This example is exceedingly useful in groined arches.
2 Let y = o, then will o = a2 + x-, whence x = a which is half of the greater axis of the curve. Suppose y infinite, then will y or y = a x, therefore a : b y : x, that is GF : FE : y ; whence GE is an asymptote to the curve. If a=b, as is generally the case, we shall have = + ; and in this case, if y=o, we shall have (a (a— d); therefore, this particular value of x will be a mean proportional between the sum and difference of the two radii.
Of the various Forms of Joint* for Doors and Shutter*, and of the Alethode of Hinging them together.
The forms of joints for folding and hinging is essential to the beauty of the work. Such joints ought to be so made, as to preserve the uniformity of the door or shutter on both sides ; and to exclude as much air as possible front rushing through between the edges of the two bodies to be hinged, and thereby rendering the apartments cold in winter.
In the joints of doors which are to be hinged together, both angles of one of the bodies arc usually beaded, in order to conceal the open space which would be seen from every point of view ; and to preserve the regularity of the work, the hinges employed to couple them together are made exactly to the size of the bead, on the side on which the knuckle is to be placed ; so that, when they arc hung, the knuckles of the hinges and the wooden bead form one continued staff or cylinder.
Plate CCCXXXIV. Fig. 1. Nos. 1. and 2, is a section of part of the door style and part of the hanging style at the joint. In this, the centre of the bead on each side is in the line of the straight part of the joint from the opposite side. To form the joint exactly, let C be the centre of the bead, AG part of the joint in a line with the edge of the same, join AC, and draw AB perpendicular to AC; the other part BH is perpendicular to EF, the face of the door or hang ing style. This joint will be found sufficient for many pur poses, and may be hung with common hinges, and, being crooked, will break the current of air.
Fig. 2. Nos. 1. and 2. is a plane joint, beaded alike on both sides. Here, as the plane of the joint is a tangent to the cylindrical surfaces of both beads, the margin will be alike on both sides, and, therefore, affords no check in pre venting the cold air from rushing through the aperture or seam into the apartment. Another inconvenience is, that
the hinge which is to connect the two bodies must not be made in the usual manner, but in the form shown in No. 3, where the bodies are hinged together.
Fig. 3. In Nos. 1. and 2. the plane of the joint from the One side is directed to the axis of the bead on the other. This joint is upon the same principle as Fig. I. and there fore may also be hinged with common hinges, as at No. 3. where the two parts are coupled together. These forms of Figs. 1 and 3. are applicable to those cases where a part of the margin is concealed on one side of the door.
Fig. 4. In Nos. 1. and 2, the beads are of the same size on both sides, and are exactly opposite to each other, and the joint is broken by indenting a part which terminates by a plane directed to the axis of the two opposite beads. This form is not only beautiful and strong, but tends to make the apartment comfortable, and the hinges only re• quire to be of the common form. No. 3. shews the two parts hinged together.
Fig. 5. In Nos. 1. and 2. the beads are placed alike on both sides, but not on the same piece as in Fig. 4; and though the uniformity is preserved, the bead which projects the whole of its thickness is weakened.
Fig. 6. Shews the method of hinging shutters and con cealing the hinges. A is the inside bead of the sash frame, B the inside lining, C part of the style of the shutter. To find the form of the joint, let of be the face of the shutter perpendicular to a r, the face of the inside lilting. Bisect the angle by the straight line a a ; then, taking any point c in the centre, draw d d perpendicular to a a, cut ting a a at c; and c is the centre of the hinge. From c de scribe an arc a m, which must be hollowed out of the inside lining of the sash, from the whole height of the shutter ; and the internal right angle must be cut out of the edge of the shutter to the breadth of the hinge, in order to make way for the binge in the act of opening and shutting. Here the toils of the hinges are shown of different lengths, in order to strengthen them.