1895.] 



Octonions. 



173 



If X is root of the cubic that cubic can always be put in the form 

 (0 — X)(0 2 — N'0+N) = 0, even when there is no second root. If X 

 corresponds to a single root of the 0! cubic (0 2 — N'0-fN)E = 0, if E 

 is any motor which intersects a certain line perpendicularly. If X 

 corresponds to a repeated root of the X cubic, the statement is not 

 always true. 



Except when all the roots of the 0! cubic are equal there is always 

 some line such that if E be any motor coaxial with the line, 0E is also 

 coaxial with the line. [When all the roots of the 0i cubic are equal 

 the statement is sometimes true and sometimes untrue.] 



Let, now, stand for a real commutative self-conjugate. It can 

 then always be put in the form — 



0E= ^XiSm-X'jSEj-X"kSEk, 



where X, X', X" are three real scalar octonions, and i, j, h are three 

 mutually perpendicular unit intersecting rotors. The cubic is 

 (0— X)(0— X')(0— X") = 0, so that it always has three real roots. 

 These may be taken as X, X', X", even when it has an infinite number 

 of roots. In this last case the X, X', X" of the equation 0E = 

 — XiSEz— .... have definite values which are called the principal 

 roots of the cubic. Thus, there are always three mutually perpen- 

 dicular intersecting lines such that if E be a motor coaxial with any 

 one of them, 0E is coaxial with E. 



If two, but not three, of the roots of the X cubic are equal, and 

 the two corresponding principal roots of the cubic are unequal, 

 cannot be put in the form — 



0E = MAEB + YE, 



where A and B are constant motors and Y a constant scalar octo- 

 nion. In all other cases can be put in this form, and A, B, and Y. 

 are real. The cubic is 



{(0_Y) 2 -A 2 B 2 }{(0-Y) + SAB} = 0. 



When neither A nor B is a lator and they are not parallel, the prin- 

 cipal axes (the three lines mentioned just now), of are the shortest 

 distance of A and B, and the two lines which bisect this shortest 

 distance and also bisect the angles between A and B. 



A scalar octonion may be of any one of five types ; one type is zero 

 and the other four may be called positive and negative, scalar octo- 

 nions and positive and negative scalar convertors. If A and B are 

 two motors (not parallel and neither a mere lator), such that SA0B, 

 = 0, A and B are said to be fully conjugate with regard to 0. If 

 A, B, C are three such motors of which each pair is fully conjugate 

 the number of the scalar octonions SA0A, SB0B, SC0C, of any 

 type is the same as the number of the principal roots of the same 



