Ball — Contributions to the Theory of Scretvs. 661 



LXXXIX. CoNTRIBtTTIONS TO THE TheOEY OF SCREWS. By RoBEET S. 



Baxl, LL.D., r.R.S., Andrews Professor of Astronomy in the 

 University of Dublin, and Royal Astronomer of Ireland. 



[Eead, November 13, 1882.] 



The theory of " emanants " in modem algebra (Salmon's Higher 

 Algebra, 3rd ed., p. 108) is specially appropriate for throwing light 

 on screw co-ordinate transformations. The present Paper relates to 

 this subject. 



Let ai, . . . ag denote the six co-ordinates of a twist or of a wrench. 

 If we regard the amplitude of the twist or the intensity of the wrench 

 as unity, then the six co-ordinates become the co-ordinates of the 

 screw. For our present purpose we require the six co-ordinates to be 

 independent variables, and therefore we shall regard them as the co- 

 ordinates of the Dyname itself, not merely of the screw on which it 

 reposes. To specify the screw five constants are required ; one con- 

 stant more gives the intensity of the Dyname, making six in all. The 

 Dyname can thus be completely expressed by the six co-ordinates, of 

 which each one is absolutely independent of the rest. 



Let a' be the intensity of the Dyname on a ; then a! is proportional 

 to tti, . . . as insomuch, that if the Dyname be replaced by another on 

 the same screw a, but of intensity a;a', the co-ordinates of this new 

 Dyname will be xa.^, . . . xa^. 



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

 its position and as to its intensity fB'. Let the co-ordinates of /5, re- 

 ferred to the same screws of reference, be /Si, ... /3^. If we suppose 

 a Dyname of intensity g/3' on the screw (3, then its co-ordinates will 

 be g(3i, . . . g/Sg. Let us now compound together the two Dynames of 

 intensities xa' and y/3' on the screws a and (3. They will, according 

 to the laws for the composition of twists and wrenches {Theory of 

 Scretvs, p. 11), form a single Dyname on a third screw lying on the 

 same cyUndroid as a and /3. The position of the resultant screw is 

 such that it divides the angle between a and (3 into parts whose sines 

 have the ratio of g to x. The intensity of the resulting Dyname is also 

 determined (as in the parallelogram of force) to be the diagonal where 

 X and g are the sides, and the angle between them is the angle between 

 a and /3. It is important to notice that in the determination of this 

 resultant the screws of reference bear no part ; the position of the re- 

 sultant Dyname on the cylindroid as well as its intensity each depend 

 solely upon the two original Dynames, and on the numerical magni- 

 tudes X and g. 



We have now to form the co-ordinates of the resulting Dyname, or 

 its components when decomposed along the six screws of reference. 

 The first Dyname has a component of intensity xai on the front screw ; 

 and as the i^ceoud Dyname has a component g/3i, it follows that the 



