Mathematical Monthly Notices. Résumé de Leçons de Géométrie Analytique et de Calcul Infinitésimal, Comprenant sur la Trigonométrie, sur l'Expression des Lieux Géométriques par leurs Équations, sur le Calcul différential et sur les Calcul intégral, l'Exposition des Connaissances nécessaires aux Ingénieurs pour l'Intelligence de la Mécanique rationnelle, de l'Hydraulique et de la Théorie dynamique des Machines. Par J. B. BELANGER, Ingénieur en Chef des Ponts et Chaussées, Professeur de Mécanique à l'Ecole impériale Polytechnique et à l'Ecole Centrale des Arts et Manufactures. Seconde Edition. Paris: Mallet-Bachelier. 1859. pp. 295. Tigr. 104. To this full title-page but little need be added to give a good idea of the work before us. It is simply intended to supply all the elementary knowledge in Trigonometry, Analytic Geometry, and the Calculus, which the student requires to read the course of rational mechanics as taught in the French schools of engineering. The subjects are clearly treated, and the only peculiarity we notice is, that the subject of Trigonometry is made to depend upon the theory of projections. Théorie des Fonctions doublement Périodiques et, en particulier des Fonctions Elliptiques. Par M. BRIOT et M. BOGUET. Paris: Mallet-Bachelier. 1859. pp. 342. The subjects discussed in this book are sufficiently indicated by the title. The method of discussion, which, in fact, involves a completely new "theory of functions," is one, probably, almost wholly unknown to American students. The authors have developed and extended the theory of imaginaries named by the illustrious CAUCHY "les quantités géométriques;" of which a full discussion may be found in his "Exercices d'Analyse et de Physique Mathématique," Tome IV.; and of which some application is made by M. V. PUISEUX in his "Recherches sur les Fonctions Algébriques," in LIOUVILLE's "Journal de Mathématiques,” Tome XV., p. 365. This view of V 1 is founded on the theories of ARGAND, MOUREY, etc. CAUCHY's "geometric quantity," which he denotes by r is algebraically expressed by x+y1r (cos + sin 0 . √ − 1) = r e3√-1, and these quantities may be added, multiplied, etc. just as coplanar quaternions would be. The modulus r corresponds to the tensor, and the argument to the angle of the quaternion. (See p. 133 of the present volume of the Monthly, equation (34).) We have no doubt that this book, especially if read with the two articles we have cited, presents an excellent opportunity to form an acquaintance with the recent progress of the higher branches of analysis among French mathematicians, just as SALMON's books and BOOLE's "Differential Equations" (heretofore noticed in the Monthly) do for the widely different paths of discovery pursued of late, more especially by British investigators. We scarcely need add, that the Elliptic Transcendants appear here in a totally different dress from that given them by their first great expounder, LEGENDRE; or even in the refined analysis of ABEL, although, as with the latter, the inverse function forms the base of the investigations. A second volume is promised, to contain the requisite tables, with practical applications, etc.-B. Leçons sur les Coördonnées Curvilignes et leurs diverses Applications. Par G. LAMÉ. Paris: Mallet-Bachelier. 1859. pp. 368. In some of his other books, the author has introduced, to a greater or less extent, the use of these peculiar coördinates invented by himself. If we conceive a family of similar ellipsoids depending on the variable parameter a, such that when a varies we pass from one ellipsoid to the next, etc.; and in the same way a family of one-sheeted hyperboloids depending on a parameter 8, and one of two-sheeted hyperboloids depending on y; then a, B, and y will be curvilinear coördinates of the common intersection-points of these three surfaces. This and similar systems have been found convenient, if not necessary, in many physical investigations, especially those involving the consideration of level surfaces, isothermal surfaces, etc. The present volume is designed to explain in their most general form the properties of such systems, with sufficient applications to the "equations of motion," the properties of heat, elasticity, etc., to make their use familiar to the student. No one who wishes to keep himself informed on all valuable advances in recent science should be inattentive to these investigations of Lamé. — B. Tracts, Mathematical and Physical. By HENRY LORD BROUGHAM, LL. D., F. R. S., Member of the National Institute of France, Royal Academy of Naples, Chancellor of the University of Edinburgh. London and Glasgow: Richard Griffin & Co. 1860. 