March 2, 1899] 



NATURE 



411 



treatise written thirty years /2^«tt? by the young electrician 

 of to-day be as much up to date iken as is " The 

 Application of Electricity to Railway Working " in 

 iSqq? "P. D." 



OCTOmONS. 

 Octonions ; a Development of CliffonVs Bi-quaternions. 

 By Alex. McAulay, M.A. Pp. xiv + 253. (Cambridge : 

 University Press, 1898.) 



FROM a purely formal point of view, apart from any 

 question of geometrical or physical application, 

 the mathematical method known as quaternions may be 

 described as a system of shorthand for dealing with the 

 algebra of certain complex numbers. 



Let/,y, k be three independent entities which obey the 

 relations 



ij = - ji 



!;, jk = - kj =i, ki = - ik =j. 



(i.) 



and those derived from them ; and let w, x,y, s be any 

 four real numbers. Then the totality of complex numbers 

 of the form 



IV + xi + yj + zk 



evidently constitute a self-contained system ; in the 

 sense that the result of combining two or more such 

 numbers by addition or multiplication is another number 

 of the system. Moreover, it may be easily shown that 

 the result of dividing any number of the system by 



«■ -F xi -F yj -I- ~J; 

 is a definite number of the system unless w, x^ y and ' 

 are all zero. 



Quaternion analysis is a method of shorthand, and an 

 extremely compendious one, for dealing with this system 

 of complex numbers. 



Hamilton himself considered, under the name of bi- 

 quaternions, an extension of this particular algebra in 

 which each real number lo, . . . , is replaced by 



7f , -I- W.j \' - I , . . . , 

 where rt'j, "w.^, • • ■ , are real numbers. 



This is equivalent to dealing with the self-contained 

 system 



^'■1+ '"i''+J'i,/ + =i'(- + ».2"-l-.r,>-t-7„/w-t-c.,^'a. .... (A) 

 in which /, /, /(•, to obey the relations (i.) and the further 

 relations 



^, = .,/,y» = <^, ^. = 0,^. ^..j 



Clifford introduced two distinct extensions of the 

 algebra of quaternions. In each of them the complex 

 number is of the form (.A). In one, /, j, k, a obey the 

 relations (i.)and (ii.), except that the last equation of (ii.) 

 is replaced by 



In the other, i\j\ k, a> again obey the relations (i.) and 

 {ii.) with 



0)- = o 



in the place of the last equation of (ii.). To both of these 

 algebras Clifford gave the name biquaternions. 



It may be noticed that the formal algebra of Hamilton's 

 biquaternions is quite independent of the supposition 

 that o) is the v - i of ordinary algebra; it depends 

 purely on the laws implied by (i.) and (ii.). 

 NO. I 53 I, VOL. 59] 



The three algebras thus obtained are the only distinct 

 extensions of the algebra of quaternions that result from 

 introducing a single new unit or entity which is permut- 

 able with i, j and k, while its square is an ordinary real 

 number. 



What one may call the geometrical counterpart of 

 quaternion algebra is the geometry of rotation round 

 a fi.xed point, and the parallelism between the algebraical 

 and the geometrical theory is complete. To the general 

 complex number in the algebra corresponds the most 

 general operation on rotations round the point, viz. the 

 operation which will change any one such rotation into 

 any other. There are also geometrical theories standing 

 in the same relation to the three extended algebras, each 

 containing as a part, as it should do, the theory of rota- 

 tion round a fi.^ed point. 



It was, in fact, from the geometrical side that Clifford 

 approached the subject in his published writings. His 

 point of view may be presented briefly as follows. 



A velocity system in space {i.e. the mode in which a 

 rigid body is moving at any instant) is completely 

 specified by an axis AB, the magnitude a of the velocity 

 of rotation about AB and the magnitude V of the 

 velocity of translation along AB. From the doubly- 

 infinite set of operations which will change any velocity 

 system given by AB, a, V into any other given -by A'B', 

 a', V, a particular one may be chosen as follows. Let 

 CD be the common perpendicular to AB and A'B' ; and 

 let n = pa, and V = qW There is a definite twist with 

 CD for its a.xis which will bring AB to A'B', and at the 

 same time the direction of V along AB to agreement 

 with the direction of V" along A'B'. The operation which 

 changes the one velocity system into the other may be 

 made up of (i.) this twist, (ii.)an operation which merely 

 changes the magnitude of the rotation velocity in the 

 ratio / to I, (iii.) an operation which changes the mag- 

 nitude of the translation velocity in the ratio ^ to i ; and 

 these three may be carried out in any order. The 

 operation involves in its specification eight distinct 

 numbers, since a twist involves six. 



Having thus obtained a definite view of the operation 

 which changes one velocity system into another, Clifford 

 goes on to discuss the laws according to which such 

 operations combine. These of necessity depend on the 

 nature of the space in which the motions take place. He 

 only glances very briefly at the case of ordinary Euclidean 

 space, and develops the theory, so far as he carries it, for 

 elliptic space. He shows, in effect, that the formal laws 

 involved for elliptic space are those of the extended 

 quaternion algebra, for which 



«- = I. 

 The carrying out of the theory for hyperbolic space, in 

 which case the formal laws are those of the extended 

 quaternion algebra where 



or = - I, 

 Still awaits treatment. 



Prof McAulay's book deals with the theory for ordinary 

 space, which is found to correspond to the remaining 

 case, viz. 



0,2 = o. 



An oclonion (the author gives reasons for preferrmg 

 this word to biquaternion) is in fact, from the algebraical 



