Piton. VI. The base and one of the planes of a prism being given, to find the section of the prism oblique to the base, but at right angles to the plane of the side given, the line of inclination being given in position upon the given side of the prism.
Place the base contiguous to the given side of the prism, so as to join to their common side, or line of con course, or line of junction; take as many ordinates in the base as may be thought sufficient, and produce them to the line of inclination; front the intersected pohits in the line of inclination draw perpendiculars, which make equal to their corresponding ordinates in the base ; and if the points in the base where the lines proceed are the junctions of straight lines, join every two adjacent points by straight lines, and the section will be formed; or, if the points proceed from a curve, a curve must be traced through the points found in the section.
Examples. In Plate CXV1. Fig 1. the base ABC!) being a rectangle, the two sides AE and BF of the given plane in a straight line with AD and BC, two of the side.i of the base answer the purpose of ordinates, and there fore we have only to complete the section to the length of the inclination El'', and to the breadth BC or AD of the base.
In Fig. 2. the base is bounded by several straight lines, and therefore every two contiguous points in the section are also joined by straight lines.
In Fig. 3. the base is the arc of a circle, therefore the remaining boundary of the section must be a curve pass ing through the several points: The section here is the segment of an ellipse.
Pion. VII. To find the section of a prism, the base and one of the adjoining planes being given, also a sec tional line on the given plane, and the angle which the common side makes with the line of intersection of the • prism.
Produce the line of concourse BA. (Fig. 4.) and the sectional line CD, till they meet in E; make the angle LIEF equal to the angle which the section line is to make with the line of concourse ; from any point Fin El' (haw FG perpendicular to Ell, cutting Ell in G, and Gfl per pendicular to EC, cutting EC at 1 ; from E, with the distance EF, describe an arc cutting GI I at I-I, and join Ell; in the boundary of the base take any point k ; dr..w k / parallel to EF, cutting AB, or All produced at 1; draw I in parallel to the side BC of the plane, cutting CD at 7/1 ; draw in n parallel to Ell ; make in n equal to k, and n will be in the boundary of the section required : In the same manner all other necessary points are to be found.
Examples. In Fig. 4. the base being a rectangle, the determination of the point n corresponding to one of the angles is sufficient; for being joined to the point C in the opposite side, two contiguous sides of the section will then be formed : The whole section M ill be in closed by drawing the other two sides parallel to these sides.
In Fig. 5. the base consists of straight lines, and there fore the ordinates are taken from the angles, and the points found in the section joined by straight lines.
In Fig. 6. the base is wholly a curve, excepting the line of concourse; therefore drawing a curve through the points found in the section completes the boluidaiy or inclusurc.