8vo. pp. 304. The distinguished author's dedication, "To the University of Edinburgh, these tracts, begun while its Pupil, finished when its Head, are inscribed by the Author, in grateful remembrance of benefits conferred of old, and honors of late bestowed," suggest the long period (62 years) within which these tracts were written. Their titles will give the best general idea of their character: General Theorems, chiefly Porisms in the Higher Geometry; KEPLER'S Problem; Dynamical Principle, - Calculus of Partial Differences, - Problem of Three Bodies; Greek Geometry, Ancient Analysis, Porisms; Paradoxes imputed to the Integral Calculus; Architecture of Cells of Bees; Experiments and Investigations on Light and Colors; Inquiries, Analytical and Experimental, on Light; On Forces of Attraction to Several Centres; Meteoric Stones; Central Forces, and Law of the Universe Analytically Investigated; Attraction of Bodies, or Spherical and Non-spherical Surfaces Analytically treated; SIR ISAAC NEWTON,— Grantham Address; Notes. These Essays possess a peculiar value for the student of mathematics and physics; and even those profoundly versed in the subjects of which they treat will not find them devoid of interest. Editorial Items. THE following ladies and gentlemen have sent us solutions of the Prize problems in the February number of the Monthly : HARRIET S. HAZELTINE, Worcester, Mass., Probs. I., II. AMANDA M. BENNETT, Saline, Washtenaw Co., Mich., Probs. I., II. T. E. TOWER, Amherst College, Probs. I., II., III., IV., V. D. G. BINGHAM, Ellicotteville, N. Y., Probs. I., II. M. K. BOSWORTH, Marietta College, Ohio, Probs. I., II., V. CARLOS, West Point, N. Y., Prob. I. JAMES F. ROBERSON, Indiana University, Bloomington, Probs. II., IV. J. KENDRICK UPTON, New London Institute, N. H., Probs. I., II. DAVID TROWBRIDGE, Perry City, N. Y., Probs. I., II., III., IV., V. W. H. SPENCER and C. Y. BALDWIN, Madison University Grammar School, each Prob. I. CHARLES H. ANDREWS, Prob. I. H. C. COREY, Exeter, N. H., Probs. I., II. ASHER B. EVANS, Madison University, Hamilton, N. Y., Probs. III., IV., V. GUSTAVUS FRANKENSTEIN, Springfield, Ohio, Prob. IV. FRANK N. DEVEREUX, Boston, Mass., Prob. II. THE MATHEMATICAL MONTHLY. Vol. II.... JUNE, 1860.... No. IX. PRIZE PROBLEMS FOR STUDENTS. I. THE sum of the radii of the inscribed circle and of the circle touching the hypothenuse and extension of the two legs of a righttriangle is equal to the sum of the said legs. II. The side of a regular octagon, inscribed in a circle whose radius is unity, is √ (2 — √2), and its area is 2√2. Required the proof. III. The perpendicular to the normal of a parabola at its intersection with the principal axis, meets the curve in two points through which tangents are drawn; find the locus of their common point. - Communicated by ARTHUR W. WRIGHT, New Haven, Ct. IV. Required the radii of three equal circles tangent to each other, and cutting off equal areas from a given circle, so that the sum of these areas may be a maximum. — Communicated by G. B. VOSE. V. Two great circles are drawn at random on a sphere. What is the probability that their mutual inclination, taken less than 90°, will be contained between any given limits, as n° and mo. Solutions of these problems must be received by August 1, 1860. REPORT OF THE JUDGES UPON THE SOLUTIONS OF THE PRIZE PROBLEMS IN No. VI., Vol. II. THE first Prize is awarded to GEORGE A. OSBORNE, JR., Lawrence Scientific School, Cambridge, Mass. The second Prize is awarded to DAVID TROWBRIDGE, Perry City, Schuyler Co., N. Y. The third Prize is awarded to Miss AMANDA M. BENNETT, Saline, Washtenaw Co., Mich. PRIZE SOLUTION OF PROBLEM I. By JOHN J. CARTER, Nunda Lit. Institute, N. Y. Find x from the equation, (a + x) + √ (a− x)=b, by quadratics. Raising the given equation to the fifth power and reducing, it becomes 2a +563 (a2 — x2)3 — 5 b (a2 — x2)2 — b5 ; or = PRIZE SOLUTION OF PROBLEM III. By GEORGE A. OSBORNE, JR, Lawrence Scientific School, Cambridge, Mass. Inscribe the maximum rectangle between the conchoid and its directrix. (1) x2 y2 = (a2 — y2) (b + y)2. Since the curve is symmetrical with respect to the axis of Y, the inscribed rectangle will evidently be a maximum when xy or x2 y2 is a maximum. (2) (3) Differentiating (1) gives D ̧ (x2 y2) = 2 (b + y) (a2 — by — 2 y2) D ̧2 (x2 y2) = 2 (a2 — b2 — 6 by — 6 y2). Putting D, (x2 y2) = 0 gives (4) y + b = 0, or a2-by- 2 y2 = 0. It is evident that the value y = - b cannot correspond to a maximum, since the rectangle then vanishes. The roots of (4) are found to be Substituting in (3) the value of y2 from (4) gives D3 (x2 y2) = — 2 (2 a2 + b2 + 3 by). 2 Whatever the relative values of a and b, y is essentially positive, and renders D2(x2 y2) < 0. Hence there will always be a maximum rectangle between the directrix and the superior branch of the conchoid. The negative 2 a2 + b2 b root y1⁄2 will render D,2 (→2 y2) < 0, when 1⁄2 +√√/ (;)2 + 2 · 4 < 3b which condition is found by reduction to be identical with b <ď. Moreover, when ba the numerical value of 2 is less than a and greater than b. It therefore lies between CO and HO. Hence we conclude that when b<a, that is to say, when the inferior branch of the conchoid forms a loop CH around the axis of Y, a maximum rectangle may be inscribed between the directrix and the curve, whose lower base is inscribed in this loop; but when |