50 On some Extensions of Quaternions. 



m~ l , m~ l , and 0, respectively. This coefficient X vanishes also, 

 when we equate x [ \x l \d ' 3 x' 4 to a?, x 2 x 3 x 4 respectively; and hence, 

 or from (179), we may infer, 5th, that "in this system of 

 quines (A), the square of every quadrinomial vector vanishes." 

 And finally, by an easy combination of the formulae (182) (185), 

 or of the 3rd and 4th of the foregoing properties of this system, 

 we see, 6th, that, in it, " every product of three quadrinomial 

 vectors vanishes ;" or that 



(A 7 ).. W.-=j"=0, zt.tz'v" = (186) 



[29.] The associative property (164) is therefore verified for 

 the system (A), by showing that, in it, each of the two ternary 

 products of vectors, which ought to be equal, vanishes. In the 

 system (B), it is easy to see that any such ternary product must 

 be itself a vector; because, in (B), no binary product of vectors 

 involves i 4 , nor does any such product involve a scalar part, 

 except what arises from i 4 . We have, therefore, here, this new 

 result, 



(B 3 ).. S(W. OT ") = S( CT .wV) = 0. . . . (187) 



And when we proceed to develope these two ternary products, 

 the associative property of multiplication is again found to be 

 verified, under the form, 



(B 4 ).. ^u'.^" = ^.w'^" = a 4 (^x' 4 x" 4 -is'x" 4 x 4 + ^"x 4 x' 4 ); (188) 



where it is worth observing that, by the laws of the system in 

 question, the result may be put under this other and somewhat 

 simpler form : 



(B 5 ).. otct '. ot "= ct .-57' ct "=wS ot ' ot "- ot 'Sot"ot+ot"S OT ot'. (189) 



Indeed, this last expression might have been foreseen, as a con- 

 sequence from the general principles of this whole theory of asso- 

 ciative pohjnomes*, combined with the particular property (187) 

 of the quines (B) . For, by that property, each of the two ter- 

 nary products is equal to its own vector part; but by (101) we 

 have, generally, in the present theory, as in the calculus of qua- 

 ternions, the following expression for the vector part of the pro- 

 duct of any three vectors, of any such associative polynomes as we 

 are considering : 



V . /30-t=joSctt — a8rp + rSpa; . . . (190) 



* It will hereafter be proved generally that for all associative polynomes 

 which satisfy the law of conjugation (though not exclusively for such asso- 

 ciative polynomes), the tensor, as defined in [6.], is also a modulus; which 

 theorem can be verified "without difficulty for the quines (A) and (B), and 

 for the quadrinomes and tetrads so lettered in [13.], as well as for the 

 irinomes [11.]. 





