US L = L and therefore a a; = This last equation will easily give us the depth of the vaulting, or thickness, d, of the arch, when the curve is given. For its fluxion is dy IP t' t x and d t x± a t x which is all expressed in known quantities : ± for we may put in place of t any power or function of x or of y, and thus convert the expression into another, which will still be applicable to all sorts of curves.
t' "Instead, of the second member t we might employ p , where p is some number greater than unity. This will :r evidently give a dome having stability ; because the original 2 formula d r ? be greater than x. This will s d y + p a give d = . Each of these forms has its advantages when applied to particular cases. Each of them also gives a :r d when the curvature is such as in precise Y ? equilibrium : and lastly, if d be constant, that is, if the vault ing be of unifin.m thickness, we obtain the form of the curve, because then the relation of x to x and to y is given.
"The chief use of this analysis is to discover what curves are improper for domes, or what portions of given curves may be employed with safety.
"The chief difficulty in the case of this analysis arises from the necessity of expressing the weight of the incumbent part, or s d y + This requires the measurement of the colloidal surface, which in most eases can be had only by approximation, by means of infinite series. We cannot expect that the generality of practical builders are familiar with this branch of mathematics, and therefore will not engage in it here ; but content ourselves with giving such instances as can be understood by such as have that moderate mathematical knowledge, which every man should possess, who takes the name of engineer.
"The surface of any circular portion of a sphere is very easily had, being equal to the circle inscribed with a radius equal to the arch. This radius is evidently equal to V + " In order to discover what portion of a hemisphere may be employed (for it is evident we cannot employ the whole) when the thickness of the vaulting is uniform, we may recur dy.i.v ±
to the equation or formula = sdy Let a be the radius of the hemisphere. We have 92 and :r =a y Substituting these values in the formula, we obtain the equation = . We easily obtain the fluent of the second ? a2 member = and y=avi+ Therefore, if the radius of the hemisphere be one-half, the breadth of the dome must not exceed 4 I, or 0.786, and the height will be 618. The arch from the vertex is about 51° 49'. much more of the hemisphere cannot stand even, though aided by the cement, and by the friction of the coursing joints. This last circumstance, by giving connection to the upper parts, causes the whole to press more vertically on the course below, and this diminishes the outward thrust ; but at the same time diminishes the mutual abutment of the vertical joints, which is a great cause of firmness in the vaulting. A Gothic dome, of which the upper part is a portion of a sphere not exceeding 45° from the vertex, and the lower part is concave outwards, will be very strong, and not ungraceffd.
"Persuaded that what has been said on this subject con vinces the reader that a vaulting, perfectly eqllifibrated throughout is by no means the best form, provided that the base is secure from separating, we think it unnecessary to give the investigation of that form, which has considerable intricacy, and shall merely give its dimensions. The thick ness is supposed uniform. The numbers in the first column of the table express the portion of the axis counted from the vertex, and those of the second are the length of the ordinates.
"The curve formed according to these dimensions will not appeal' very graceful, because there is an abrupt change in its curvature at a small distance from the vertex. 11, how ever, the middle be occupied by a lantern of equal, or of smaller weight than the part whose place it supplies, the whole NVIII be elegant and free from defect.