Proceedings of the Royal Irish Academy. 



If 



1 1 1 



= 0, 



a P y =0, 



a/3y 

 the vectors, if coinitial, terminate on a line. In fact, 



/?-a y-a 



fi-a y-a 

 or /8 - a is parallel to y - a. 

 Consider the quaternion 



= 2 (a' r^y + /3' Fya + y' Fa^) : 





a' IB' i 



2' = 



a 13 y 





a (3 y 



its conjugate is 





a /3 y 



X^ = 



a ^ y 





a' /3' y 



and its scalar naiay 





a ^ y 



Sq = 



a' 13' i 





« ^,r 



= 2(F/5y.a'+ Fya.y3'+ Vafi.y'); 



= (ya'/3 - ^a'y) + (a/3'y - y^'a) + (^y'a - ay'^), 



the terms teing grouped so that the pairs within the hrackets are 

 scalars. This may serve as a particular example of the effect of 

 interchanging the rows. 



If a', (3', y' are regarded as the points of application of the forces 

 F)3y, Fya, and Vafi, respectively, Vq = expresses that the sum of 

 the (vector) moments of these forces with respect to the origin of 

 vectors is zero, or that the resultant of the forces is a single force 

 through the origia ; Sq = expresses that the virial of the forces 

 with respect to the origin is zero ; and generally q = expresses, in 

 Hamilton's phraseology, that the forces are equivalent to a single 

 force, and that the origin is the centre of the forces, heing that point 

 for which their total moment q vanishes, or, more generally, is a 

 minimum.^ 



1 "Elements of Quaternions," Art. 414 (16). "What is now called the virial, 

 was called by Hamilton the total tension. By Art. 7 of the present Paper the 

 relation of these six vectors may be illustrated by means of a quadric. 



