If only a portion of the section passing along the axis be given, it is always supposed to be terminated by the same surfaces, which also terminate the surface to be covered.
The following is a more general method of finding the envelope I if a solid by means of points.
In this description it will only be necessary to have the seats of three points given on the base, and the heights of the points on the cylindric surface from their seats. Let A B C (Plate 11. Figures 1, 2, 3) be the part of the base of the cylinder, and A, 13, c the seats of the three points; join A C; draw c E and A D perpendicular to A C; make A D equal to the height of the point above A, c E equal to the height of the point above c, and c F equal to the height of the point above the seat 11 ; i0111 E D ; draw F a parallel to c A, and a. II parallel to c cutting A c at 11 ; and join II B ; prOdllec A C to I ; divide the curve A 13 c into :my number of equal parts; and extend those parts upon c ; through the points of division in the curve, A B C, draw lines parallel to B intersecting B e; from the points of intersection in A c draw lines parallel to o n, to meet D is ; from the points of intersection draw lines parallel to A C ; and from the divisions in C I draw lines parallel to e E, so as to intersect the other parallels last drawn ; through the points thns found, draw the curve ciKE C, and it will be the envelope required.
Figure 1, is the case w here the rectangular plane makes a right angle with the elliptic section : Figure 2, that Ns here the angle made by the rectangular plane and the elliptic section is acute : Figure 3, where the angle formed by these two planes is obtuse.
In Figures 2 and 3, n M E is the orthographical projection of the curve ; with which the curve, E s, Of the envelope would coincide. This is found by drawing parallels to a El through the points of division in the curve A n c, to meet the parallels of A c, as shown by the dotted lines.