Page:The Kinematics of Machinery.djvu/64

 assume in the first instance that the bodies possess complete rigidity, and shall pay no attention to their size,—in other words we take all questions belonging to what we have called the special study of machinery and the study of machine-design as solved, so that only  properties remain for us to consider.

. 4.

Now in order that any moving body $$A$$, of given form, may remain continually in contact with a stationary one $$B$$, we must give to the latter a special form. This can be found if the body $$A$$ be caused to take up consecutively the series of positions which it is intended to occupy relatively to $$B$$, and the figure which envelopes all the positions of the outline of the body $$A$$ determined. If for example $$A$$ be a parallelopiped (Fig. 4), of which one surface remains during its motion in a plane, the figure $$B$$ will become a curved channel. The geometrical form thus found for $$B$$ is called the  of the moving figure $$A$$. $$A$$ has to $$B$$ also the same relation as $$B$$ to $$A$$, that is, $$A$$ is the envelope to $$B$$, or at least those points in the figure $$A$$ with which $$B$$ comes into contact form the envelope of the body $$A$$ in respect to $$B$$. The relation, therefore, is.

Many such reciprocal envelopes can be actually constructed, If a moving body be surrounded by stationary ones containing its envelope in such a way that at no instant is more than one motion of the body possible, then its motion must necessarily be such as belongs to the envelopes, and is determined by their form.

We see at once that at least one other body is necessary for the envelopment of a moving form. If it be found necessary to use several—perhaps because the one first found, while actually forming an envelope, does not exclude all motions but the one required,—then these can be united with the first into  body. Thus for instance we can suppose the upper and lower half brasses of a plummer block joined together. We find, that is to say, that in