500 The Rev. T. P. Kirkman on Pluquaternions, 



for Q'4 in the second line, and Q b in the third above written 

 line, are forms of Q„, which have (a— b) zero constituents; 

 and Q' c in the last written line is a form of Q b . Hence, if 

 ltt£ + 9U„ be the sum of the squares of the real quantities in 



(to w 2 terms). 



Leta = 7 = &=c= . . =n; then B^ = 0, and we have the 

 theorem, 



* The product of any two sums, each qfSn squares, is always 

 reducible to 8n 2 squares. 



(ttt, 5 -f-2Sj 5 ) is always forty-six squares, which is therefore 

 not the least number to which the product of two sums of 

 sixteen squares is reducible. It is thus proved also, that 

 (Jc not < ; h not < 0, not > 2). 



The product of ' any two sums, each ofw (6k + 2h + 8) squares, 

 is reducible to a sum of n 2 [(3kf + 3k(2h + 5) +/z*(A+7)+8] 

 squares. 



Let now 



? = 2?« + 7 = 6 , w+l— 2e, 



where e is either =2 or =0. We know that 

 Q'.Q.=<a t + R ra , 



ll m being of the second degree as to its imaginaries. Let 



^m i A m , . . . . L m , Lt m , 



represent functions of the third, fourth, .... n— lth, wth de- 

 gree as to their imaginaries, the functions having no real term ; 

 that is, let S m be a sum of terms such as b Q i p B^, and T m 

 of terms such as c Q l r Q v Q Ci n . Then, paying attention only 

 to the form of the products in the second members of the equa- 

 tions which follow, 



* This theorem and its demonstration were suggested by the following 

 proof, kindly communicated to me by its discoverer Prof. J. R. Young, that 

 the product of two sums of sixteen squares is reducible to a sum of thirty- 

 two squares; a proof certainly simpler and more elegant than that of the 

 same property given in the text, and containing, in fact, the whole theorem. 



Let 2 8 q denote a sum of eight squares : 



V 2 8 q2 s q'=z2 e q" .'. 

 2i 6? % 6 «f = (2 8 y + 2 8 q') < V+ 2?"') 



=2 8 q2 8 q"+2 s <j%q">+ 2 8 q%q» +2 e q2 s q'» 



=2 8 q, + 2 8 7/ ( + Stf/M + Stf/i/i 



=2 32 y. 



