The conditions of the problem are clearly shown in the plan and side elevation of Fig. 400, in which Z B C is the elevation and X C' D' Y is the plan of the level portion of a cold air passage joining a furnace just above the floor line. The inclined portion of the air passage or box is required to join the level portion at the angle Z A E of the side elevation, and at the angle Y A' E' when viewed in plan. These conditions are in many respects similar to those given in Problem 95, with the difference, however, that in this case the joint or miter between the level and the inclined portions does not appear as a straight line in the plan. It may be here remarked that the solution of this problem is more a matter of drawing than of pattern cutting, as nothing can be more simple than the cutting of a miter between two pieces of rectangular pipe when the required angle between them is known. This problem is capable of two solutions, both of which will be given, leaving the reader to choose which is the more adaptable to his requirements.

First Solution. - As above intimated, before the pattern can be developed it will be necessary to make careful drawings, in the preparation of which a knowledge of the principles of orthographic projection is necessary. (See Chapter III (Linear Drawing)).

To proceed, then, with the drawings, first draw a plan and elevation of as much of the furnace as is necessary to show its connection with the cold air box, placing each part of the plan directly under its corresponding part in the elevation, so that as soon as any new point is determined in either of the views its position can be located in the other by means of a perpendicular line dropped from one view to the other. Upon the plan set off the width of the box b and draw parallel lines from the side of the furnace body to the right indefinitely, and upon the elevation set off its hight, a, from the floor line up, and draw A Z. A vertical line from the point X of the plan will give the point Z upon the elevation, or, in other words, show how far the curve of the furnace body cuts into the top and bottom surfaces of the cold air box. Next, upon the elevation locate the point A the required distance from the side of the body according to specification and find its position in the plan by means of a vertical line, as shown. From the point A in both views lines must be drawn to represent the angle or deflection of the pipe as it would appear in those views. Thus the ele-vation would show the slant, which is determined by the two dimensions e and d. Therefore from the point A of the elevation erect a perpendicular line equal to the required hight c, from the top of which draw a horizontal line to the right of a length equal to the amount of slant d, thus locating the point E, which connect by a straight line with A. Then will A E represent the angle of the inclined portion of the pipe as it appears in the elevation. But according to the requirements the pipe is also to have an offset a distance equal to e - that is, the point E of the elevation is nearer the observer than the point A. Therefore from A' of the plan draw a line forward the amount of the offset, from the end of which draw a line to the right, in length equal to d, or in other words till it comes directly under the point E of the elevation, thus locating that point in the plan, and draw A' E', which will show the apparent angle in the plan.

The depth and width of the oblique portion of the box will next demand attention. At right angles to the line A E of the elevation set off the depth of the box a and draw a line to represent the lower near corner of the box, which continue downward until it cuts the floor line, as shown at D; then draw A 1), which represents the miter cut for the side of the box. At right angles to A' E' of the plan set off the width b, as shown, and draw a line parallel to A' E' intersecting the line from X at B', as shown, and draw A' B', which gives the plan of the miter cut across the top of the box. As the point D of the elevation is in the same vertical plane as A it may now be dropped into the plan, intersecting with the line showing the front side of the box in that view, as shown at D'; and the point B' of the plan, being on a level with A', may be projected into the elevation, where it would intersect with the line showing the top of the box at B. A line drawn from D' of the plan parallel to A' E' (shown dotted) will then show the position of the lower near angle of the inclined portion of the box, and a line from B of the elevation parallel to A E will show the position in that view of the further top corner of the box.

The position in the two views of the remaining angle of the inclined portion of the box may be ascertained in several ways: The width h may be set off from D' of the plan and a line drawn which will intersect with X B'continued, as shown at C'; thence it may be projected into the elevation a1 C, as shown; or the width a may be set off from B of the elevation, thus locating the line which intersects with the floor at C, which point may be dropped into the plan, thus locating the point C; or, again, B G may be drawn parallel to A D, or D' C' may be drawn parallel to A' IV, all producing the same result.

In the case in Problem 95, above referred to, it was noted that if the normal profile is adhered to in the level arm, the profile of the gable mold must be changed or "raked" before a perfect miter joint can be ob-tained. What is true in the case of the gable miter is equally true in the case of the furnace pipe - a correct profile or cross section of the box must be developed in order that a correct stretchout may be obtained for use in cutting the miter of the inclined arm of the pipe. As neither the plan nor the elevation, which have been correctly obtained, gives the true length of the inclined piece - that is, the true distance from A to E - it will be necessary to obtain still another elevation, in which such distance is correctly shown. As A' E' of the plan gives the horizontal distance between the points A and E, and c represents the vertical distance between them, if a right angled triangle be constructed with A' E as a base and the hight c as the perpendicular, its hypothenuse will then give the desired measurement. Such a triangle properly forms part of an oblique elevation which may be projected from the plan in the following manner: Parallel to A' E', at any convenient distance away, draw a line to represent the level of the floor, as shown; above which, at a distance equal to a, draw another parallel line, X2 A2, representing the hight of the horizontal arm of the pipe. Above the line X2 A2 at a hight equal to c, draw still another line, upon which the point E is subsequently to be located.