I 73 .] 



IV. 



THE DIFFERENTIATIOlSr OF QUATERNION FUNCTIONS. 

 By K. T. WANG. 



[Read May 8. Published July 13, 1911.] 



Differentials of quaternion functions of algebraical and transcendental forms 

 containing no quaternion constants are precisely analogous to those of ordinary 

 or scalar functions, if dq///q. But in the general case, without the coplanarity 

 dqlllq, the differentiation becomes far more complex. I have never yet seen 

 any author deal with much of it. 



But recently I found that there is a general expression for the differentiation 

 of such functions. 



In the general case dq can always be resolved into two components : the 

 one is a quaternion jjlq, and the other a vector ± Axq ; the perpendicular 

 vector component can be expressed in the form 



- Vq. V{Vdq: Vq), 



and the coplanar quaternion component, 



dq+Vq. V{Vdq: Vq). 



For every set of values of q and dq there is only one set of components which 

 satisfies the above definition. 



Let F{q,dq) denote the differential oifq; hence it is a linear function of 

 dq : it is distributive, or 



F{q,dq) = F \q,dq + Vq . V {Vdq : Vq)\ + F [q,- Vq . V {Vdq : Vq)]. 



Because dq + Vq . V(Vdq : Vq)///q and its functions, containing no 

 quaternion constants ; so, similar to the special case dq///q, 



F {q, dq+Vq. V {Vdq : Vq)] =f'q .[dq+Vq. F( Vdq : Vq)\ 



(/' signifies the ordinary form of derivative of/), and it is also Ijjq oi'fq. 



The other term F {q, - Vq . V(Vdq: Vq)], though it is not so simple 

 as the first one, is always a sum of products of coplanar quaternions and 

 the vector, that ± their axis, - Vq . V{Vdq: Vq). All such products are 

 vectors -L the axis of the quaternions ; therefore the sum, i.e. 



F\q,-Vq. V{Vdq: Vq)], 



is also a vector ± Axq or Ax/q. 



R.I. A. PKOC, VOL. XXIX., SECT. A. [lOJ 



