662 Proceedings of the Royal Irish Academy. 



sum of these Itto must be, the component of the resultant. Thus we 

 have for the co-ordinates of the resultant Dpiame the expressions 



xa.i -f y^i, . . . xa^ -T y/3e. 



Let us suppose that without in any paiiicular altering either of the 

 Dynames a and /S we make a complete change of the six screws of re- 

 ference. Let the co-ordinates of a with regard to these new screws 

 be Xi, . . . Ag, and those of (3 be /xj . . . //.g. Precisely the same argu- 

 ment as has just been used will show that the composition of the 

 Dynames xa' and y/3' will produce a Dyname whose co-ordinates are 

 xXi + y/j.i, . . . a-Xg f y/zg. We thus see that the Dyname defined by the 

 co-ordinates xai -f ^jSi, . . . xa.^ 4 y/3g, ref eiTed to the first group of re- 

 ference screws is absolutely the same Dyname as that defijied by the 

 co-ordinates xX^ + y/^i, . . . xX^ -f y/ig referred to the second group of 

 reference screws, and that this must remain true for every value of 

 X and y. 



In general, let 6i, ... 6^ denote the co-ordinates of a Dyname in the 

 first system, and 91, . . . <^6 denote those of the same Dyname in the 

 second system. Let f{0\, . . . 6^) denote any homogeneous function of 

 the first Dyname, and let F{4>i, • • • ^e) be the same function trans- 

 formed to the other screws of reference. Then we have 



/(^i. . . . ^g) = F{4^„ . . . c/)g) 



as an identical equation which must be satisfied whenever the Dyname 

 defined by Oi, . . . d^is the same as that defined by 0i, . . . 4>e- "We must 

 therefore have 



f{xai + y/3u . . . xos^ y/Se) = F{xXi -f y/j.^, . . . xX^ + y/y^). 

 These expressions being homogeneous, they may each be developed 



in ascending powers of -. But as the identity must subsist for every 



value of this ratio, we must have the coefiicients of the various powers 

 equal on both sides. The expression of this identity gives us a series 

 of equations which are all included in the fonn — 



d d Y ! d d \»„ 



The functions thus arising are well known as •• emanants '' in the 

 theoiy of modem algebra, and we have now proved that they are co- 

 variants of the original quantic. It is instinctive to notice how inti- 

 mately this branch of algebra is connected with the Dynamical con- 

 ceptions in the theory of screws. The cases which we shall consider 

 are those oi n = I and 71 = 2. In the former case the emanant may be 

 written 



dai ' ^ doe 



