Managed in this way, the butt-joint is in a plane pierced perpendicularly by the tangent. But if the block be not sawed square through, but vertically from the edges of the face-mould, then, especially, care is required in locating the joint. The method of sawing square through is attended with so many advantages that it is now generally followed; yet, as it is possible that for certain reasons some may prefer, in some cases, to saw vertically, it is proper that the method of finding the position of the joint for that purpose should be given. Therefore, let A, Fig. 169, be the plan of the rail, and B the elevation, showing its side; in which kz is the direction of the butt-joint. From k draw kb parallel to lo, and ke at right angles to kb; from b draw b f, tending to the centre of the plan, and from f draw fe parallel to bk; from l, through e, draw li, and from i draw id parallel to ef; join d and b, and db will be the proper direction for the joint on the plan. The direction of the joint on the other side, a c, can be found by transferring the distances x b and od to xa and oc. Then the allowance for over wood to cover the butt-joint is shown as that which is included between the lines ox and db. The face-mould must be so drawn as to cover the plan to the line b d for the wreath at the left, and to the line ac for that at the right. Bv some the direction of the joint is made to radiate toward the centre of the cylinder; indeed, even Mr. Nicholson, in his Carpenter s Guide, so advised. That this is an error may be shown as follows: In Fig. 170, arji is the plan of a part of the rail about the joint, s u is the stretch-out of a i, and gp is the helinet, or vertical projection of the plan arji. This is found by drawing a horizontal line from the height set upon each perpendicular standing upon the stretch-out line su. The lines upon the plan arji are drawn radiating to the centre of the cylinder, and therefore correspond to the horizontal lines of the helinet drawn upon its upper and under surfaces.
Bisect rt on the ordinate drawn from the centre of the plan, and through the middle draw cb at right angles to gv; from b and c draw cd and be at right angles to su; from d and e draw lines radiating toward the centre of the plan; then do and em will be the direction of the joint on the plan, according to Nicholson, and cb its direction on the falling-mould. It must be admitted that all the lines on the upper or the lower side of the rail which radiate toward the centre of the cylinder, as do, em, or ij, are level; for instance, the level line wv, on the top of the rail in the helinet, is a true representation of the radiating line j i on the plan. The line b h, therefore, on the top of the rail in the helinet, is a true representation of e m on the plan, and kc on the bottom of the rail truly represents do. From k draw kl parallel to cb, and from h draw hf parallel to be; join l and b, also c and f; then cklb will be a true representation of the end of the lower piece, B, and cfh b of the end of the upper piece, A; and fk or h l will show how much the joint is open on the inner, or concave, side of the rail.
To show that the process followed in Art. 287 is correct, let do and em (Fig. 171) be the direction of the butt-joint found as at Fig. 169. Now, to project, on the top of the rail in the helinet, a line that does not radiate toward the centre of the cylinder, as jk, draw vertical lines from j and k to w and h, and join w and h; then it will be evident that wh is a true representation in the helinet of jk on the plan, it being in the same plane as jk, and also in the same winding surface as wv. The line ln, also, is a true representation on the bottom of the helinet of the line j k in the plan. The line of the joint e m, therefore, is projected in the same way, and truly, by ib on the top of the helinet, and the line do by ca on the bottom. Join a and i, and then it will be seen that the lines ca, a i, and ib exactly coincide with cb, the line of the joint on the convex side of the rail; thus proving the lower end of the upper piece, A, and the upper end of the lower piece, B, to be in one and the same plane, and that the direction of the joint on the plan is the true one. By reference to Fig. 169 it will be seen that the line lt corresponds to xi in Fig. 171; and that e k in that figure is a representation of fb, and ik of db.
288. - Scrolls for Hand-Rails: General Rule for Size and Position of the Regulating Square. - The breadth which the scroll is to occupy, the number of its revolutions, and the relative size of the regulating square to the eye of the scroll being given, multiply the number of revolutions by 4, and to the product add the number of times a side of the square is contained in the diameter of the eye, and the sum will be the number of equal parts into which the breadth is to be divided. Make a side of the regulating square equal to one of these parts. To the breadth of the scroll add one of the parts thus found, and half the sum will be the length of the longest ordinate.