294? Rev. T. P. Kirkman on Bisignal Univalent Imaginaries. 



giving 



bc = hi 9 

 or, comparing abc 9 



a = hi; 



a pair of results that can be reconciled on no less violent sup- 

 position, than that the duad imaginary hi retains its value 

 when it changes its sign : and I have shown, at page 450 of the 

 thirty- third volume of this Magazine, that no one of the 

 twenty-seven triplets following ahi can be admitted along with 

 it, if we interpret each to express three such conditions as 

 a=hi 9 h=ia 9 i=ah. It is there proved impossible to make 

 consistently the 15x7 suppositions, such as ab=c 9 ac=—b 9 

 ad=e 9 &c, which are necessary to enable us to eliminate the 

 duads ab 9 ac 9 ad 9 &c. from the product of two pluquaternions 

 of fifteen imaginaries, 



Q 15 =w + tfa+6b + cc + dd + £e+/f+gg + #h + n + &k + Zl 



-f mm + nn + oo +pp 9 

 and 



Q'15 = w / + aa i + bb t + . . +pp { , 

 so as to reduce the product to the form 



Q"is = w u + a H + bh u + • • • +jPVir 

 From this consideration may be framed a proof of a celebrated 

 negative, which has not yet lost its interest, although the 

 question has been some time ago set at rest by a master of 

 analysis, who has achieved the laborious task of establishing 

 the negative by ordinary algebra. See a memoir in the 

 Transactions of the Royal Irish Academy, vol. xxi., " On an 

 extension of a Theorem of Euler, &c, by J. R. Young, Pro- 

 fessor of Mathematics in Belfast College," a gentleman in 

 whose recent and most cruel wrongs every cultivator of science 

 in this empire has been bitterly insulted. The proposition 

 meant is, that " Two sums each of sixteen arbitrary algebraic 

 squares cannot have for their product a sum of sixteen alge- 

 braic squares." 



Let it be supposed, for a moment, that they can, or that 



(w 2 + a 2 + b 2 +...+p^)(w / 2 + a / 2 + b / 2 +... + p / 2 ) 

 = (w/ + a // 2 + b/+... + p/), 



the number of squares in each factor being sixteen. The 

 function 



w 2 + a 2 + b 2 +... + p 2 



is the product of two pluquaternions, 



Qi 5 =w + aa + 6b+...-}-^p, 



