1895.] 



Octonions. 



177 



a given energy function, the following n real independent motors, 

 forming a conjugate set with regard to sr, can always be found, and 

 in general uniquely ; (1) Ai . • . . «i . . . • , positive norms with regard to 

 w; (2) B x . . . . fa.. , negative norms with regard to w ; (3) Ci . . . . , 

 zero norms with regard to w. The numbers in the two groups «i . , . . , 

 fa . . . . are the same. The motors A : . . . . , B x . . . . ,Ci . . . • , a.\ + fa.. , 

 ai—fa,..., form a conjugate set (not norms in general) with regard to 

 yjr t and of these aii—fa...., are zero norms with regard both to w and 

 ty. When there are any a' s and /3's there are not n real independent 

 motors forming a common conj ugate system of w and yjr. If is a 

 complete energy function there are no a's and /3's. Several particular 

 cases of the general theorem are examined, those of especial importance 

 being when «r = 1. 



If -ar and w' are two general self- conjugates such that sE-srE = 

 sEVE for every motor E of a given complex, then sEwF = sEisr'F 

 where E and F are any two motors of the complex. In particular, 

 any motors of the complex which form a set conjugate with regard to 

 sr are also conjugate with regard to -sr'. 



If (n) is a given complex of order n and (6— n) a given independent 

 complex of order 6— n, and if -ar is given then -set' can be determined 

 uniquely, so that sE^Ei = sE'-srEj, -sr'E 2 = where Ex is any motor of 

 (n) and E 2 any motor of (6 — n) . Also, whatever motor value E 

 have -sr'E belongs to the complex reciprocal to (6— n). 



If is a general (not necessarily self- conjugate) function, such that 

 it reduces every motor of (n) to a motor of (w), and reduces every 

 motor of (6 — n) to zero, satisfies an «-tic. If the roots of this equa- 

 tion are a repeated r times, b repeated s times, &c, where a, 6, ... . 

 are all different, there are certain definite independent complexes (r), 

 (s), .... included in and making up (n), of orders (r), (.§),.... cor- 

 responding to these roots. In (r) is included a complex (_p), (not 

 always definite) of order p, where p is any positive integer not 

 greater than r, such that (0— a)^A = for every motor A of (p). 



If p be any positive integer and e be a scalar different from both a 

 and b, (0— e)i>E is not zero, and belongs to the complex consisting of 

 (V) and (s) if E itself belongs to that complex. A similar statement 

 is true of the complex consisting of any number of the complexes (r), 



(«),.... 



If (0— e)P~E = (p any positive integer) for every motor E of a 

 complex of order q included in (n), e is a root of the %-tic repeated at 

 least q times, and the complex is included in the complex correspond- 

 ing to this root. 



(0— b) s (0— -cY . . . . E where all the roots occur their full number 

 of times, except a which is absent altogether, and where E is a motor 

 of (n) belongs to the complex (r), and by giving a suitable value to E 

 it may be made any motor of (r). 



