504- The Rev. T. P. Kirkman on Pluquaternions, 



terms in — lH m l£t m are the positive squares in — R TO R,„ and 



— S,„S,„, and that the imaginary terms in it, of the fourth, 

 fifth, and sixth degrees, mutually destroy each other in pairs. 

 It is shown also by reasons like those adduced in the discus- 

 sion of <8fcS,„ and of R m S m , that the imaginaries of the sixth 

 as well as those of the seventh degree in 



every term of which is imaginary, destroy each other in pairs. 

 Of — T TO T TO , we can prove, as before in the treatment of 



— S m S ni , that the only real terms are of the form 



— (blpvblpv)W lpv . 

 Now 



because 



wherefore 



— ipohVo v -)(b 'l p v )=l p v 'b *-l p v = -l p v o 'l p Q v o 



= -(-l) = l. 



— T TO T TO is equal therefore to a certain number of positive 

 squares ( + B\ v &c.) + certain imaginaries of the eighth de- 

 gree, which must destroy each other in pairs. 



Of <5t/T TO we can prove, by considerations like those em- 

 ployed in discussing ^,.S m , that every term is imaginary. 

 Hence all the imaginaries of the fifth degree in 



must vanish in pairs; and, whatever be Q/'", if 



fX m +T m =fX' m , 

 whose modulus is 3$ /y , we obtain 



By pursuing this analysis we shall at length obtain ; if 



R M +s M +T w -f ... +Y m =it£- 4) ; 



Q(n-l)9t(n-4) = §Ct(«-4)Q(M-l) 

 i mw m mn m —i » 



