When the centre line of the pipe is inclined to that of the cone (Fig. 220), then the determining of the joint curve is a more difficult matter. The only real difference, however, between the construction in this problem and the last is in the arcs d 1", b 2", and a 3"; for whereas in the former case they were arcs of circles, in this example they come out as parts of ellipses. The difference of construction, then, lies in obtaining the shapes of the elliptic arcs. To do this all that is necessary is to first get the two diameters of the respective ellipses, and then set the small arcs out by the method shown in connection with Fig. 213. It will perhaps be sufficient to explain how to get the diameters of the ellipse of which the arc d 1" is a part, as the method will be the same for each arc. Draw the line 1 e parallel to the centre line of the pipe, and bisect 1" e in q. Draw g f square to the axis of the cone, and on h f describe a quarter-circle, producing it a little beyond the point where it meets the cone axis. Now draw a line through g parallel to the centre line of the cone, to meet the quarter-circle produced in k. Then the line q k will be half the small diameter, and the line g e half the large diameter of the ellipse. These two lengths are set along a trammel, as previously explained, and the two points slid along the lines g 1 and g I, thus obtaining the curve d 1". Now mark off g m equal to 1° 1, and draw a line down parallel to g 1, and so fix the point d. The line d 1' is then drawn square to d m or g 1, and thus the point 1' on the joint curve is found. In the same way the other points 2' and 3', can be determined.
The striking out of the patterns is not shown, as this part of the work will be done in an exactly similar manner to that illustrated by Fig. 219.