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



containing h or i Q , and one of the seven imaginaries a b c 

 d e ofogo> none of which duads are in the system of seven 

 triplets. 



From what has preceded, we conclude that it is impossible 

 for pluquaternions of more than seven imaginaries to have a 

 product of the same form, if there are no limitations upon the 

 values of the real quantities, ww p aa t . . . .rr t . 



If we now suppose that there are eleven imaginaries, we 

 can form a fifth triplet beginning with a oi viz. a k l ; and 

 it can be proved, as in the case of nine imaginaries, that b Q l , 

 c ohi d l oi e l ,f l Q , g l o , have each of them an equivalent 

 duad, but that they cannot any of them be put equal to any 

 single imaginary, without a contradiction. The equal values 

 of a , h i Q and k l , give also 



of which neither h l Q nor h Q k can be equal to any of the 

 seven a b c Q d e o f o g ; for this equality is what we have just 

 shown to be impossible ; nor can h Q l be equal to either i Q or 

 k , nor h k Q to either i Q or l . Wherefore the product of two 

 pluquaternions of eleven ( = 2*2 + 7) imaginaries must of ne- 

 cessity be of the form 



where the condition-function R 2 contains, besides the six terms 

 of R 15 the six following, 



+ b l B l + c l C l + d l T> l + eJ E l +f l F lo +g l G lo , 



with the additional pair of terms +h l U l + i l I l . 



The six B,,, &c. differ from the six B,-, &c. only in this, that 

 for the real quantity h or i in the latter, is substituted k or / 

 in the former ; and, in the same way, for h t or i, is put k t or l r 

 Hi and I ( are formed from equations (D.), and k u and / /y , the 

 two additional terms in Q fl// , with the aid of the triplet a k l . 



It can also be shown, as in (C), that the additional duads 

 in R 2 , b Q l Q , &c, have the property that b o l o +l o b o =0. 



Suppose next that we have thirteen ( = 2'*3 + 7) imaginaries. 

 There is now nothing to hinder uc from forming another system 

 of seven triplets, with ahiklmn, of the same type or not as the 

 former. We may, e. g. have the fourteen triplets following: 



+ abc + ahi 



+ ade — bdf+ cdg + akl—hkm -f ikn 

 + a fS + b e g + ce f> + amn + M n + M m 5 

 for, although we have proved that no one of the seven ab . . .g 

 can form a tenth triplet with any two of hikl, such a triplet can 



