Yet, though dome-vaulting, in this ',articular, agrees with common arching, they differ materially in several ether iloints. For, in order to equilibrate the figure of the former. after the convexity has been carried to its full extent of equilibrium around, and equidistant from the summit en the exterior side, the curvature may be changed into a concavity : here the Ulterior circumference of the courses is less than the exterior, and theref,ire, whatever the pressure towards the axis, the course cannot fall inwardly, without squeezing the stones into a less compass. I Tenee a vault may be executed with a con vex surthee inwardly, and a concave surface outwardly, and be sufficiently firm.
The strongest form of a circular vault, required to bear a weight on its top, is that of a truncated cone, similar to the exterior dome of St. Paul's, London, of which it is impossible to conceive any force acting on the summit, that would be capable of disturbing Its equilibrium : for the pressure being communicated in the sloping right line of the sides of the cone, perpendicular to the joints, the conic sides have no tendency to bend to one side more than to another; the gravity of the materials towards the axis, being counteracted by the abutting vertical joints.
In dome-vaulting, the case is very different : for here the contour being convex, there is a certain load, which, if laid On the top, !mist burst it outwardly, which weight becomes greater in proportion as the contour approximates towards chords of the arches of the two sides, or to a conic vaulting on the same base, carried up to the same altitude, and ending in the same circular course. For example, suipose a hori zontal line, tangent at the vertex, proceed from the key-stone downwards, course by course; it will be evident, that every successive oantsing-j,,int may he made to slant as much, and consequently, that the pressure of the arch.stenes or any course towards the axis will be so great, as to be more than adequate to the resistance of the weight of all the super ineumben t parts. I knee it may be clearly deduced, that there is a certain degree of curvature to be given to the contour, which will just prevent the stones in an• succeeding course from beine- treed outwardly.
The circular vault. thus balanced, is indeed an equilibrated dome ; but, instead of the strongest, it is the weakest of all between its ow n contour and that of a cone upon the same base, rising to the same height. in a key-stone. or in an equal circular course. The equilibrated dome has therefore the Wriest contour ; but is the limit, of an infinitude of inscribed circular vaults, all of them stronger than In other respects, eircelar vaulting differs from straight vaulting in being built with Courses in circular rings ; and in having the stones in each course of equal length, which press ing equally towards the axis, cannot slide. inwardly. Circular vaults may theretlwe be open at the top ; and the equilibrated dome, which, as we have just observed, is the weakest of all, may be made to bear a lantern of equal weight with the part that would otherwise have completed the whole. Domes of flatter contours will bear more, in as they approach nearer to that of a cone : and circular vaults. that are either straight or concave on the sides, if chained at the bottom, may be loaded to any degree, without giving way, until the materials of which they are built be crushed to powder.
The foregoing description of the equilibrium and pressure of domes may be comprehended without any acquaintance with either algebra or flexions, and will be of use to the ordinary workmen ; for the satisfaction of more scientific readers, we subjoin Dr. Robison's theory.
Problem.—To determine the thickness of dome-vaulting when the curve is given, or the curve when the thickness is given.
Plate 1. Figure 1.—" Let B It A be the curve which pro duces the dome by revolving round the vertical axis A D. We shall here suppose the curve to be drawn through the middle of the arch-stones, and that the coursing or horizontal joints are everywhere perpendicular to the curve. We shall sup pose (as is always the ease) that the thickness s L, a &e. of the arch-stones is very small in comparison to the dimen sions of the arch. If we outsider any portion it A it of the dome, it is plain that it presses on the curve of which IT L is an arch-stone, in a direction It e, perpendicular to the joint it 1, or in the direction of the next superior element 0 It of the curve. As we proceed downwards, course after course, we
see plainly that this direction must change, because the weight of each course is superadded to that of the portion above it to complete the pressure of the course below. Through draw the vertical line B C a meeting b, produced in c. We may take It c to express the pressure of all that is above it, propagated in this direction to the joint is L. We may also suppose the weight of the course II L united in 1, and acting on the vertical. Let it be represented by It F. If we form the parallelogram It F o c, the diagonal It o will represent the direction and intensity of the whole pressure on the joint We have seen, that if It o, the thrust compounded of the thrust It c exerted by all the courses above n 1 L h, and if the force It F. or the weight of that be everywhere coin cident with It B, the element of the curve, we :Ilan have an equilibrated dome ; if it fall within it, we have a dome which will bear a greater load, and if it fall without it, the dome will break at the joint. We must endeavour to get analytical expressions of these conditions. Therefore draw t he ordinates d b", B D B", c cl c". Let the tangents at It and b" meet the axis in NI, and make at o, at e, each equal to It c, and complete the parallelogram M o N P. and draw o Q perpendicular to the axis, and produce It F. cutting the ordinates in 1.: and e. It is plain that m N is to m o as the weight of the arch n A h to the thrust It c, which it exerts on the joint K a (this thrust being propagated through the course of ti 1 t. N,) and that NI Q, or its equal b e, or d d, may represent the weight of the half A II. `• Let A D be called and ID a he called y. Then It e = and e c = (because It c is in the direction of the (dement (3 b). It is plain if we make Ti constant, B C is the second fluxion of .r, or a c = and It e and It E May be eonsidered as equal, and taken indiscriminatidy for We have also c x, y,; let d be the depth or thickness of It t of the arch-stones. Then :a,: will represent the trapezium it L ; and since the circumference of every course increases in the proportion of the radius y, ti y %%ill represent the whole coe•se. If s be taken to represent the sum or aggregate of the quantities annexed to it, the formula will be analogous to the (Lent of a fluxion, and s d will represent the whole mass, and also the weight of the vaulting down to the joint n Therefore we have this proportion : br=be: co= d y y' d: co=x: c a. Therefore c a= s d y + " If the curvature of the dome be precisely such as puts it in equilibrium, but without any mutual pressure in the vertical joints, this value of c a must be equal to c B, or to the point o coinciding with 13. This condition will be d y expressed by the equation = x or more con s d y ± d y + veniently by = . But this form gives only a tottering equilibrium, independent of the friction of the joints and cohesion of the cement. An equilibrium, accom panied by some firm stability produced by the mutual press ure of the vertical joints, may be expressed by the formula. d + d y + y' i t sdyvx 2, or by sdy? Sr ± x t , where t y is some variable positive quantity which increases when x increases. This last equation will also express the brated dome, if t be a constant quantity, because in this case : is = 0 y + " Since a firm stability requires that sdy + shall be greater than a', and c c must be greater than c B. Hence we learn that figures of too great curvatures, whose sides descend too rapidly, are improper. Also since stability d y V ± requires that we have greater than sdyV + we learn that the upper part of the dome must not be made very heavy. This, by diminishing the proportion of 1' F to c, diminishes the angle c G a, and may set the point G above 13, which will infallibly spring the dome in that place. We see here also, that the algebraic analysis expresses that peculiarity of dome-vaulting, that the weight of the upper part may even be suppressed.
z "The fluent of the equation = — —is most s d y ± easily found. It is Lsdy + = L X where L is the hyperbolic logarithm of the quantity annexed to it. If we consider y as constant and correct the fluent, so as to make it nothing at the vertex, it may be "expressed thus : Lsdy — La = L x—L L t. This gives s d y + fig