506 The Rev. T. P. Kirkman on Pluquatemions, 



These results, formidable as they appear in extent, and in 

 the labour which would be required to verify them, are capable 

 of demonstration, if it be true that every multiplet imaginary 

 is equivalent, either with or without a change of sign, to every 

 permutation of itself; and if it be allowable to substitute, in 

 any multiplet, for any two contiguous imaginaries which be- 

 long to the same system of seven triplets, that monad from the 

 same system to which their product is equivalent. 



I will not undertake to demonstrate that the first is true, 

 and that the second is allowable. I will not even say that the 

 argument is conclusive whereby I have endeavoured to prove 

 that, in any imaginary triplet, 



b • np e= n* pb m p • bn. 



It is certain, however, that to deny this property of odd mul- 

 tiplets in general, implies a contradiction. 



Let it be supposed, for the present, that it is possible to 

 determine the signs of all the permutations of any irreducible 

 multiplet, or that the signs of certain of them may be so as- 

 sumed, as congruously to fix those of all the rest. 



Let the i ( = 6n-\- l = 2m + 7) imaginaries be a and then 

 sixes, bcdefg, b x c x ..g v Va-'&j' • v *»-iC»-i...&-i, with 

 which n similar systems of seven triplets are constructed, a 

 being the imaginary common to all the systems. 



Let A =f x c 3 c 4 g 6 e 7 be any quintuplet and irreducible imagi- 

 nary arising in the product of five pluquatemions of the order i. 

 The multiplication of these by any sixth pluquaternion will 

 include the operation Q.A, of which it will be here sufficient 

 to attempt {b + c + b x +c x +b, 2 + Cc l + b 3 + c 3 )A: the triplets being 



abc ab x c x .... c x e x f x , 



ade - bdf+ cdg ab# % .... c 2 ^/ 2 



ofg + beg + cef y ab 3 c 3 .... c 3 e 3 f 3i &c. 



The equal values of a give a series of conditions 



b» Cfi =■ Co, bji be ejs = c a dp> 



b u bp, = —Co, c/i b u dp, = —C* eg, , &c. 



If A were/jCg, a duad in R m , the operation b'f x c 3 , which 

 by definition of J\c 3 is b-f x 'c 3i would be of doubtful sign ; for 

 there is no reason why b'f x c 3 should be considered either as 

 b'f\H o r as bf x 'c& the one rather than the other; and it has 

 been shown, that bf x 'c 3 — —b'f v c Si is a consequence of 



b 'fi c 3= : fi'C3b=c 3 'bf 1 . 



In all probability, the sign of b' A, or b-f x c 3 c 4 g 6 e v when 



