Tidal Researches. By William Ferrel, A. M. (U. S. Coast Survey Report, Appendix.) Washington, 1874. 4to. 268 pp. Elements of the Differential Calculus, founded on the Method of Rates or Fluxions. Part First. By J. Minot Rice and W. Woolsey Johnson. J. Wiley. New York. 1874. 8vo. 66pp. A Treatise on Electricity and Magnetism, by James Clark Maxwell. Two Vols. Macmillan. New York. 1873. 8vo. 425, 444 pp. $10.00. A History of the Mathematical Theories of Attraction and the Figure of the Earth, from the time of Newton to that of Laplace. By I. Todhunter. Two Vols. Macmillan. New York. 1873. 8vo. 476, 508 pp. $12.00. Researches in the Calculus of Variations, principally on the Theory of Discontinuous Solutions. By I. Todhunter. Macmillen. New York. 8vo. $3.50. An Elementary Treatise on Curve Tracing. By Percival Frost. Macmillan. New York. 8vo. $6.00. A Treatise on the Theory of Friction. By John H. Jellet. Macmillan. New York. 8vo. $4.00. Introduction to Quaternions, with numerous Examples. By P. Kelland and P. G. Tait. Macmillan. New York. 8vo. $3.00. Papers on Electrostatics and Magnetism. By Sir Wm. Thomson. Macmillan. New York. 8vo. $9.00. B. Westermann & Co., 524 Broadway, New York, are the agents for the sale of the Smithsonian publications, of whom each paper can be got separately if desired. NOTES.-Dr. David S. Hart, Stonington, Conn., writes--"I think the alteration from a monthly to a bi-monthly a good measure. The first periodical of this kind, published in this country, was the "Mathematical Correspondent" edited by George Baron and issued annually; the second was the Analyst edited by Dr. Robert Adrain, also published annually; the third was "Nash's Diary," edited by Michael Nash, also issued annually; the fourth was the "Mathematical Diary," edited at first by Dr. Adrain, and afterward by James Ryan, issued semi-annually; the fifth was the "Math'l Miscellany," edited by. Prof. Charles Gill, also issued semi-annually; the sixth was the “Mathematical Monthly," edited by J. D. Runkle, issued monthly. Besides these there were issued two or three ephemeral serials, one of them by John D. Williams, who published 14 Challenge Problems, directed against Samuel Ward, (Editor of an American edition of "Young's Algebra), James Ryan, Patrick Lee, and others. Six of these problems are impossible. Besides the above named periodicals, there are now issued quite a number of Educational periodicals, each having a Math'l Department, and, last, but not least, the "Yates County Chronicle," a weekly paper, having a Mathematical Department under the direction of Dr. Samuel H. Wright." Prof. N. R. Leonard, of Iowa State University at Iowa City, writes—“A very large and brilliant fire-ball passed this place at 10-30 yesterday evening (Feb. 12th)-its size apparently half that of the moon-its course slightly north of west, and marked by a brilliant and broad train of light and by three separate explosions, and followed at an interval of three minutes by a report that some compared to a fusilade of musketry, others to the rumbling of a train of cars. SOLUTIONS OF PROBLEMS IN NUMBER 1, VOL. II. Solutions of problems in No. 1 have been received as follows: From J. M. Arnold, 52; R. W. Ryan, 52; Prof. A. B. Evans, 54, 55 & 58; Edgar Frisby, 56; E. S. Farrow, 51, 52, 53, 54, 55, 56, & 57; J. M. Greenwood, 54 & 56; Henry Gunder, 52, 56 & 58; G. W. Hill, 55 & 56; Prof. E. W. Hyde, 58; H. Heaton, 52, 53, 56, 57 & 58; Phil. Hoglan, 52; Prof. W. W. Johnson, 58; Artemas Martin, 52 & 56; O. D. Oathout 52 & 56; E. B. Seitz, 52, 54, 56, 57 & 58; Walter Siverly, 52, 54 & 56 Werner Stille, 54 & 56. The following credits, due, were omitted, by an oversight, in the January No. Prof. J. Scheffer, 41, 42, 43, 44, 47 & 49; E. B. Seitz, 41, 43, 44, 45 & 48; L. Regan, 41, 42, 43 & 45; August Zielinski, 43 & 45. Also, Th. L. De Land, in addition to the solutions for which he was credited in the Jan. No., sent a correct and very elegant solution of 48. It must not be inferred that, when solutions are not selected for publication, they are thought unworthy. On the contrary, many solutions that are not published are among the very best that we receive, and are passed, in making our selections, because our space will not permit us to introduce the details which they contain. For instance, we have on hand very elegant solutions of No. 48, by Prof. Hyde, Prof. Evans and Prof. Johnson, but, at this time, our space will not permit us to publish them. Although very brief, it is believed that the published solution of 58, and also that of 54, will be found sufficient to guide the student in making out a solution in detail if he desires it. 51.