18 Proceedings of the Royal Irish Academy. 



scalars, with the single restriction, that the order of the vectors in 

 the expansion shall take the order of their positions in the columns. 



"We thus conclude that the AusdeJmungslehre, as adapted to a space 

 of n dimensions, may be regarded as a part of the associative algebra of 

 n + \ dimensions limited by more or less arbitrary restrictions. "We 

 see that the conceptions of progressive and regressive multiplication 

 are partial aspects of a complete operation which alone is entitled to 

 the name multiplication. In this respect the Ausdehnungslehre 

 resembles several systems of vectorial analysis, and for very similar 

 reasons. The point symbol is introduced by the artifice of leaving 

 the origin arbitrary, not as Hamilton did in the space under investi- 

 gation, but in a space of one dimension higher. 



