Extensions of Quaternions. 47 



a x . . a 4 , and of the six other symbols b x . . c 3 , when values of 

 /j i . «g have been found, which satisfy the eighty conditions. 

 And then, if we denote the quine itself by the following expres- 

 sion (compare [1.]), 



which is a little more symmetric than the form (111), the laws 

 of multiplication of any two such quines, P, P', will be sufficiently 

 expressed by the formulae 



tf=a lt i* = a 4 , Si 2 i 3 =b u Si } i 4 -c v ^ 



Yi 2 t 3 =i 1 l l + i 2 m 3 + i 3 n^+t 4 p 1) j. . (159) 



Vtjt 4 = i l r l + £ 2 Sj + tgrfj + i 4 u x j J 



if we remember that 1, 2, 3 may still be cyclically permuted, 

 and that the law of conjugation (32) gives 



Ki'i = u', Si'i=Su', Vl'c=-Yh'. . . (160) 

 For in this manner, by (41), if «r denote, as in (14), the vector 

 part of P, so that 



©■=*,#, + t 2 a? 2 + %Z3 + ^4, .... (161) 

 we shall have 



Sctct' = a 1 x l xS 1 + a^x^x'^ + o 3 x 3 x' 3 + a 4 x 4 x' 4 



+ b x (x 2 x' 3 + x 6 x'^ + &c. + c x {x x x' 4 + x 4 x x ') + &c, (162) 

 VW = ( t, /j + i 9 m 3 + c 3 n 2 +i 4 Pi) {x Q x' 3 — x 3 xj) + &c . 



+ (t 1 r 1 + i 2 s 1 + t 3 £ 1 + t 4 w,)(« 1 «' 4 — # 4 ^j) + &c., . (163) 

 each " &c." representing terms obtained by the permutations 

 already mentioned ; and if the constants abclmnprstu have been 

 chosen so as to fulfill the conditions above developed, we may 

 then conclude (compare (51)) that the following equations of 

 association hold good, for the multiplication of any three such 

 vector-units 1, or quadrinomial vectors vs, or quinquinomial expres- 

 sions P, whether equal or unequal among themselves : 



i.i<J' = u' .l"; <B.v'v» = mm , .<a"; P. P'P" = PP'.P" ; (164) 



which it has been the main object of our recent investigations to 

 establish. 



[27.] Without pretending to do more, on the present occa- 

 sion, than merely to exemplify the possibility of satisfying, for 

 quines, the foregoing equations of association, I may here remark 

 that if we restrict the question by assuming (with the usual 

 permutations), 



(A, B)*... «, = >»„ Pl = 0, m, = 0, . (165) 



* This line is lettered thus, because it contains the conditions common 

 to the two systems (A) and (li) of associative (mines, which are deduced a 

 little further on. 



