different aspect, in order that it may be subservient to the construction of catenarian arches of equilibration, whether for bridges or for powder magazines.

It is very well known with regard to arches of equilibration, when the substance of the structure presses vertically downwards, by the force of gravity, that for a parabolic arch assumed as the intrados, the principles of equiponderance require a similar and equal parabolic curve, situated throughout at the same vertical distance from the inner curve as the extrados; and in that case the curves may be easily constructed and the joints of the voussoirs found, by the methods explained in p. 165 and 167. But it is equally true, that upon the same general principles of equipollence a catenarian curve may be assumed for the arch, and the intrados and extrados be two similar and equidistant curves, provided that equidistance be measured upon the radius of curvature at each point, and be but small compared with the span of the arch. Constructions founded upon the knowledge of this fact have been long rejected, but have of late been re-introduced; on which account Table III. is given, that the time of practical men may not be wasted in needless calculation.

Suppose that the marginal figure in the lower part of p. 164 represented a catenary, in which, as indicated in the table, A P represented any portion of the curve A G, and G P, the corresponding abscissa and ordinate, and G K, the subnormal, p representing the parameter, or constant horizontal tension at the vertex of the curve. Then, in addition to the equations exhibited in p. 175, &c., it is known that A P = √ (A G)2 + 2 p a G.

Now, the table presents values of A G, G P, G K, corresponding to several values of A P, the arch, on the supposition that P, the parameter, or tension at the vertex, is 25.

Suppose, as an illustration of the use of the table, it were required to construct a vault whose semibase should be 8 (yards, for example) and the height 10; we shall find its model by searching that part of the table where the abscissa, A G, and the ordinate, GP, are in the ratio of 10 to 8. This is, where the arc A P is 80; for there the abscissa and ordinate are 58.81 and 46.99, which are in the ratio of 1001 to 8, sufficiently near for practical purposes. Hence, then, the arch may be regarded as divided into 80 equal parts, and the table will present the computed proportions for 80 voussoirs, on each side of the vertex, or rather for 79 voussoirs and half the key stone. Or they may be reduced to 40, or to 20, or to 10, on each side the crown of

the structure. The actual values to the dimensions 10 and 8, will be found by multiplying each number in the table by

8

46.99'

or its equivalent 0.1702; or, to have the dimensions in feet, let the multiplier be 3 x 0.1702 or 0.5106. Thus, we should have the values of A G, G P, A P, for every voussoir from the vertex of the curve downwards, while the corresponding values of A G+ G K would give the points K, from which the line K P must be drawn to give in each case the direction of the joint. In the example assumed, the value of the parameter, p, would be 25 × 0.1702 or 4·255; and this would be the measure of the horizontal thrust.

The curve thus sketched will be posited in the middle of the arch, half way between the intrados and the extrados. Trace above and below this, at the distance (half the thickness at the crown), measured upon the respective positions of the joints, two curves parallel to the mean catenary; so shall you obtain the proposed arch of equilibration.

In cases where the proposed ratio of the height and semi-span of the arch, cannot be found with sufficient accuracy in the tables, it may be approximated to by the usual methods employed in reference to proportional parts. As, suppose the semi-span and the altitude were to be equal; this condition lies between 60 and 61 in the values of the arch; and, by the method just alluded to, we shall find that when the arc is equal to 60-44, the height and half base are each 40'40, the horizontal thrust being 25.

If an equilibrated circular arch were to be erected, of the same span and height, and the same thickness at the crown, the horizontal thrust would be 31.6. Whence, by the way, it appears that an equilibrated catenarian arch in these proportions, would produce a less horizontal thrust than an equilibrated circular arch contrary to the opinion of Bossut, Equilibres des Voutes.

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THE END.