Extensions of Quaternions. 266 



consequence, by simple additions, of the form ulse (95) and (97). 

 Thus, after assigning any two unequal values to the indices/ 

 and ff, we see that the two symbols, {fy), [gf) ; the two others, 

 r^], [gf] ; the n—2 symbols, [fge] ; and the n — 2 symbols, 

 Igfe], are indeed all equal to each other: but that the 2n—l 

 equations between these 2n equal symbols are connected by a 

 relation, such that any 2n — 2 of them, which are distinct among 

 themselves, include the remaining one; and that therefore, after 

 the elimination of [fg) and {gf), there remain only 2(n— 2) 

 distinct equations of condition, as was otherwise shown in [14.] . 

 But, in that paragraph, we proposed to form those resulting con- 

 ditions on a plan which may now be represented by the formulae 



U9] = U9e'\, [^/] = [^M; • • • 033) 



whereas we now prefer, for the sake of the convenience gained 

 by the disappearance of certain terms in the subtractions, to 

 employ that other mode of combination, which conducted in 

 [18.] to the formula (123), and may now be denoted as follows : 



U9l = lgM, [9n = if9e'\. ■ ■ ■ (134) 

 Summing these last with respect to e, we find 



(«-2)[/^]=2'[^/«]> {»-2)[^/]=2'[^«]; . (135) 



and therefore, by the identity (132), 



('»-3)[<;/] = (»-3)[/^] (136) 



If, then, n^S, we are entitled to infer, from (123) or (134), 

 the following formula, which is equivalent to (126), 



Q?/] = M; (137) 



and therefcre also by (134) this other type, equivalent to (124)> 

 LM^lfffil (138) 



which includes n—2 equations, when/ and ^ are given, and 

 conducts, reciprocally, by (132), to (137). In general, therefore, 

 if we adopt the type (134), we need not retain also either of these 

 two latter types, (137), (138). But in the particular case where 

 n=S, that is, in the case of quadrinomes, the identity (132) 

 reduces the two equations (134) to one, after/ and ^ have been 

 selected ; and with this one we must then combine either of the 

 two equations (137) or (138), which in this case become iden- 

 tical with each other. 



[20.] In particular, for this case of qudrinomials (^^ = 3), we 

 have with the notations (114) (128), the four following values 

 for (23), or for ^j (compare again a note to [12.]) : 



