This periodical was evidently got up as a rival of The Mathematical Diary. His opponents were numerous, and the contest was carried on with some bitterness, till finally Mr. Williams issued his 14 famous "Challenge Problems," directed against all the mathematicians in America, excepting only Dr. Bowditch, Prof. Strong and Eugene Nulty. Six of these problems are impossible. Some of the others are somewhat difficult, but have all been solved by several persons. Besides the ANALYST, for which this paper is a contribution, and which is devoted exclusively to mathematics, there are many periodicals at present in America which appropriate a portion of their space to mathematics. "The Yates Co. Chronicle," a newspaper published weekly at Penn Yan, Yates Co. N. Y., by S. C. and E. Cleveland, has a mathematical department edited by Dr. Samuel H. Wright, containing problems and solutions in nearly every branch of mathematical science and is undoubtedly the best of its kind in the country. "The Railroad Gazette," published weekly at N. Y. and Chicago, has mathematical problems relating to the construction of Railroads, Engines, Cars, &c. There are also several monthly Educational periodicals which have a department devoted to mathematics. "The Schoolday Magazine,” published at Philadelphia has a mathematical department which is ably edited by Artemas Martin Esq., of Erie, Pa. It has done much good to the class of advanced students for whom it was intended. "The Normal Monthly," edited by Prof. Edward Broaks at Millersville, Lancaster Co. Pa. has a mathematial department in which Artemas Martin has a series of articles on the Diophantine Analysis, which he well illustrates and in a style adapted to general comprehension. "Educational Notes and Queries," edited by Hon. W. D. Henkle at Salem, Ohio, has also a department for mathematical Notes and Queries which bids fair to be interesting and useful. There are also the "Illinois Schoolmaster," and Morton's Monthly, published at Chicago and Louisville respectively, which contain many excellent solutions of mathematical problems. The above are all the serials, having a mathematical department which have come under the notice of the writer. NOTE ON THE REACTIONS OF CONTINUOUS BEAMS. BY MANSFIELD MERRIMAN, C. E., NEW HAVEN, CONN. As a matter of purely mathematical interest I wish to give here, without demonstration, the relations between the reactions of continuous girders of equal spans resting on level supports. Let a girder of 8 equal spans be subjected to any number of concentrated loads, and the distance of any load P from the nearest left hand support be denoted by kl, 7 being the length of a span; let P1, P2, Pr, etc., denote concentrated loads on the 1st, 2nd, rth, etc., spans, and R1, R2, R„, etc., the reactions at the points of support. Then the relation between the reactions is given by the equations, 6R1 + R2 = P1(6—6k+k3) + ΣP2( 1 — k)3 1 19 4R2+ R ΣP1(6k — 2k3) + ΣP,(4—6k2 + 3k3) + ΣP3(1 - k)3 R2+4R+R1 = ΣP1k3+ΣP2(1+3k+3k2-k3)+ΣP3(4—6k2+3k3) ΣΡ 2 +ΣP1(1 — k)3 ΣP,-2k+ Σ'P,-1(1 + 3k + 3k2 ——— k3) + ΣP,(4 — 6k2 + 3k3) + ΣPr+1(1 — k)3 * * R ̧,+4R, = [P ̧-2k3 + ΣP,-1(1 + 3k + 3k2 — 3k3)+ΣP ̧(4—6k2+2k3) R.+6R+1=XP,_,k+ΣP, (1+3k+ 3k2 - k3) 8-1 If the spans be uniformly loaded throughout with loads w1, w2, w,, etc., per unit of length, these equations become, and for a uniform load of w per linear unit over the whole beam, they are From the time of Navier to the discovery by Clapeyrou of the Theorem of Three Moments, the method of investigation of continuous girders consisted in first determining, by long and tedious equations, the reactions at the supports, and then from there deducing the shears and moments for any required section. This was undoubtedly the most logical method—to find all the exterior forces under which the beam was held in equilibrium and then pass to the internal strains. But so long and difficult was the labor of finding the reactions that the theory was but slightly advanced until the happy discovery of Clapeyron of the relation existing between the moments