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ENGINEER AND MACHINIST’S DRAWING-BOOK.

then becomes different from the preceding, but the mode
of construction is the same.

Fig. 4.—When the object is situated in the angle
formed by the junction of two vertical planes inclined to
each other, and to the vertical plane of projection, its
shadow will be thrown upon both surfaces, and will be
expressed by two distinct parallelograms having a com-
mon side ef in the line of intersection of the planes. The
construction necessary for determining these figures will
be easily understood by inspection of our illustration, by
which it will be observed that the only part of the slip
which casts a shadow upon the plane X Y is the rectangle
A C F E, and that the shadow of the rest is thrown upon
the plane Y Z, which being inclined to the plane of pro-
jection in a direction contrary to the first, necessarily
causes the shadow also to lie in the contrary direction to
the part a efc.

Figs. 5 and 6.—To find the shadow cast by a straight
line A B upon a curved surface, either convex or concave,
whose horizontal projection is represented by the line
Xer Y.

We have already explained that the shadow of a point
upon any surface whatever is found by drawing a straight
line through that poinf, parallel to the direction of the
light, and marking its intersection with the given surface.
Therefore, through the projections A and A' of one of the
points in the given straight line, draw the lines A a, A a',
at an angle of 45°; and through the point a, where the
latter meets the projection of the given surface, raise a
perpendicular to the ground-line; its intersection with
the line A a, is the position of the shadow of the first point
taken; and so for all the remaining points in the line.

If it be required to delineate the entire shadow cast by
a slip A B C D, as before, upon the surfaces under consi-
deration, we shall be enabled, by the construction above
explained, to trace two equal and parallel curves a e b,
cfd, representing the shadows of the sides A B and C D;
while those of the remaining sides will be found denoted
by the vertical straight lines a c and b d, also equal and
parallel to each other, and to the corresponding sides of
the figure, seeing that these are themselves vertical and
parallel to the given surfaces.

Fig. 7.—When the slip is placed perpendicularly to a
given plane X Y, on which a projecting moulding, of any
form whatever, is situated, the shadow of the upper side
A' B' which is projected vertically in A, will be simply a
line A a, at an angle of 45°, traversing the entire surface
of the moulding, and prolonged unbroken beyond it. This
may easily be demonstrated by finding the position of the
shadow of any number of points such as D', taken at plea-
sure upon the straight line A' B'. The shadow of the
opposite side, projected in G, will follow the same rule,
and be denoted by the line C c, parallel to the former.
From this example we are led to state as a useful general
rule: that in all cases where a straight line is perpen-
dicular to a plane of projection, it throws a shadow
upon that plane, in a straight line, forming an angle of
45° with the ground-line.

Fig. 8 represents still another example of the shadow
cast by the slip in a new position; here it is supposed to
be set horizontally in reference to its own surface, and
perpendicularly to the given plane X Y. Here we see
that the shadow commences from the side D B, which is
in contact with this plane, and terminates in the horizon-
tal line a c, which corresponds to the opposite side A C of
the slip

Plate LI. Fig. 1.—Required to find the shadow cast
upon a vertical plane AY by a given circle parallel to it.

Let C, O' be the projections of the centre of the circle,
and R, R' those of the rays of light.

We have already shown that when a figure is parallel
to a plane, its shadow cast upon that plane is a figure in
every respect equal to, and symmetrical with it; therefore
the shadow cast by the circle now under consideration
will be expressed by another circle of equal radius ; con-
sequently, if we can find the position of the centre of this
new circle the problem will be solved. Now the position
of the shadow of the central point C, according to the
rules we have already folly developed, is easily fixed at c ;
from which point if we describe a circle equal to the
given circle, it will represent the outline of the required
shadow.

Fig. 2.—If the given circle be horizontal, its shadow
cast upon the vertical plane X Y becomes an ellipse which
must be constructed by means of points, as indicated by
the figures referred to above; that is to say, that in the
circumference of the circle a certain number of points are
to be taken, such as A', D', B', &c., which are to be pro-
jected successively to A, D, B, on the line A B, and through
each of these points, lines are to be drawn parallel to the
direction of the rays of light, and their intersection with
the given plane determined. The junction of all these
points will give the ellipse adb, which is the contour of
the required shadow.

Fig. 3.—When the circle is perpendicular to both planes
of projection, its projection upon each will obviously be
represented by the equal diameters A B and C' D', both
perpendicular to the ground-line. In this case, in order
to determine the cast shadow, we must describe the given
circle upon both planes, as indicated by the figures, and
divide the circumference of each into any number of equal
parts; then, having projected the points of division, as
A2, C2, E2, &c., to their respective diameters AB and C' IT,
draw from them lines parallel to the rays of light, which,
by their intersection with the given plane, will indicate so
many points in the outline of the cast shadow.

Fig. 4 represents a circle whose plane is situated perpen-
dicularly to the direction of the luminous rays. In this
example the method of constructing the cast shadow does
not differ from that pointed out in reference to Fig. 2,
provided that both projections are made use of. But it is
obvious that instead of laying down the entire horizontal
projection of this circle, all that is necessary is to set off
the diameter D' E', equal to A B; because the shadow of
this diameter, transferred in the usual way, gives the
major axis of the ellipse which constitutes the outline of