260, 261] EMANANTS AND PITCH INVARIANTS. 275 



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 x^ on the first screw ; and as the 

 second Dyname has a component y^ 1} it follows that the sum of these two 

 must be the component of the resultant. Thus we have for the co-ordinates 

 of the resultant Dyname the expressions 



261. Emanants. 



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

 Dynames a and ft we make a complete change of the six screws of reference. 

 Let the co-ordinates of a with regard to these new screws be \, ... \ 6 , and 

 those of (3 be //, 1} ... //,. Precisely the same argument as has just been used 

 will show that the composition of the Dynames xo! and yfi will produce a 

 Dyname whose co-ordinates are x\^ + y/j, lt . . . x\ 6 + y(j, s . We thus see that the 

 Dyname defined by the co-ordinates x^ + yfti, ... xa 6 + y{3 6 , referred to the 

 first group of reference screws is absolutely the same Dyname as that defined 

 by the co-ordinates X,j + y^, ... x\ 6 + yv 6 referred to the second group 

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

 x and y. 



In general, let 1} ... # 6 denote the co-ordinates of a Dyname in the first 

 system, and &amp;lt;j&amp;gt; l} ... &amp;lt;/&amp;gt; 6 denote those of the same Dyname in the second system. 

 Let/(0!, ... 6 ) denote any homogeneous function of the first Dyname, and let 

 JP (&amp;lt;/&amp;gt;!, ... &amp;lt; 6 ) be the same function transformed to the other screws of refer 

 ence. Then we have 



as an identical equation which must be satisfied whenever the Dyname de 

 fined by lt ... 6 is the same as that defined by &amp;lt; u ... &amp;lt;/&amp;gt; 6 . We must there 

 fore have 



f(xa.i + yfi-i , . . . aJOe + y/3 6 ) = 



These expressions being homogeneous, they may each be developed in 



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

 oc 



this ratio, we must have the coefficients 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 form* 



d d 







* 



* See Proceedings Roy. Irish Acud., Ser. n. Vol. in. ; Science, p. 601 (1882). 



18-2 



