JoLY — Quaternion Invariants of Linear Vector Functions^ 8fc. 3 

 and 



= 6 (a/S'/JyS - (3SayS + ySajSB - hSafiy) = 0. 



Determinants of this type enter largely into the treatment of hyper- 

 space hy means of a symbolic algebra analogous to qnaternions. 



Generally, also, the determinant of the fourth order whose rows 

 are identical, and whose constituents are quaternions, vanishes iden- 

 tically. For, if p, q, r, and s are four arbitrary quaternions, the 

 transformation 



a/3 



y 



8 



a (3 



y 



S 



a /3 



y 



8 



a(3 



y 



8 



p q r s 





p q r 1 



p q r s 





p q r 1 



p q r 8 





p q r I 



p q r s 





p q r 1 



aSp ^-h8q + cSr + dSs 

 d 



(in which 



aVp + hVq + cVr + dFs = 0) 



is the result of adding the first, second, and third columns multiplied 

 by a, h, and c to the fourth multiplied by d, and then dividing by d. 



Expanding the transformed determinant by the minors formed from 

 the first and second rows, it is seen to vanish identically. 



Again, if ^i, <J32, 4>5, and ^^ are any linear vector functions, 



= 0. 



If 



^itt ^1^ ^ly ^iS 



^2a ^2^ <l>2y <^2S 

 <^3a ^3/3 (f>3y ^38 

 <jiia <lii/3 ^47 (fiiS 



5. Geometrical interpretations concerning vanishing determinants. — 



a P 

 a' yS' 



= 0, or afi' = ^a', 



the four vectors are coplanar ; the angle between a and ^' is equal to 

 that between fi and a' ; and, if the vectors are coinitial, the triangle 

 determined by a and ^' is equal to that determined by j3 and a'. 



B 2 



