Rev. T. P. Kirkman on Bisignal Univalent Imaginaries, 299 



h'i'k'l'm'n'o'p', all having the property h l2 =i ,2 =,..=p ,2 = — 1, 

 and such that m'n 1 -f- ?i'm' = for every pair ; and ask how many 

 conditions are to be added to the seven already found in order 

 that Q 23 .Q' 23 = Q' / 23 . If we add to the groups of triplets A 

 and B the following group, 



fak'i' bh'k' -ch'l' dh'm) eh'ri -fh'o f gh'p' 

 J ak'l' — bi'l' —ci'k' —di'n' ei'm' fi'p' gi'o' 



I am'n' -\-bm'o' cm'p' —dk'o' ek'p' —fk'm 1 —g/c'n' 

 K-adp* —drip 1 cji'o' dl'p 1 eVo f fVii —gVm\ 



we shall obtain, as before, seven congruous values of each of 

 the eight h'i'k , l'm'n , o'p / ; and by defining every duad imaginary 

 made with these as bisignal, we shall remove all contradiction 

 from the values of abcdefg deduced from the groups A and B'. 

 The equation to be satisfied, Q 2 3-Q , 23 =: Q"235 or 

 ( w + aa + 6b + . . +/f -\-gg + hh + h'h' + i\ + i f ¥ + kk 

 + l'k , + ....+pp+p'p') 

 X (w i + aa k + bb l + .. +A+gg, + hh t + h'h, 1 + i\ i + *'i/ + kk t 

 + #k/ + .... +pp,+rfpl) 



= w u + a H + bh u + • • +fit +gSu + hh u + h,h! u + H + *V + k K 

 +#k f u + r . , f +pp u +p'p' ip 



requires that the sixty-four duads, hh' hi' hk' &c, made each 

 from both groups hi...p and h'i'...p', should be eliminated or 

 otherwise disposed of. Elimination by substitution of a monad 

 imaginary is impracticable ; for hk 1 e. g, cannot be equal to any 

 of the seven abcdefg, nor to any of the eight hik . .p, nor to 

 any of h'i'k . .p', without introducing a linear relation between 

 two of our twenty-three imaginaries, contrary to hypothesis. 

 Thus the supposition hk'=f, combined with either of the ex- 

 isting conditions —ho=f, m'k'=f 9 would give a linear relation 

 between two monads. These sixty-four duads must then per- 

 force appear in the product Q^Q/gg, unless they are made to 

 vanish by null coefficients. From our definition of h'i', h'k', &c. 

 as bisignals, it will follow, as before in the case of hi, hk &c. 

 and their coefficients, that the seven equations 



h'th'^i'ri'^k^k'^.^p'rp', 

 must be satisfied; but there is no necessity, from these con- 

 ditions, that the ratios h : h y and h' : h' t should be equal. Now 

 the duad hk' will introduce into the product of Q 23 and Q' 23 , 

 with one sign or other, the term hk',— k'h /5 and no more. 

 This will not vanish, unless h : h y = k' : k'^h' : h', ; but does 

 vanish, if the ratios h : h i and h' : h' y are equal. As this equa- 

 lity of ratios, being admitted, will cause all the sixty-four duads 

 hh' 9 hi', hk'> &c. to disappear, we have our product Q^.Q^s 



