elm 



-fl* 



ghp 



eim 



fiP 



gio 



ekp 



—fkm 



—gkn 



elo 



fin 



-glm. 



Rev. T. P. Kirkman on Bisignal Univalent Imaginaries, 293 



possesses the property expressed by the equation mn-\-nm = 0, 

 we can form thirty-five triplets thus, so as to employ once 

 every duad that can be made with the fifteen symbols : 



;abc 

 ade — bdf cdg 

 afg beg cef 



f~ahi bkk — chl dhm 



I aid — bil — cik —din 



j amn bmo cmp — dko 



\jaop — bnp cno dip 



If we consider every triplet taken with its sign to be equal 

 to negative unity, as 



abc= — bdf= — 1, 



and to imply three equations, such as 



a—bc 9 b=ca, c=ab; —b = df, —d=fb, —f=bd; 

 equivalent to the equations 



a=—cb, b=—ac 9 c=—ba; b=fd, d=bf, f=.db; 



we shall find that the three equal values of each of the seven 

 imaginaries, abcdefgh, deduced from the first seven triplets A, 

 are all congruous with all ; and that the seven equal values of 

 each of the eight imaginaries, /iiklm?iop, deduced from the 

 twenty-eight remaining triplets B, are all congruous with all. 

 Thus the two values of h, given by the triplets ahi and — chl, 



7i=ia=cl 9 



are congruous with the values of k, given by the triplets — cik 

 and akl, 



k = ic=Ia; 



for ic — la follows from ia==cl. 



But if we compare the values of the first seven imaginaries, 

 as given by the first seven triplets, with those deduced from 

 the other twenty-eight, we find them contradictory. From 

 the triplets dhm and eim, we have 



m = dh=ei 9 



de = ih=—hi 9 



giving 



or, comparing ade, 



#= — hi; 



from bhk and —cik, we get 



k=bhz=ic, 



