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LXVI. On PluquaternionS) and Homoid Products of Sums of 

 n Squares. By the Rev. Thomas P. Kirkman, A.B., Rector 

 of Croft with Southworth, Lancashire*. 



THE following analysis is the fruit of my meditations on 

 Professor Sir W. R. Hamilton's elegant theory of qua- 

 ternions, and on a pregnant hint kindly communicated to me, 

 without proof, by Arthur Cayley, Esq., Fellow of Trinity 

 College, Cambridge, about the connexion between a system 

 of triplets having no duad in common, and the property that 

 the product of two sums of In squares shall be a sum of 2n 

 squares. In a note with which the latter gentleman recently 

 favoured me, he writes, " The complete test of the possibility 

 of the product of 2 ra squares by 2 ra squares reducing itself to a 

 sum of 2 n squares, is the following : — Forming the complete 

 system of triplets for (2 n — 1) things, if eab ecdfacfdb be any 

 four of them, we must have, paying attention to the signs only, 



( ± eab) ( ± ecd) = ( ±fac) ( ±fdb) ; 

 that is, if the first two are of the same sign, the last two must 

 be so also, and vice versa. I believe that, for a system of seven, 

 two conditions of this kind, being satisfied, would imply the 

 satisfaction of all the others. It remains to be shown, that the 

 complete system of conditions cannot be satisfied for a system 

 of fifteen things." 



In the following investigation, the truth and completeness 

 of Mr. Cayley's test of the property in question for sums of 

 2" squares are established, and the contradiction anticipated 

 by him for a system of fifteen is deduced ; while ulterior results 

 about homoid products of sums of n squares are obtained. 



Let a Q b c . . . r be In— 1 imaginary units, among which 

 no linear relation exists, and let 



a*=b *=...=r *=-l. 

 Let 



Q a =w+a a+b b+... +r Q r, 



Q a =w l +a a l + b b l + . . . + r Q r ti 



ww l aa t . . . rr t being any real quantities : it is proposed to con- 

 sider under what conditions it is possible that the product of 

 two such functions shall be of the same form. Suppose, then, 

 that, if possible, 



Q«Q* = Qa„= Vo«//+M// + • • • + r o r ifl 

 * Communicated by the Author. 



