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so wo 2|cos a sin a sin a cos 3 Whence, by Lagrange's Theorem, neglecting powers of a above the first, we have, by equation (60),
Equations (62), (63), (65) determine the values of P and Y, that is, the pressure upon the key-stone and the positions of the points of rupture, for every condition of loading, and every form of the gothic and circular arch.
Assuming 6 = 0 and eliminating the value of tan;, we have, by
equations (62), (63), and (65),
It has been supposed that the load X is collected over a single point in the arch, or rather in a single line stretching across it in a direction parallel to the axis of the cylinder of which it forms part. Let it now be imagined to be distributed in any way over the extrados, but symmetrically on the two opposite semi-arches. Find the center of gravity of the load on either semi-arch, and let a be its distance from the vertical which passes through the center of the circle of which the semi-arch forms part. Imagine the whole load X to be collected in this point, and on this hypothesis determine Y and P; the values thus determined will evidently be their true values. To find the line of resistance, substitute in equation (46) the value of P, and for X and a substitute their values in terms of 6; that is, for X substitute the load incumbent upon that portion of the arch which subtends the angle 6, and for a the distance from the vertical through the center, of the center of gravity of that portion of the load. The resulting equation will be the true equation to the line of resistance. Thus the point where the resultant pressure of the arch intersects the supporting surface of the abutment will become known; and its direction being found, as in equation (57), all the circumstances which determine the equilibrium of the abutment will be known, and the conditions of its equilibrium may be determined by the equation given in section 5 of this paper. The analytical discussion of these conditions, and of that case in which the arch being overloaded at the haunches, its rupture takes place by the elevation of the crown, is yet wanting to complete the theory of the arch.
A very simple expression for P offers itself in the case in which a = 0, or in which the thickness of the arch is considered evanescent in comparison with its radius. In this case
When e = 0 we obtain the segmental arch, and P then equals
If the weight of the arch itself be imagined to be included in that of its loading, that is, in X, and if a be determined on the same hypothesis; if, moreover, R be substituted for r, this expression for P will give in every case a useful approximation to its true value. It is a limit which the pressure on the key can never exceed, and to which it approximates more nearly as the radius of the arch is greater in comparison to its thickness. It possesses, moreover, this advantage to the practical man, that it admits of an easy geometrical construction.
13. Let us suppose that the arch were supported at its springing on the edge of its joint at the extrados (see Fig. 19). Instead of assuming p, in equation (46), equal to r, we must now assume it equal to R at the springing, since the line of resistance will manifestly pass through the point of support. By this supposition we obtain, taking p = R cose,
...(69). In the case in which €3 = 0 and X = 0,
• The author has verified this formula, and a corresponding formula, for the case in which the arch is supported at its springing on the inferior edges of its extreme voussoirs, by experiments of which the results were communicated to the Mechanical Section of the British Association of Science, at their Meeting in 1837.
Vol. VI. PART III. 3 R
XXV. Account of Observations of Halley's Comet. By R. W. RothMAN, Esq., M.A. Fellow of Trinity College.
[Read December 11, 1837.]
THE accompanying observations of Halley's Comet were made on the great tower of Trinity College, with a 30-inch achromatic telescope, of 2% inches aperture, to which was adapted a ring-micrometer. The power used with this micrometer was about 25: the radius of the inner ring was found by the transits of stars near the meridian = 1258": of the outer = 1710".
The observations here detailed form the whole of those that I was able to make on this Comet: I saw it on the 18th of Sept. and 1st of Nov., but was unable to get an observation. Whenever an observation has been rejected, the circumstance has been noted in its proper place and the reasons for doing so assigned.
The time employed is Greenwich mean solar time. The seconds watch I used was compared every evening with the clock by Molyneux in the Reading Room, and that again the next day with the transitclock at the Observatory.