152. Draw c c', o c', perpendicular to the projections of the ray, and make them respectively equal to the distances of the centre of the sphere from the co-ordinate planes ; e's, c' T, drawn through the points in which the given ray cuts the co-ordinate planes, will represent that ray brought into the co-ordinate planes by the turning round of its projecting planes on its projections ; draw a' d' e', perpendicular to c' T, c' 8, making them equal to the diameter of the sphere ; then lines drawn through a', b', parallel to c' T, will represent the two rays, touching the surface of the solid and lying in the projecting plane of the given ray brought into the co-ordinate plane along with that ray : these lines will cut CT In Q n, the vertices of the major axis of the elliptic outline of the shadow of the sphere on the co-ordinate plane. The conjugate axis q e will be given by drawing lines parallel to C T tangents to the projection of the sphere; for these last parallel tangents will be the boundaries of the projections of the cylinder of rays. Lines drawn through a', b', parallel to C c', will cut c T in the vertices a, n, of the conjugate ails of the elliptic projection of the great circle, sepa rating the illuminated hemisphere from that in shade ; a diameter D z to the circular projection of the sphere, drawn through c perpendicular to o T, will be the major axis of this ellipse.
153. For the plane of the great circle, of which ADES is the pro jection, is obviously by the construction perpendicular to the given ray, and the plane of this circle is cut by the projecting plane of the given ray CT in the original of AB, while the diameter Da is the projection of the intersection with the plane of the same great circle, by a plane passing through the given ray e s, co, and perpendicular to the plan projecting plane (9) of the given ray. This perpendicular plane must, therefore, be the elevation-projecting plane of the given ray.
154. By the same construction applied the other projection, the elliptic elevation adbe, of the circle separating the light and shade on the sphere, and the elliptic shadow of the sphere on the vertical co-ordinate plane, may be obtained.
155. It is clear that in this example the two elliptic outlines of the shadows of the sphere on the co-ordinate planes, must cut T z in two common points ; because the segments of the ellipse on either side of T Z of each outline is the projection on the one co-ordinate plane of that portion of the cylinder of rays which forms on the other co-ordi nate plane the portion of the outline of the shadow on the same side of T Z. t L, perpendicular to Y z, lA the trace of the elevation pro jecting plane of Cs, cc; o, 0, is the point in which this same plane cuts the trace of the given plane, consequently L o is the plan of the inter section of those two planes, and in which this line is cut by the plan of tho ray c s is the intersection of that ray, and the given plane ; the elevation t' of the same Intersection may be obtained by applying the same constructions to the other traces and projections.
156. The two pair of parallel planes, which are respectively perpen dicular to the co-ordinate planes, and therefore to each other, and which are parallel to the given ray, touch the sphere iu the points A, a ; B, b ; D, d ; E, e. These four planes will be cut by the plane L u n in a parallelogram, the sides of the projections of which must be parallel to those of the ray c s. c, a, and to the lines a o, w n. Draw t L' per pendicular to o t, and make t a' equal to ta ; join o L', which will represent the intersectian of the projecting plane with L m n ; draw lines through e', parallel to c's, and from the points in which these parallels cut on! draw parallels to a' t to cut o t; again lines drawn through these last intersections parallel to se Is will be the two sides of the elevation of the rectangle above mentioned ; the parallel tangents at a and b will complete the figure ; and o t. w n, will cut the opposite aides in the points in which the elliptic outline of the shadow of the sphere will touch those sidce, or the points which represent the shadows of d, e, a, and b.
157. The plan of this parallelogram may he determined in the same manner, or by the other constructions explained for determining the projections on the other co-ordinate plane from those already deter. mined on the first, and which are sufficiently indicated in the figure to render further description of them unnecessary.
158. If a represent a luminous body, and r a point, then, by imagining a plane to pass through them, the intersection of that plane with the plane on which the shadow Is cast will cut the ray L P In g, the shadow of the point. To determine this intersection, we have only to draw two parallel lines through L and P, in any direction, and deter mine the points t and p, or 1', p', in which these parallels meet the plane of the shadow : then 1 p, L p being drawn, they will cut each other in g, the shadow of the point. This is the principle employed in the following construction.
159. Let abcdefg be the perspective projection of a cube, c being the centre of the picture, c V the distance of the picture, x the vanishing line of the face a bcd, and Y z its intersecting line ; while z' is that of the face efg, parallel to the former. Let Y z and w z be given as the vanishing and intersecting lines of a plane, on which the shadow of the cube, as cast by the luminous body given in position, Is to he determined.