Extensions of Quaternions, 1^ 



dimension, between the symbols of the form {fgh). And with- 

 out any such elimination, the formula (96) gives immediately 

 ^n^[n — \)[n—2) other equations of the same kind between the 

 same set of symbols ; because after choosing any pair of unequal 

 indices e and g, we may combine this pair with any one of the 

 n values of the index/, and with any one of the n — % values of 

 k, which are unequal both to e and to g. There are therefore, 

 altogether, ^n^[n + l)(7i— 2) homogeneous equations of the second 

 dimension, obtained by comparison of the vector parts of the 

 general formula of association, to be satisfied by the ^n\n — \) 

 symbols of the form {fgh) . 



[15.] To prove now, generally, that when the vector parts of 

 the associative formula are thus equal, the scalar parts of the 

 same formula are necessarily equal also, or that the system of 

 conditions (56) in [9.] is included in the system (57) or (59) ; 

 we may conveniently employ the notations S and V, and pursue 

 the analysis of paragraph [10.], so as to show that the system 

 of equations (65), including (68), results from the system [Q7), 

 including (69) ; or that if the formula (84) be satisfied for every 

 set of three unequal or equal vector-units, i t! t", then, for every 

 such set, the formula (89) is satisfied also. For this purpose, I 

 remark that the formula of vector -association (67),. when com- 

 bined with the distributive principle of multiplication [1.], and 

 of operation with S and V [5.], gives generally, as in quater- 

 nions, the transformation 



V^Vo-T=TSpo— o-S/dt; ..... (101) 



where p, <7, t may denote any three vectors , and the symbol 

 V/oVcrr is used to signify concisely the vector part of the product 

 p X V(<rT) ; whence also we may derive by (41) this other general 

 transformation, 



Y{Y(TT.p) = aSpT-TSpa. .... (102) 

 If then we write 



Y<TT = p', YtP = <j', YpCT = T', . . (103) 



and introduce another arbitrary vector tsr, we shall have 



V/o''sr = crSTtEr — tSct-ct; (104) 



and therefore 



YpYp''GJ = T'ST^ + a'S(T'sy; .... (105) 

 but also 



YpYp''Uj = rzSpp'^p<Sp'^!y', .... (106) 

 whence 



OTS/3/[>' = /o'S/9ts- + cr'Scr'nr + T'ST«r, . . (107) 

 and consequently 



S|0/o'=Scr(7' = STT': (108) 



but this is precisely by (103) the formula of scalar-association 



