Interpretation of Quaternions. 109 



although obtained by an entirely independent process. I have 

 stated my views as briefly as possible, because Prof. Donkin's 

 paper renders any more lengthened discussion superfluous; 

 and if any expressions occur which appear to indicate a view 

 of algebraic symbols, &c. different from his, they have been 

 used merely because they are the ordinary terms ; and I should 

 wish them to be understood as far as possible in his way, if 

 for no better reason, at least in order that the two methods 

 may be compared. 



The calculus of quaternions is a generalization of algebra, 

 in which sets of four ordinary algebraical quantities are used 

 instead of single quantities. Each such set of four quantities 

 is called a quaternion; the nature and laws of combination of 

 which are the object of the present investigations. The cor- 

 responding laws in ordinary algebra will be assumed as known. 



Let a quaternion be defined to be a set of four algebraic 

 quantities considered with reference to their order of position, 

 and let it be expressed by the following equation, 



Q = (w, x, y, z\ (1.) 



in which Q, or its equivalent, is called the quaternion, and 

 w, x, y, z its constituents. As this definition involves no law 

 of connexion between the constituents, it is clear that the equi- 

 valence of any number of quaternions must involve the equi- 

 valence of their several constituents; so that the equations 



Q=Q 1 =Q 8 = (2.) 



involve the following, 



X — X I — Xq — — • • • I 



z= z l =z 2 =...^ 



and conversely (3.) will involve (2.). The same principle 

 gives rise to the following law for the addition and subtraction 

 of quaternions: 



2Q;=(2»»„, 2*„ %„, 2.g; . . . (4.) 



particular cases of which are 



wQ = (wu 1 , nx, Jiy, nz) (5.) 



Q_Q = 0=(0, 0,0,0) (6.) 



The following consideration will assist further investigations. 

 The quaternion 



K 0,0,0) (7.) 



is a system consisting of the quantity W, followed by no other 



