Sq + V q). (Sr+ Vr)=pq.r. Equation (48) shows that in general, quaternions are associative.

Mathematical Monthly Notices.

A Manual of Spherical and Practical Astronomy, embracing Nautical Astronomy, and the Theory and Use of Fixed and Portable Astronomical Instruments. Amply illustrated by engravings on wood and steel. By Professor WILLIAM CHAUVENET, of the United States Naval Academy. In two royal octavo volumes. Price $7.50. Philadelphia: J. B. Lippincott & Co. London: Trübner & Co.

The Publishers propose to issue these volumes as soon as the number of subscriptions will warrant the undertaking; and in their "Prospectus" they say:

"There exists at present no work on Spherical and Practical Astronomy in the English language, adapted to the wants of the Practical Astronomer, or even of the advanced university student. While there are many elementary treatises designed as text-books in a collegiate course, some of them admirably adapted for this use, there are none which are intended to carry the student beyond the Elements, and to give him that insight into the general theory, and that familiarity with the practical details of the subject, which are indispensable to the working Astronomer. "PROFESSOR CHAUVENET, who is well known to the scientific world as an exact investigator and clear expounder of mathematical and astronomical subjects, has undertaken to supply this want. His work will not only be the most complete reference-book on this subject that exists in the English language, but will cover the whole ground occupied by the best modern German treatises on both Spherical and Practical Astronomy. The most recent investigations of American as well as European Astronomers will be incorporated in the work. All the most useful problems will be fully illustrated by numerical examples, based upon numbers derived from actual observation, and carried out in the forms which appear to be most approved among experienced computers."

Our readers need no assurance from us of Professor CHAUVENET's peculiar fitness for the task he has undertaken.

We learn from the "Synopsis of the Table of Contents," that the first part, on "Spherical Astronomy," will contain thirteen Chapters, and the second part, on the "Theory and Use of Astronomical Instruments." ten Chapters, with an Appendix, containing the "Method of Least Squares."

We ask every one at all interested in the subject of these volumes to send their names to the Publishers, and aid in securing their early issue. CHAUVENET's Trigonometry is, of itself, an ample guaranty that the proposed volumes will combine the highest scientific and literary excellence.

Tables of Victoria. Computed with Regard to the Perturbations of Jupiter and Saturn. By F. BRÜNNOW, Ph. Dr., Professor of Astronomy in the University of Michigan, and Director of the Observatory at Ann Arbor. Printed by order of the Board of Regents. New York: B. Westermann & Co. London: N. Trübner & Co. 1857. 4to. pp. 77.

These Tables embody the results of the author's labors upon the asteroid Victoria (sometimes called Clio) since its discovery, September 13, 1850; and are based upon the observations made at the oppositions in 1850, 1852, 1853, 1854, 1856, and 1857. The elements derived from the first five of these oppositions, and the numerical expression of the perturbations produced by Jupiter and Saturn, may be found in the Astronomische Nachrichten, No. 1077; and the corrections of these elements, as given by the observations of the opposition of 1857, in the Astronomical Journal, No. 108. The smallness of these corrections showed that the resulting elements were sufficiently accurate to form the basis of a set of Tables which are now before us. These Tables, by means of 37 different arguments, give the perturbations of M, the mean anomaly, of log r, the logarithm of the radius vector, and of §, the perpendicular to the plane of the asteroid's orbit. These are HANSEN's co-ordinates, and the general expressions for their perturbations are found in the Astronomische Nachrichten, No. 799. Having found these perturbed co-ordinates, we readily pass to the rectangular co-ordinates, referred to the plane of the equator, and the line of the equinoxes. Abbreviated tables are given for finding an approximate place when only five principal terms of the perturbations are used, by means of which an ephemeris sufficiently accurate for all the purposes of observation can be rapidly computed. When it is remembered that an accurate ephemeris is only needed for a month or so before and after the opposition, while a rough one for the purpose of identification is needed for all the rest of the year, the advantage of this feature will be apparent. The Tables are well arranged for use, and printed in fine style. The author has placed astronomers, as well as the science, under renewed obligations by the publication of this timely and valuable work.

THE following gentlemen have sent us solutions of the Prize Problems in the November Number, Vol. II., of the Monthly:

JAMES F. ROBERSON, Senior Class, Indiana University, Bloomington, answered Problems III. and IV.

GEORGE B. HICKS, Cleveland, Ohio, answered Problems III., IV., and V.
WILLIAM HINCHCLIFFE, Barre Plains, Mass., answered Problems III. and IV.

D. M. HUDSON, Paris, Jennings Co., Indiana, answered Problems I. and II.

DAVID TROWBRIDGE, Perry City, Schuyler Co., N. Y., answered Problems III., IV., the first part of V.

HENRY E. PRINDLE, Ashtabula, Ohio, answered Problems III. and IV.

and

We would also acknowledge valuable solutions from ASAPH HALL, Esq., Assistant at the Observatory of Harvard College, and JAMES CLARK, Esq., Wayne, Me.

II. If a circle be described touching the base of a triangle and the sides produced, and a second circle be inscribed in the triangle; prove that the points where the circles touch the base are equidistant from its extremities, and that the distance between the points where they touch either of the sides is equal to the base.

III. Inscribe the maximum rectangle between the conchoid and its directrix. - Communicated by Prof. DANIEL KIRKWOOD.

IV. Given a cask containing a gallons of wine. Through a cock at the bottom of the cask wine flows out at the rate of b gallons per minute, and through a hole at the top water flows in at the same rate. Supposing the water, as fast as it flows in, to mingle perfectly with the wine, how long before the quantities of wine and water in the cask will be equal? and how much wine will be left in the cask at the end of t minutes? - Communicated by Prof. C. A. YOUNG.

V. Two circles being given in a plane, find geometrically the locus of the points from which chords of similar arcs in the two circles will be seen under the same angle, the chords being perpendicular

to the lines of vision drawn through the centres of the given circles. -Communicated by Prof. WM. CHAUVENET.

The solutions of these Problems must be received by the 1st of May, 1860.

REPORT OF THE JUDGES UPON THE SOLUTIONS OF THE PRIZE PROBLEMS IN No. III., Vol. II.

THE first Prize is awarded to JOHN Q. HOLLISTON, Sophomore Class, Hamilton College, Clinton, N. Y.

The second Prize is awarded to F. E. TOWER, Senior Class, Amherst College, Amherst, Mass.

The third Prize is awarded to FRANK N. DEVEREUX, Boston, Mass.

PRIZE SOLUTION OF PROBLEM I.

By FRANK N. DEVEREUX, Boston, Mass.

If two circles touch each other, any straight line passing through the point of contact cuts off similar parts of their circumferences.

The line joining the centres C and C' will pass through the point of contact B. Let A and A' be the points in which the line passing through the point of contact meets the circumferences. Join A and C, A' and C'. The triangles A B C and A' B C' thus formed are isosceles and similar, as is easily seen. The angles at the centres, A CB and A' C' B, are therefore equal, and are measured by similar parts of the circumferences. Hence the proposition is true. This proposition applies whether the circles are tangent externally or internally, a fact not noticed by any of the competitors.