Extensions of Quaternions. 263 



where h, under S'\ receives only «— 4 values, being distinct 

 from each of the four unequal indices, efgk. 



[18.] By changing e to f'm (95), and attending to the pro- 

 perties of the symbols {fgh), we obtain the expression 



(fg)='Z{fgh){hff); ..... (120) 



where / and g are unequal, and the summation 2 extends from 

 h-=\ to h=n. The term for which A =/ vanishes, and the for- 

 mula (1 20) may be thus more fully written : 



{f9)-{f9eWf) = {f9g){sff] + S(fgh){hff); . (131) 



where the letters efg denote again some three unequal indices, 

 and the summation 2^ is performed as in the foregoing para- 

 graph. But also, by (97) and (100), 



{fy)''iM{eff) = (fee){gee)-^T{feh){ghe); . (122) 



subtracting, therefore, (122) from (121), we eliminate the sym- 

 bol {fg)f and obtain the type 

 II. (fee) {gee) - (fgg){gff)=X { (fgh) [Jiff) -(/eh)(ghe) }; (123) 



which represents in general a system oi n(n — l)(n—2) distinct 

 and homogeneous equations of the second dimension, containing 

 each 2{n—2) terms, and derived by eliminations of the kind last 

 mentioned, from the formulee (95), (97), (100), in a manner 

 agreeable to the analysis of paragraph [14.]. Indeed, it was 

 shown in that paragraph, that the equation 



{fff)={fg), (100), 



though known from earlier and simpler principles to be true, 

 might be regarded as included in (95) and (97) ; but this need 

 not prevent us from ming that equation in combination with 

 the others, whenever it may seem advantageous to do so : and 

 other combinations of them may with its help be formed, which 

 are occasionally convenient, or even sometimes necessary, although 

 all the general results of the elimination of the symbols {fg) are 

 sufficiently represented by the recent type II., or by the for- 

 mula (123). For example, a subordinate type, including only 

 ^n{n—l){n—2) distinct equations, of 2(7* -—2) terms each, be- 

 tween the symbols {fgh), may thus be formed, by subtracting 

 (95) from (97), under the condition that efg shall still denote 

 some three unequal indices ; namely, 



0='S.{ifeh){ghe)-{geh){fhe)}; . . . (124) 



or more fully, but at the same time with the suppression of a 

 few parentheses, which do not appear to be at this stage essen- 

 tial to clearness, 



{fge)(eff+egg) = X(geh.flie-feh.ghe): . (125) 

 this last formula admitting also of being obtained from (122), 



