502 The Rev. T. P. Kirkman on Pluquatemions, 



of the constituents ; that is, whatever be Q,!' or R m , of these 

 forms, 



Q"R =R Q" ., 



the same triplets being employed. Further, 



Q"-a'Q=Q."<&,. + a"R =<St +R +s • 



q o! -a" -<st a" -r a" . 



=<0t_ i a"_-Q/'R m =^_-R m -S ro . 



/•^V^^a^.-^R^SJ + lR^ + SJia.-CR^ + S.) 2 , 



F **-t- *A, + S TO <Sl_ - R m R w - R m S„ - S m R w - S m S w , 

 if jw,, (jl p ju, y/ be the moduli of Q i? ft/, Q./'. 

 OtjiSt.i is real, and =m 2 , if Itt be the modulus of <Bt f ; 



— R TO R m we know to be =i$ 2 + certain imaginary terms 

 which all vanish, where 5$ 2 is the sum of the squares of the 

 constituents of R ; and it is evident, that all the imaginary 

 products in the five terms containing S^ must destroy each 

 other likewise. 



For in ^S TO every term is of the form c 'b Q n p ' C, which 

 is not real ; since, if 



c 'b o n Q p o - +1, 



-b n p = ±c c) 



and b n p o would be reducible to a simpler form, contrary to 

 the supposition which we have a right to make concerning all 

 the imaginary triplets in S m . 



In R m S OT also every term is imaginary ; for if, e. g., 



c h'b n p =±U 



—l 'b ?i p =±c, 

 and 



!> n p =±l c oi 



a contradiction of a similar kind. 



In — S m S m , the only real terms are of the form 



-b n p 'b n p o B 2 np . 

 Now 



h'n p =— n o p -b o , 



