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



m /y =mw / + wm y + na / — an^ob/ — boy + pc^— cp / + dh / ~hd / 



+ ei / -ie / + kf / -fk / + lg y -g] /5 

 n // =nw y + wn / +am i — may + bp^— pb ; + 0^—00/4- id/— di ; 



+ eh / -he / + fl y -lf / + kg y -gk / , 

 o u = o w y + wo, + pa y — ap/ + bm / — mb y + en, — nc y + kd y — dk, 



+ el/-le y + hf y -fh y + gi / -ig /5 

 p yy =pw y + wp y + ao y — oa y -}-nb y — bn y -f cm y — mc y + dl y — ld y 



+ ek y -ke y + fi y -if y + gh y ~hg y . 



The bracketed terms, of which there is one, and one only, 

 introduced by each bisignal, must severally vanish, giving the 

 seven conditions, 



h :h y =i : i y =k : k y = l : l y =m : m y = n : n y =o:o y = p : p y , 



as those which must be fulfilled in order that our results may 

 be consistent with our definitions ; and these seven conditions 

 ought to be sufficient, if our reasoning is valid, in order that 



(w 2 + a 2 -fb 2 + c 2 + d 2 -fe 2 + f~ + g 2 + h 2 + i 2 + k 2 + l 2 + m 2 + n 2 

 + o 2 + p 2 ) 

 X ( w y 2 + a 2 + b y 2 + c y 2 + d y 2 + e y 2 + f y 2 + g/ 2 + h/ 2 + i, 2 + k, 2 + l, 2 



+ m 2 + n 2 + o / 2 + p 2 ) 

 should be identical with 



w /y 2 + a / 2 + b / 2 + c/4-d / 2 -fe // 2 + f/ + g/ + h /y 2 + i / 2 + k/ + l / 2 

 + m / + n/ + o/ + p/; 

 or that the product of two sums, each of sixteen algebraic 

 squares, should be a sum of sixteen algebraic squares. 



That they are sufficient, is evident from inspection, and is in 

 fact a known truth, discovered by Prof. J. R. Young, and esta- 

 blished by ocular demonstration in the memoir already referred 

 to; although the author gives no account of the process by 

 which he arrived at the result. Should the reader hesitate to 

 accept this conclusion on the strength of my arguments, from 

 a very natural suspicion that the latter are more to be admired 

 for their luck than their learning, I trust that he will allow 

 the result to be an apology for the reasoning, and be lenient 

 to the logic for the conclusion's sake. Is there not room to 

 indulge the hope, that when the theory of these imaginaries 

 is perfectly understood, as instruments of the algebra of time 

 and order, what is here offered on bisignal univalents may 

 amount to something better than fortunate nonsense and two 

 useless words? 



Let us now consider two pluquaternions, each of twenty- 

 three imaginaries, viz. the fifteen before us and the eight 



