1895.] 



Mr. A. McAulay. Octonions. 



169 



Aristodesmus, which suggests this link, is at present placed in the 

 Procolophonia, a group separated from, its recent association with 

 Pareiasaurus and restored to its original independence, because it has 

 two occipital condyles, with the occipital plate vertical, and without 

 lateral vacuities, and has the shoulder girdle distinct from Pareia- 

 sauria in the separate pre-coracoid extending in advance of the 

 scapula. 



III. 44 Octonions." By Alex. McAulay, M.A., Lecturer in 

 Mathematics and Physics, University of Tasmania. Com- 

 municated by Kev. N. M. Ferrers, D.D., F.R.S. .Received 

 November 28, 1895. 



(Abstract.) 



Octonions is a name adopted for various reasons in place of 

 Clifford's Bi-quaternions . 



Formal quaternions are symbols which formally obey all the laws 

 of the quaternion symbols, q (quaternion), x (scalar), p (vector) 

 (linear function in both its ordinary meanings), 0' (conjugate of 0), 

 *» h ^> -K-S'j Tgs Ug, Yq. Octonions are in this sense formal 

 quaternions. Each octonion symbol, however, requires for its 

 specification just double the number of scalars required for the 

 corresponding quaternion symbol. Thus, of every quaternion formula 

 involving the above symbols there is a geometrical interpretation 

 more general than the ordinary quaternion one, an octonion interpre- 

 tation. The new interpretation, like the old, treats space impar- 

 tially, i.e., it has no special reference to an arbitrarily chosen origin 

 or system of axes. 



If Q is an octonion and q a quaternion, the symbols, which in 

 octonions correspondjto K^, Sg, Tq, ~Uq, Vq in quaternions, are denoted 

 by KQ, SQ, TQ, UQ, MQ. KQ is called the conjugate of Q, SQ the 

 scalar octonion part, TQ the augmenter, UQ the twister, and MQ the 

 motor part. 



If A is a " motor " whose axis intersects the axis of Q perpendi- 

 cularly, QA is also a motor intersecting Q perpendicularly. UQ.A is 

 obtained from A by a combined translation along and rotation about 

 the axis of Q; i.e., by a " twist " about the axis of Q. TQ.A is 

 obtained from A by increasing the " rotor " part of A in a definite 

 ratio, and by increasing the " pitch " of A by a definite addition. 

 UQ = U X Q U 2 Q, where when Q is thus regarded as an operator, UxQ 

 is a " versor," i.e., it effects the rotation mentioned ; and U 2 Q is a 

 " translator," i.e., it effects the translation mentioned. Similarly, 

 TQ = TxQ.T 2 Q where TiQ is a "tensor," i.e., it effects the ratio- 



VOL. LIX. N 



