and Homoid Products of Sums ofn Squares. 449 



so as to have no duad in common, and that generally in various 

 ways (vide Camb. and Dub. Math. Journ., N.S. vol. ii.p. 191), 

 upon the comparison of which it is not necessary here to enter, 

 although the restriction might thus be further narrowed. 



When w = 2, or when there are only three imaginaries, Q„ 

 is the quaternion invented by Sir W. R. Hamilton, and pro- 

 foundly discussed by him in the twenty-fifth volume of the 

 Philosophical Magazine. When » = 4, it is the biquaternion 

 or octad, of which Professor John T. Graves has made men- 

 tion at the end of a paper On the Connexion between the 

 General Theory of Normal Couples, and the Theory of 

 Quadratic Functions of two Variables, ibid. vol. xxvi. As it 

 will be useful to have a name for Q a in cases where n > 4, I 

 venture to propose the appellation Pluquaternion, as generally 

 applicable to Q„ when there are more than three imaginaries ; 

 not dreading here the pluperfect criticism of grammarians, 

 since the convenient barbarism is their own. 



Further, ifc , a Qi b Q be any three imaginaries forming one 

 of the triplets, c a Q b cannot differ in sign from a b c ; for 

 if it does differ, 



c o ff o 6 o=-«o 6 o c o=+£ c o fl o=-- c o fl! cA; Q-E.A. 



•*• c o a o & o = fl, o b o c o=^o c o »o= + 1 i 



which is the property of every triplet of the system. 

 From these we deduce 



«cA=± c o> 

 c a Q =±b Q9 



*o c o=±«o5 



whence also 



«o^o c o«o = c o 6 o = a o«o 6 o c o=~ & o c o» 

 Co«o&o c o=& «o= c o c o«o 6 o=--«cA> 

 *o c o"o ft o = «o c o = Mo C o«o=- C o«o; 



for, because b c a , z=a b c Qi is real, 



«o 6 o- c o«o = a o- & o- c o a o = fl! o^o c o fl! o= : «o- fl! o i o c o= flr o- c o-Vo : 



and so the other two of these three assertions may be esta- 

 blished likewise. Wherefore 



™ o n o + n ° m o = 



is true of every duad m n that can be made with the(2« — 1) 

 imaginaries in the equation Q a Q a ==Q a/ , by virtue of our 

 definitions. 



Let Q„ Q a; and Q a/I be biquaternions, that is, let there be 

 seven imaginaries a b Q c d Q e f g Q . The number of triplets 



