The Rev. T. P. Kirkman on Pluquaternions, 



be any two of these ; then, neglecting subindices, + dg will be 

 equal to some sixth, and + eg to some seventh imaginary, say 



±dg—c t + eg=b. 



It follows that 



cd= + be 



cb = ±ed= +a. 



If then cdg and beg are of the same sign, abc is a triplet in the 

 pluquaternion system, of the same sign with ade and afg ; and 

 since by equations (A.), which here apply, 



dg=ef and eg =fd, 



it is plain that the imaginaries abcdefg form some one of eight 

 complete systems of seven triplets. If cdg is of a different 

 sign from beg, acb is a triplet in the system, of the same sign 

 with ade and afg ; so that, if we put b' for c, and d for b, we 

 know that aUc'defg form one of the same eight systems. 

 Let it be either of these two, 



+ abc 



+ afg L + %+«/J 

 and let +a/iibe any fourth triplet in the system beginning 

 with a. Then the following are proved to coexist, like the 

 equations (A.), being deduced from the equal values of a: 



bc=hi t bi=ch y bh——ci\ "J 



de—hi t dizseh, dh — —ei i K . . . (B.) 



fg=M> fi=g h > fh=—gi' J 

 Let now k be the imaginary equivalent to bi in the system. 

 We have then, since bi=ch, 



^bd/^bik, 



+ df=ik, 



+ dk=ft=gh; 



and 



+ cdg=chk y 



+ dg—hk, 

 ± dk=gh; 



or 



dk=-dk; Q.E.A. 



If the system of seven triplets be of the other type, or one 

 of these, 



+ abc 



+afs \+W-«sS 