at the points of support. This Theorem with its later extension to concentrated loads and variable moments of inertia serves as a starting point from which the whole theory is easily deduced and put in to shape for practical Those familiar with the history of this subject before the time of Clapeyron's discovery will at once recognize the simplicity of the above equations compared with the complicated method of Navier. NOTE ON THE DIVISION OF SPACE. BY PROF. HALL. This question occurred to me several years ago in reading an account of Bertrand's method of treating the doctrine of parallel lines. I have not seen Prof. Cayley's solution, but of course so obvious a question could not be new. Lately I have found that this question and several kindred ones are very completely discussed by Steiner in the first Vol. of Crelle's Journal, published in 1826. For the number of parts into which n planes can divide space Steiner finds an expression which is eqnivalent to (n3+5n+6); and he shows also that of these parts CORRECTION OF AN ERROR IN BARLOW'S THEORY BY ARTEMAS MARTIN, ERIE, PA. In Barlow's Theory of Numbers, page 299, it is stated that "the equation 5658y2 = 1 x=166100725257977318398207998462201324702014613503, y= I will show that these values are not correct, and then compute the true numbers. The units figure of the square of Barlow's value of x is 9; the units figure of the square of his value of y is 4; the units figure of 5658y2 is 2, and 9 -2=7. If his values were correct the units figure of 22 would be 1 greater *For a full presentation of ready methods for finding moments and reactions the reader may consult the Journal of the Franklin Institute, March and April, 1875. than the units figure of 5658y2 unless the units figure of a2 was 0, in which case the units figure of 5658y2 would be 9. Let A = 5658, then √(5658) = √ A = r + 1 where r is the greatest integer contained in √A. The last quotient of every complete period is 2r. of quotients in a complete period, and pm9m the Let m be the number last convergent in the first period; then, when m is even, x = Pm, y=qm, and when m is odd x = P2m) Y = 12m. Let √(A) + an bn Pm Y =um + &c. and √(4) + an+2 two consecutive complete quotients, then ba+1 = Un+1 + &c. be any A-an+1 = b If Pn÷In Pn+19+1 be any two consecutive convergents and Un+1 the quotient corresponding to Pa+19+1, then P2+1 PR+2 Un+1Pn+1 + Pn 9n+2 Un+19n+1 + In r + 75 =r, 1 r+ 57 33 = 4 + = ~1, 73 = 1 + = U2, 74 = 1 + =U8; 31 Ид = 3 + 42 = U sr As 18, the number of quotients in a complete period, is even, therefore x = p18 and y = 918° = = 677 = 121555 P10 41 96 762201 P11 = 437 98 99 1616 10133' 911 910 NOTE ON THE SOLUTION OF MR. HOLBROOK'S QUESTION. BY THE EDITOR. Prof. Eddy (see p. 126) has given the equation of a solid whose surface is generated as described in the question proposed by Mr. Holbrook on page 72; but it appears from a subsequent clause that Mr. Holbrook had a different question in view, viz.; he asks for a "demonstration that no two ellipses can be parallel.” The surface of a solid with an elliptical base, horizontal sections of which shall be bounded by curves parallel to the periphery of the base, may be generated by a straight line which makes a constant angle with the normal to the ellipse while the extremity of the line describes the periphery of the ellipse. That no horizontal section, above the base of such a solid, can be an ellipse, is what we understand Mr. Holbrook to assert and desire to have demonstrated. Horizontal sections of the solid of which Prof. Eddy has given the eqnation, are obviously ellipses; but that no section, above the base, of the solid whose surface is generated as above described, can be an ellipse, may be demonstrated as follows: Let ABP (see diagram on next page) represent an ellipse whose semi-axes are AC a and BC b, and the normal of which, at any point P, is PO = N. Let A'B'P' be a parallel curve within the ellipse; then will the portion PP' of the normal, be of the same length for all points of the ellipse. We may therefore put PP' =c, a constant. |