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 a l} ... a 6 n 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 of is a factor in each of 

 !, ... a 6 , and if the Dyname be replaced by another on the same screw 

 a, but of intensity #a , the co-ordinates of this new Dyname will be 



#!, ... XO.Q. 



Let ft be a second Dyname on another screw quite arbitrary as to its 

 position and as to its intensity ft . Let the co-ordinates of ft, referred to the 

 same screws of reference, be ft 1} ... /3 6 . If we suppose a Dyname of intensity 

 yft on the screw ft, then its co-ordinates will be yft 1} ... yft 6 . Let us now 

 compound together the two Dynames of intensities oca and yft on the screws 

 a. and ft. 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 ft. The position of the resultant screw is such that it 

 divides the angle between a and ft 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 ft. 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. 



