As an example of the application of the foregoing method to the purposes of calculation, let the case of the Britannia Bridge be taken, and let the large span be supposed to be divided into five, and the small span into three equal parts, and let the moments of inertia of the sections and loads per unit of length be supposed constant within each part and equal to their mean values. 1857E' 1664E 1746E' 1520E' -0.002035. H1=3.65, μ2=3·49, μ=3.57, μ1=3.31, and, from symmetry of loading, T= t = − 2 Applying equations (30) and (32) to spans (1.0) and (1.2) respectively and eliminating T and T' by adding them, we obtain 0.1888,+004827,-10481=0; and applying equation 32 to span (21), whence 0.048270, +0.087654,- 5420=0, 1=46206, 2=36387. Taking these values of o, and 2, and applying equation (33) to the calculation of the deflection at the middle of the large span, Y=0.375 ft.=4.5 inches. If, now, the values of 1, 2, and Y be calculated from equations (26) and (19), on the supposition that the moments of inertia of the section and the loads are constant throughout each span and equal to their mean values, they are which are almost identical with the values ascertained by Mr. Pole. If the variation of section alone be considered, the load being taken at its mean value, =46382, 4, 34465, Y=4.52. It therefore appears that the amount of variation in the section and load which occurs in each span of the Britannia Bridge, when taken strictly into account, produces scarcely any effect on the values of the bending-moments and deflections, which are practically the same as those resulting from their mean values considered as constant; and it may be considered demonstrated that, for most ordinary cases of large bridges, calculations founded on equation (26) may be confidently relied on. It need scarcely be remarked that these are much more simple and easy than those founded on the more exact but complex equations above given. In smaller bridges, however, the error of the approximate process will be more considerable, and the process above given may be applied with advantage to its correction. In concluding this paper, the author desires to record his thanks to his young friend, Mr. Henry Reilly, for the patience and skill with which he made, in detail, all the intricate calculations of the numerical values of the various functions involved in the above demonstration. "Remarks on Mr. Heppel's Theory of Continuous Beams." By W. J. Macquorn Rankine, C.E., LL.D., F.R.S. 1. Condensed form of stating the Theory.-The advantages possessed by Mr. Heppel's method of treating the mathematical problem of the state of stress in a continuous beam will probably cause it to be used both in practice and in scientific study. The manner in which the theory is set forth in Mr. Heppel's paper is remarkably clear and satisfactory, especially as the several steps of the algebraical investigation correspond closely with the steps of the arithmetical calculations which will have to be performed in applying the method to practice. Still it appears to me that, for the scientific study of the principles of the method, and for the instruction of students in engineering science, it may be desirable to have those principles expressed in a condensed form; and with that view I have drawn up the following statement of them, which is virtually not a new investigation, but Mr. Heppel's investigation abridged. Let (x=0, y=0) and (a=l, y=0) be the coordinates of two adjacent points of support of a continuous beam, a being horizontal. Let y and the vertical forces be positive downwards. At a given point x in the span between those points let μ be the load per unit of span, and El the stiffness of the cross section, each of which functions may be uniform or variable, continuous or discontinuous. In each of the following double and quadruple definite integrals, Phil. Mag. S. 4. Vol. 40. No. 269. Dec. 1870. 2 H When the integrations extend over the whole span 7, that will be denoted by suffixing 1; for example, m, n,, &c. Let -P be the upward shearing-force exerted close to the point of support (x=0), 4, the bending-moment, and T the tangent of the inclination, positive downwards, at the same point. Then, by the general theory of deflection, we have, at any point of the span l, the following equations: moment, deflection, Let be the moment at the further end of the span 7, and suppose it given. This gives the following values for the shearing-force P and slope T at the point (x=0):— Consider, now, an adjacent span extending from the point of support (x=0) to a distance (-a) in the opposite direction, and