Extensions of Quaternions. 289 



CONSTANTS ARBITRARY ill this theory of ASSOCIATIVE QUINES. 



We may indeed vary in many ways, consistently with the same 

 general theory, and by the assistance of the other recent types VI. 

 and VII., the system of the sixteen equations of condition which 

 are to be satisfied, and the choice of the eight constants which 

 are to be regarded as still remaining arbitrary and undetermined : 

 and it may not be useless, nor uninteresting, to make some 

 remarks hereafter, upon the subject of such selections. But in 

 the mean time it appears to be important to observe, that if 

 some of the recent results, especially the formulae (210), (228), 

 be combined with some of those previously obtained, and more 

 particularly with the equations (112), (121), of Section III., the 

 following very simple expressions are found, for the ten remain- 

 ing constants of multiplication, the discussion of which had been 

 reserved : 



(/)=*/; (f<j)=v f .v e ; ..... (251) 

 or, with the notations abc, and with the usual cyclical permuta- 

 tion of the indices 1, 2, 3, 



a,=»i 2 , « 4 =»4 2 , b l =v 2 v 3 , c l =v 1 v 4 . . . (252) 

 If then we write for abridgement, 



v = Vjfej + ivt'a + %? 3 + v 4 x 4 , \ 



v' = v x x' , + v<^\ + v 3 x' 3 + v 4 %' 4 , J 

 the square of any quadrinomial vector •&, and the scalar of the 

 product of any two such vectors, will take these remarkably 

 simple forms : 



^- = v 2 ; Svnx' = v.v'; .... (254) 

 this latter scalar thus decomposing itself into a product * of two 

 linear functions of the constituents, namely those here denoted 

 by v and v 1 . And because it is easy to prove, from what has 

 been already shown, compare (244), that in the present theory 

 the constants v e are connected by relations of the form 



-v e .efe=v e .fee = v f .eff+v g .efc/ + v h .efk, . (255) 



we find, by multiplying this equation by v g , and attending to 

 (251 ), the following theorems for those general associative quines 

 which have been in this section considered: 



* A similar decomposition into linear factors takes place for the quadri- 

 iioiiics I A) of par. [l.'l], but at the expense of one of the six arbitrary con- 

 stants / : /., l t /,/, m , Hij, when we establish between those symbols the relation, 

 him* + l 2 ni.>- -f ? s m s s = /,/,/ 3 + •2m 1 ?ii. 2 m s . 



Tn general, I find that it is possible to satisfy all the conditions of asso- 

 ciation for polynomes, and at the same time to secure a decomposition of 

 Sbjot' into linear factors, while yet preserving so many as3n— 4 constants 

 of multiplication arbitrary. (For quaurinomee, fti'— 4—9— 4—5$ for 

 quines, 3» — 4=12 — 4=8.) 



Phil. Mag. S. 4. Vol. 9. No. 59. April 1855. U 



