296 Rev. T. P. Kirkman on Bisignal Univalent Imaginaries. 



pendent of its sign, provided that we take care in our result, 

 to reduce to zero every term which is affected by the value or 

 sign of A. 



Let me beg the reader's forbearance for a little while, and 

 his permission to lay down, for brevity's sake, the definition 

 following : — 



A bisignal univalent, or more briefly, a bisignal, is a quan- 

 tity whose value is independent of its sign ; and such that it 

 may be employed in the same argument with contrary signs, 

 and yet remain always equivalent to one defined value. 



In real arithmetic there can be no bisignals, except zero 

 and its reciprocal ; in imaginary arithmetic we may establish 

 any number, each having a value neither null nor infinite, 

 if they are not suffered to affect any finite terms of our results. 



We shall then venture to define, that all the twenty-eight 

 duads, hi, hk, hi, &c, made with the imaginaries hiklmnop are 

 bisignal imaginaries, whose values are given by the twenty- 

 eight lower triplets : and we have now the power of asserting, 

 at one point of our argument, that a — hi and not = ih; and 

 again, at another point of the argument, that a = ih and not 

 = hi. Through all this we must retain the definition that 

 hi + ih = Q 9 which is the common property of all our imagi- 

 naries, by an extension of Sir W. ft. Hamilton's definition of 

 the three imaginaries of his quaternion theory. I am con- 

 vinced that it is in vain to attempt to prove this property, at 

 least in the case of those imaginaries which are not combined 

 into a congruous system of seven. We are to conceive, then, 

 that whenever we change for our convenience the sign of hi, 

 we change by the same supposition the sign of ih\ and that 

 these obliging duads are always of contrary signs. 



By this definition of the twenty-eight duads, hi, &c, we 

 remove all contradiction among our triplets, if only the reader 

 will grant that the definition itself is not a flat contradiction. 

 There is no need that we should swallow the absurdity that 

 hi is of two opposite signs at once, i. e. in the same circum- 

 stances: when we are about to compare, as in a preceding 

 paragraph, the four triplets dhm, eim, ade and ahi 9 we establish 

 the proposition a = ?h-~ —hi ; and the result of our comparison 

 agrees with this: at another moment, in other circumstances, 

 when we compare together the four triplets bhk, —cik, abc and 

 ahi, we first lay down that a = hi=—ih, and our way is clear 

 again. We are in a condition to bid defiance to every diffi- 

 culty about the comparison of the first seven triplets with the 

 lower group of twenty-eight ; and we proceed without fear to 

 the formation of our sixteen functions, w^a^b,, &c, which are 

 to appear on the right side of the equation 



