Page:The Kinematics of Machinery.djvu/350

 328 KINEMATICS OF MACHINERY.

the conic one, but with certain differences in their relations. The principal of these relates to the relative lengths of the links, which would vary if they were measured upon spheric surfaces of different radii, if they were taken, that is, at different distances from the point of intersection. The ratio, however, between the length of a link and its radius remains constant for all values of the latter, and these ratios are simply the values in circular measure of the angles 11/2, 2 M 3, 3^4, and 4 M 1, subtended by the links. Instead of the link lengths therefore we must consider the relative magnitudes of these angles, which we can also indicate by the letters a, I, c, and d.

The series of alterations in these lengths which we supposed in the former case, and which we carried on until all the links be- came infinite, are here represented by corresponding angular changes. The infinitely long link corresponds to an angle of 90. For the case where two links are infinite but have a finite difference ( 73) we have now one subtending a right, and the other an obtuse, angle. As however we must always imagine the axis of the links prolonged through and beyond the centre of the sphere, the obtuse angle between two axes gives on the other side also an acute angle between them, so that no real difference exists between acute and obtuse-angled links. A similar simplification affects the centroids and axoids. The infinitely distant points of the centroids in the chain (C^'), of which we had illustrations in 8, are here represented by the points in which the common normal to the fixed axes cuts the sphere. The axoids here are consequently cones (circular or non-circular) upon some closed base.

Keeping these points in view we may now proceed to examine the mechanisms formed from the conic quadric crank-chain, which we shall do as far as possible in the 'same order as before.

A. Conic quadric crank-chain (C f f-) Fig. 257. All links subtend less angles than 90. We obtain from it, as from (C^), eight mechanisms for its eight principal special cases or posi- tions ; to these we can give the same names as before, only pre- fixing the word conic in each case. Their formulae, also, are analogous to the former ones, the form-symbol for oblique replacing that for parallel. I do not know of any applications of these mechanisms, but it is quite possible they may exist, disguised under dissimilar constructive forms.