-"Given (x2 + y2)y = 39, .... (1) x + y = 97, to find x and y by quadratics." SOLUTION BY C. Put y = ах. EDWARD S. FARROW, WEST POINT, N. Y. Substating this value in (1) and (2), multiplying (1) by x and (2) by a, d adding, we shall have Applying Mr. Hill's method for can easily find from (3) that x = stituting this in (2), we have y responding value of x is 2. 52. "A hare 45° with the merid. (α5 a3)x + 39x = 97a. the solution of biquadratic equations, we a; ... y = ax = x2, or x2=y. Sub+ 16y=97: whence y = 3. The cor rods north of a hound and runs at an angle of he hound pursues in a straight line and in such a direction that h vill intercept the hare without changing his course. Supposing the hound to run n times as fast as the hare, how far will he run before he catches the hare?" SOLUTION BY O. D. OATHOUT, READ, IOWA. Put 10 rods = a, and 45° = 0. Let x = the distance ran by the hare, then nx = distance ran by the hound. whence 2(n2 — 1)x 2a cos 0 = ±√[4a2cos20 + 4a2(n2 — 1)]. x= a cos 0 ± √(a2cos20 + a3n2 — a2) n2 - 1 Substituting the values of a and 0 and reducing, we get x= 5√/2.[1 ± √(2n2 — 1)] •. nx = 5n√/2.[1 ± √(2n2 — 1)] 53. If a ball of 6 inches diameter is discharged from a canon at the rate of one mile in 7 seconds, how much greater force will be required to throw a ball of double the weight with the same velocity, taking into account the resistance of the air and the diameters of the balls?" SOLUTION BY EDWARD S. FARROW. If we suppose the balls of the same material, or density constant, the diameter of the 2nd ball will be 62. The actual resistance of the atmosphere at any time, is composed of two terms, the one due to the inertia of the displaced particles, the other to the difference of the atmospheric pressure, in front and rear. Both terms must be variable and a function of the velocity. From a series of careful experiments, Colonel Robert has constructed this formula for spherical projectiles, viz: Resistance of air = r = ATR2(1+)2, in which A is the resistance, in pounds, on a square foot of a projectile moving with a velocity of 1 foot per second; a is linear and equal to 1427 ft.; A=.000514; R2 area of the cross section of the ball. For the ball whose diameter is 6 inches 1 16X (5280) * (1 + 5280 =87 lbs. nearly. r = .000514 X 3.1416 X For the ball whose diameter is 62 in. r=138 lbs. nearly. Consequently there is 13887 51 lbs. more resistance to be overcome. The force necessary to project a body with a given velocity, varies with the density of the body; for, the living force of a body divided by 2 is equal to the inertia of the body overcome in acquiring its velocity work of the motive forcemv2. mv2. (m=v x D). When D is given, the additional force can easily be calculated. Call it P; then P+ the force necessary to overcome a resistance of 51 lbs. shows how much greater force will be required. 54.-"Let ABC represent a spherical triangle and M the centre of the sphere. Find the three distances of the centre of gravity of the spherical triangle ABC from the three planes ABM, ACM and BDM." SOLUTION BY PROF. J. M. GREENWOOD, KANSAS CITY, MO. In the diagram M is the center of the sphere; MB, MA, MC, are the three radii; ABC represents the spherical triangle; AMB, AMC, BMC, are the three planes, and a, b, c, the sides opposite the angles. Put the distances of the three planes from the center of gravity, equal d, di, d2; and let r denote the radius of the sphere, s the area of the spherical triangle, and 8, the area of its projection on BMC, then d= 180 A+B+C Tr2 area AOB.cos B-area AOC.cos C 360 (a -c cos B-b cos c). ra b cos C -- c cos B 2 A+B+C-180 In the same way d1 and d2 may be found. 55.-"A very small bar of matter is movable about one extremity which is fixed half way between two centers of force attracting inversely as the square of the distance; if I be the length of the bar, and 2a the distance between the centers of force, prove that there will be two positions of equilibrium for the bar, or four, according as the ratio of the absolute intensity of the more powerful force to that of the less powerful is or is not greater than (a +21) ÷ (a-21): and distinguish between the stable and unstable positions." SOLUTION BY G. W. HILL. Assume the fixed extremity of the bar as the origin of coordinates and the direction of the line joining the two centers of force as that of the axis of x. Then x and y being the coordinates of a material point dm of the bar, and X and Y the forces acting on it, we have from the well known equations for the motion of a rigid body If M and M' denote the intensity of the force at the unit of distance, we have Y: ́ [(a — x)2 + y2]} ̄ ̄ ̄ [(a + x)2 + y2]} · Introduce now polar coordinates, and put x = r cos 0, y = r sin 0, and since the mass of the bar may be supposed evenly distributed along its length put dm = dr, and take the integration with respect to r between the |