let the definite integrals expressed by the formula (1), with their lower limits still at the same point (r=0), be taken for this new span, being distinguished by the suffix -1 instead of 1. Let -T be the slope at the point of support (x=0) Then we have for the value of that slope, $_19-1, m-19-1 F-1 で Add together the equations (5) and (5A), and let t=T-T' denote the tangent of the small angle made by the neutral layers of the two spans with each other in order to give imperfect continuity. Then, after clearing fractions, we have the following equation, which expresses the theorem of the three moments in Mr. Heppel's theory: O=Þ ̧(9,712+q_ ̧12 — n ̧ ll12 — n _ ‚l'l2) —§‚¶‚ ̧112 — Þ_1I~ıl2 -1 1 1 } (6) That equation gives a linear relation between the bending moments _,,, at any three consecutive points of support, and o certain known functions of known quantities. In a continuous girder of N spans there are N-1 such equations and N-1 unknown moments; for the moments at the end of most supports are each =0. The moments at the intermediate points of support are to be found by elimination; which having been done, the remaining quantities required may be computed for any particular span as follows:The inclination T at a point of support by equation (5); the shearing-force P at the same point by equation (4); the deflection y and moment at any point in that span by equations (3) and (2). Points of maximum and minimum bending-moment are of course do found by making =0; and points of inflection by making =0. 2. Case of a uniform girder with an indefinite number of equal spans, uniformly loaded; loads alternately light and heavy. The supposition just described forms the basis of the formulæ given in a treatise called 'A Manual of Civil Engineering,' page 288; and it therefore seems to me desirable to test those formulæ by means of Mr. Heppel's method. The cross section of the whole girder and the load on a given span being uniform, the definite integrals of the formulæ (1) take the following values :— The values of those integrals for the complete span are expressed by making x=1. The values of n and q are the same for every span. In the values of m and F, the load μ per unit of span has a greater and a less value alternately. Let wo be the weight per unit of span of the girder with its fixed load, w, that of the travelling load (increased, if necessary, to allow for the additional straining effect of motion); then the alternate values of μ are The moments at the points of support are all equal; that is, $=01=0_1. -1° Equation (6) now becomes the following (the common factor ľ2 having been cancelled) : giving for the bending-moment at each point of support If t be made =0, so that the continuity is perfect, this equation .exactly agrees with the formula at page 289 of the treatise just referred to; and the same is the case with the following formulæ for the shearing-forces and slopes close to a point of support, and for the moments and deflections at other points : x= Central moment, light Central moment, heavy load, Central deflection, light load, Φ y=Tx-Pq+n+F(with x= F( Central deflection, heavy load, 24 == y=-T'x-P'q+on+F(with x=— ·F ( L 8 384EI LVIII. Intelligence and Miscellaneous Articles. LECLANCHE'S MANGANESE ELEMENTS. BY J. MÜLLER. ECLANCHE'S voltaic elements have recently been extensively recommended, although no statements as to their constants have been published. I have thus been led to make a few experiments with them. The arrangement of the elements is as follows:-A plate, or rather a rod of gas-coke is placed in a porous clay cylinder, and the rest of the space filled with a mixture of equal volumes of manganese (pyrolusite, Braunite) and of gas-coke in pieces the size of a pea. The clay cylinder thus prepared is placed in a wide glass vessel filled with solution of sal-ammoniac, in which is placed an amalgamated zinc rod. To determine the constants of this combination I used Ohm's method. Three of Leclanché's cells, connected so as to form one pair of plates, produced a deflection of 13° on a tangent-compass the reduction-factor of which was 74 when this was connected with the rheometer by only short thick copper wires. This deflection was diminished to 5o•1 by the insertion of a Siemens's unit. Hence for the electromotive force of a Leclanché's element we get the value e=10.76, and for the essential resistance of one cell r=1.89, if, according to Waltenhofen's proposal, we assume as unit that current which furnishes one cubic centimetre of explosive gas in a minute, and as unit of resistance a Siemens's unit (the resistance of |