508 The Rev. T. P. Kirkman on Pluquaternions* 



in the first place. If Z m be the function of w-plets, n being 

 the number of different subindices, Q^Z m can yield only w-plets 

 and (w— l)plets, since every imaginary in Q, can be made to 

 coalesce with some one having the same subindex in every 

 term of Z m , as in the products 6 3 A and c 3 A, above considered. 



The number of irreducible e-plets in the product of e plu- 

 quaternions of 6n+ 1 imaginaries is obtained, by writing out 

 all the 3 C permutations that can be made with repetitions of 

 c, e, g, taken e together, and then under each permutation 

 writing every possible combination of e different ascending 

 indices that can be made with 1 2.. . (n — 1). The first 

 letter, if c, may then be exchanged for b, if e for d, if g forf, 

 in every permutation, which doubles the number of them. 



The no-plets made with ceg are the two first terms in & ( ; 



whence 



^. *-*-n-H-l. ..»-—. i = ,. (1+3)n=2 „ tl) 



is the number of terms in the product of n pluquaternions of 

 (6//+ 1) imaginaries. This, at the most (I apprehend that, 

 in general, this exactly), is also the number of terms in the 

 product of n pluquaternions of (6n— 1) imaginaries. 



if we now suppose the addition of two imaginaries more, 

 making 6« + 3, it is clear that the product of e pluquaternions 

 of 6?i + 3 will exhibit all the e-plets already found, besides 

 those made by the combination of all our former (£—1) plets 

 with one of the two new monads. This, reckoning the two 

 new one-p\ets, doubles our former result, giving the number of 

 terms = 2 2 " +2 ; or the number of terms in the complete pro- 

 duct of n pluquaternions of 6n— 3 imaginaries is 2 2n . 



The reasons above given will shield me from any charge 

 graver than that of a pardonable credulity, if I confess my 

 belief in the existence of this complete pluquaternion product : 

 and I hope that, instead of incurring the blame of presumption 

 in touching upon questions which can be completely discussed 

 only by far better analysts, I may even earn the thanks of the 

 mathematical reader for pointing out the connexion between 

 such a product, and the remarkable properties of products of 

 sums of squares, to which it must conduct us. For whatever 

 is obscure, unfinished, or even illogical, I trust to receive every 

 indulgent allowance to which the confessed difficulty of these 

 subjects may entitle those, who, while they are not forbidden 

 to speak on them, are yet not expected to bring to their dis- 

 cussion the powers of mind possessed by such writers as the 

 distinguished inventor of quaternions, or to exhibit his brilliant 

 results in rich and varied applications. 



