Extensions of Quaternions. 127 



the sum in the left-hand member of the formula (54) reduces itself 

 to the term {efk) : but such is also in this case the value of the 

 right-hand sum in the same formula, because in calculating that 

 sum we need attend only to the value A = ^, if ^ be still =0. 

 In like manner, if /=0, each sum reduces itself to {egk) ; and if 

 e = 0, the two sums become each ={fyk). If then any one of 

 these three indices, e, /, g, be = 0, the formula (54) is satisfied : 

 which might indeed have been foreseen, by observing that, in 

 each of these three cases, one factor of each member of the equa- 

 tion (52) becomes =1. We may therefore henceforth suppose 

 that each of the three indices, e, /, ff, varies only from I to w, 

 or that 



e>0, />0, ff^O; .... (55) 



while k may still receive any value from to n, and h still varies 

 in the summations between these latter limits: and thus the 

 number of equations, supplied by the formula (54), between the 

 constants {fyh), is reduced, as was lately stated, to (n-}'l)n^; 

 while the number of those constants themselves had been seen 

 to be reduced to {n + l)n^, by the same supposition Cq = 1. 



[9.] Additional reductions are obtained by introducing the 

 law of conjugation (32), or by supposing K . Lft,g=Lg(,f, with the 

 consequences already deduced from that law or equation in [5.]. 

 Using S' to denote a summation relatively to h from 1 to n, and 

 taking separately the two cases where A: = and where A;> 0, we 

 have, for the first case, by (54), 



^\efh){gh)^%<{fgh){eh); (56) 



and for the second case, 



(ef){gQk)-{fg){eOk)=X<{(efh)(ghk) + [fgh){ehk)}. (57) 



No new conditions would be obtained by interchanging e and g ; 

 but if we cyclically change efg to fge, each of the two sums (56) 

 is seen to be equal to another of the same form; and two new 

 equations are obtained from (5 7), by adding which thereto we find, 



0=l,<{{efh){shk) + {fgh){ehk) + {geh)(fhk)}; . (58) 

 and therefore, 



{fg){eOk)-{ef){gOk) = -^'{geh)ifhk) (59) 



When e=/, the equations (56) and (59) become, respectively, 



0^1'{fh){fgh), (60) 



and 



{fg){fOk)-(f){ffOk)=l.'{gfh){fhk); . . (61) 



which are identically satisfied, if we suppose also f=g ; the pro- 

 perties [5.] of the symbols (fgh) being throughout attended to : 

 while, by the earlier properties [2.], the symbol (eO^) or {Oek) is 



