PROCEEDINGS 



or 



THE ROYAL IRISH ACADEMY. 



PAPERS READ BEFORE THE ACADEMY. 



I. 



QUATERNION INYAEIANTS OE LINEAR VECTOR FUNC- 

 TIONS AND QUATERNION DETERMINANTS. Bx 

 CHARLES J. JOLT, M.A. 



[Eead June 8, 1896.] 



1. Introductory. — This Paper is, to a certain extent, supplementary to 

 a Paper on " The Scalar Invariants of Two Linear Vector Eiinctions," 

 wluch was puhlislied in toI. xxx. of the Transactions of the Royal 

 Irish Academy, The notation of that Paper is followed as closely as 

 possible, so as to facilitate occasional references to it. 



Thie quaternion invariants being simply exj)ressible as quotients of 

 two determinants witli vector constituents, it seems desirable to con- 

 sider briefly such, determinants, and to point out the geometrical 

 meaning of their vanishing in certain simple cases.^ 



2. Expansion of determinants ivith quaternion constituents. — Because 

 quaternion multiplication is not commutative, a determinant whose 

 constituents are quaternions is unmeaning until some convention is 

 adopted respecting its expansion. If it be agreed that the order of 

 the constituents in the expansion shall follow the order of the rows, 

 all indefiniteness is removed. 



^ Determinants, whose constituents are alternate numbers, have been considered 

 by Clifford ("Mathematical Papers," p. 277). If «i and «2 are any two consti- 

 tuents, ii^ = t2^ = 0, and i\iz + izh = 0, these being the defining formulas for 

 alternate numbers. 



E.I. A. PEOC, SEE. in., VOL. PV. B 



