Page:The Kinematics of Machinery.djvu/453

 WHEELS OF INFINITE DIAMETER. 431

machinery differential gear. This name has apparently been chosen because of the minus sign in the last formula. We shall not retain it, for it may occasion misunderstanding, but shall call the mechanism a compound (reverted) epicyclic train.

If there be an annular wheel in either of the two pairs of wheels a, b and c, d, the formula for the relative rotations will be :

If each of the two pairs contain an annular wheel* it is again

n i _ i ac n bd

Or generally, if we indicate the simple velocity ratio of the train of wheel work by a, we obtain the formula

n i i

- 1 = 1 a.

n

Here a itself is positive if there be two annular wheels or none, the minus therefore remains ; while if there be one annular wheel only a becomes negative and the sign in the formula is positive.

There are many forms and still more applications of the mechanism before us. It will be noticed that in cases where a is negative in the formula and > 1, the rotation of d is in the opposite direction to that of e. To simplify the description we shall call a the first and d the second central wheel and & the first and c the second outer wheel.

The limiting cases which occur when some of the wheels are made infinite are very important. One of these I must specially examine, it is as follows. Let us suppose that either of the wheels a or & be annular, as in the diagrams Fig. 283, in which the pitch circles only are shown ; then a is, as we know, negative, and the expression for n^ : n is

rc 0^

n f Id

Let, however, the radius of the annular wheel be infinite, then in order to gear with it the other wheel of the pair must be infinite also. The centres of the two infinite wheels lie within the finite

Sept. 17, 1875.
 * As, for example, in Moore's pulley-block, illustrated in Engineering,