Or, in the cast of a prism joining with a conoid, that the plane described by the axis of the generating figure of the prism, and the plane passing through the axis or centre of the comid at right angles therewith, and also to the plane of projection, are the two planes whose distances are re spectively a- and y.
PROB. To find the projection of the intersection of two Prob. To find the projection of the intersection of two prisms, or of a prism and conoid.
Suppose the plane of projection to pass through one ex tremity of the axis of the generating figure, and let a be equal to the axis of that figure, and consequently equal to the distance of the most remote point of the intersection of the two solids from the plane of projection ; and be equal to the distance of any point in the intersection from the said plane, x equal to the distance of that point from one of the vertical planes, and y the distance of the same point from the other vertical plane.
Find the equation of the one generating curve in term:: of x and z, also the equation of the other generating curve in terms of y ands. Find the value az or any equal pow er of z in each of these equations; put these values equal to one another ; then the equation between x and y will de termine the species of the curve.
Ex. I. Suppose two parabolic prisms to intersect each other, so that the apex line of the one prism, and the rect angle opposite the apex line of the other, may be in the plane of projection.
Fig. 20. Plate CCCXXXIII. Let CI be the apex line of the one prism, and A F the line described by the extremity of the axis which meets the double ordinate of the other. Let APDC be half of the generating parabola, of which its axis forms the line CI ; and A'P'D'C' half the generating section, of which the extremity of the axis that meets the double ordinate forms the line AT.
Draw BP parallel to CD; make AC= a, CD = b, AB = z, and BP = y. Draw B'P' parallel to ; make C'A' = a, CD' = c, C'B' = :, and B'P' = ; therefore WA' will be = a Then, by the property ofthe parabola, In the section APDC, a : z : therefore z b and in the section CUP/kr, a : a z : x= ; therefore a = a ; Consequently a whence we infer, that the curve FGHRI, which is the projection of the two prisma tic surfaces, is an ellipse.
If c, the projection will be a circle.
Ex. 2. Suppose the generating figures of both prisms to be parabolas, as before ; and that the rectangle described by the double ordinate of each is on the plane of projec tion, to find the projection of the intersection of their sur faces.
Let AFDC, Fig. 21. Plate CCCXXXIII, be the gener ating figure of the one prism, and A'P'D'C' that of the other. Make CA a, CB = z, CD = b, and BP = x, also C'A'= c, C'B' z, C'D'= d, and B'P'= y ; therefore BA a z, and B'A' = c z.
From the propel ty of the parabola we have from the figure A P D C, a : a z : : ; whence z = a also from the figure A'P'D'C, c c z : ; c y - 2 'w a x y hence z = c ; therefore a = C 2 c dr d" (c a) a therefore = --+ x ; whence we infer c that the curve is an hyperbola ; and if a and c become equal, we should then have , or y = x, and 6 2 6 consequently, in this case, the projection of the intersec tion of the two surfaces would be straight lines, and would form the figure cf a groin ; and if b and d were equal, we should have = (c a) a + which is a case that would more frequently occur in practice.
Ex. 3. Suppose the gt net ating figure of the one prism to be a semi-ellipse, of i hich the greater axis is the base, and the generating figure of the other to be a semi-circle ; and tiett the rectangle described by the greater axis of the forn.er, and the rectangle desctibed by the diameter of the latter, are in tile plane of projection.
Let APDC. Fig. 22. he the half of the generating figure of the elliptic prism, or send-cylinclroid, and A'I"D'C' be half of the ceneratitn?: figure of the semi-cylinder. Mike CA=a. C.:n=7z, CD=b. and also C' C'B' =z, u'D'=d, and B'z''=y.
Then, from the property of the ellipse, in the figure APDC, we have : : z (az) : ; whence a2s2-62 ) Therefore z 2 = A:Tain, from the property of the circle in the figure A'F'D'C, we have = (d+z)(dz..)= ; whence therefore ; whence = + Whence we infer that the curve is an hyperbola.