A Treatise on the Theory of Screws

CHAPTER XX . EMANANTS AND PITCH INVARIANTS. 260. The Dyname. If we wish to speak of a magnitude which may be a twist or a wrench or a twist velocity it is convenient to employ the word Dyname used by Pliicker* and by other writers. The Dyname a is completely expressed by its components alt ... a6 on the six screws of reference. These six quantities are quite independent. They may be considered as the co-ordinates of the Dyname. Let a' be the intensity of the Dyname on a; then a' is a factor in each of a1; ... a6, and if the Dyname be replaced by another on the same screw a, but of intensity xa', the co-ordinates of this new Dyname will be xalf ... xa6. Let ß be a second Dyname on another screw quite arbitrary as to its position and as to its intensity ß‘. Let the co-ordinates of ß, referred to the same screws of reference, be ßlt ... ß6. If we suppose a Dyname of intensity yß' on the screw ß, then its co-ordinates will be yßlt ... yß6. Let us now compound together the two Dynames of intensities xa' and yß' on the screws a and ß. They will, according to the laws for the composition of twists and wrenches (§ 14), form a single Dyname on a third screw lying on the same cylindroid as a and ß. The position of the resultant screw is such that it divides the angle between a and ß into parts whose sines have the ratio of y to x. The intensity of the resultant Dyname is also determined (as in the parallelogram of force) to be the diagonal where x and y are the sides, and the angle between them is the angle between a and ß. It is important to notice that in the determination of this resultant the screws to which the co- ordinates are referred bear no part; the position of the resultant Dyname on the cylindroid as well as its intensity each depend solely upon the two original Dynames, and on the numerical magnitudes x and y. * Pliicker, ‘Fundamental views regarding Mechanics,’ Phil. Trans. 1866, Vol. clvi. pp. 361 —380.