HIP-1100F, a roof whose ends rise immediately from the wall-plate, with the same inclination to the horizon as the other two sides of the roof have.

The hacking of a hip is the angle made on its upper edge, to range with the two sides or planes of the roof' between which it is placed.

Jack-rafters are those short rafters fixed to hips equidis tantly disposed in the planes of the sides and ends of the roof, and parallel to the common rafters, to fill up the triangular spaces, each of which is contained by a hip-rafter, the adjoin ing common rafter, and the wall-plate, between them. The seat or base of the rafter is its ichnographic projection on the plane of the wall-head, or on any other horizontal plane.

The principal angles concerned in hip-roofing are, the angle which a common rafter makes with its seat on the plane of the wall-head ; the vertical angle of the roof; the angle which a hip makes with the adjoining common rafter ; the angles which a hip makes with the wall-plate on both sides of it ; the angle which a hip-rafter makes with its seat ; and the acute angle which a hip-rafter makes with a vertical line. The principal lengths concerned are, the height of the roof; the length of the common rafters and their seats; the length of the hips and their seats ; and. lastly; the length of the wall plate contained between the lower end of a hip and the lower end of the adjacent common rafter.

The sides and angles may be found by geometrical con struction or trigonometrical calculation. It is evident, that it' the hipped end of a roof he cut off by a vertical plane parallel to the wall, through the upper extremity of the hips, it will form a rectangular pyramid, or one whose base is a rectangle. The base of this pyramid is bounded by the wall plate between the two hips on one side, and on the opposite side by the seat of the two adjoining common ratters ; on the other two opposite sides, by that part of the wall-plate on each side contained by the lower end of the hip and the next com mon rafter adjoining. One of the sides is the isosceles triangle

contained by the two adjoining common rafters with their seat ; the opposite side is the hipped end of the roof, forming also an isosceles triangle; the other two opposite sides are the right-angled triangles contained by the two hips and the two rafters on the side of the roof. This rectangular pyramid may be divided into three pyramids by the two vertical triangular planes, formed by the hip-rafters, their seats. and the common perpendicular from their vertex.

Two of these pyramids, when the plan of the building is a rectangle, are equal and opposite. In each of these equal and opposite pyramids the base is a right-angled triangle, con tained by the seat of the hip-rafter, the seat of the adjoining common rafter, and the part of the wall-plate between the hip and the adjoining common rafter. One of the sides is a right-angled triangle contained by the adjoining common rafter. its seat. and perpendicular: a second side is a right angled triangle contained by the common rafter, the hip rafters, and the wall-plate, between them ; and the remaining third side is the triangle contained by the hip-rafter, its seat, and perpendicular. With regard to the remaining pyramid, its base is a right-angled triangle contained by the seats of the two hips and the wall-plate between them, the right angle being that contained by the seats of the two hips ; two of its sides are the triangular planes passing the hip-rafter, which are also common to the other two pyramids ; its third side is the hipped end of the roof.

Given the plan of a building, or the form of a wall-plate of a hip-ro,f, and the pitch of the roof to find the various lengths and angles concerned, whether the roof is square or bevel.

EXAMPLE 1.—To find the length of the rafters, the hacking Example 1.—To find the length of the rafters, the hacking of the hips, and the shoulders of jack-rafters and purlins, geometrically.