Rev. T. P. Kirkman on Bisignal Univalent Imaginaries, 295 



and 



Q_ 15 = w— aa— bb — ..,~pp, 



Q_ 15 denoting a function differing from Q 15 only in the signs 

 of the fifteen imaginaries abc.p, which have the properties 

 above defined. We have then, as an equivalent form of the 

 above equation, 



QlsQ— 15^ 15^ -15 = Q 15Q -15 J 



or, since Q'i 5 Q'-i 5 is real, 



Q15Q 15^ -15Q-15 = Q 15Q -15« 



The left member of our supposed equation is thus divided 

 into a pair of factors, Q 15 Q\ 5 and Q_i 5 Q'_i 5 , which differ only 

 in the signs of the fifteen imaginaries; and since no such pair 

 of factors can be found to produce the right member, different 

 in form as to the imaginaries from Q" 15 and Q"_ 15 , we must 

 have 



Ql5Q 15 = Q 155 

 W,-15Q-15—Q -15> 



congruous equations, considered as functions of our imagina- 

 ries: but both these are proved above to be contradictory. 

 The negative proposition seems thus established : but all this 

 involves the petition of this other negative, that our factors Q 

 cannot be formed with less than fifteen monad imaginaries. 

 Why not with nine monads and six duads, as at page 453 of 

 vol. xxxiii. ? 



Let us return to our refractory triplets, and try to induce 

 the imaginaries to submit to the violence above hinted at. If 

 in any argument it suits our convenience, for the sake of 

 clearness or symmetry, to retain a term AZ, in which A is 

 not infinite, and Z is zero, we may exhibit it at different points 

 of the process with different signs ; for AZ is a quantity which 

 changes not its value when we reverse its sign. Suppose that 

 A is not a symbol of real quantity, but given in value by the 

 equation A — B, when B is an imaginary having one value 

 only : if there is no term in our reasoning that is affected by 

 our definition of A, except AZ, we may, on reviewing our 

 process, in which AZ appears with contrary signs at different 

 points, conceive that the property, which the term has, of re- 

 taining its value when its sign is changed, is conferred on it by 

 A, considered to be, whether affected by a positive or negative 

 sign, constantly equivalent to + B : and evidently, if it facili- 

 tates the establishment of relations between A and other like 

 quantities, considered apart from real numbers, we may en- 

 dow A by definition with the property, that its value is inde- 



