116 Mr. F. L. Hitchcock on the 



the only kind of differentiation implied in the first two terms 

 on the right is the direct application of nabla ; the other 

 term implies ordinary differentiation along the direction Yq, 

 or, if we prefer to put dr=fdp, so that/is the linear function 

 obtained by differentiating r completely, we may rewrite the 

 last term as —2fYq*. 



Second, suppose we wish to obtain, instead of the operator 

 SVgV, which is the scalar product of a vector and nabla, the 

 corresponding vector operator YYq\/, useful in the trans- 

 formation of line and surface integrals j\ Arranging the 

 work so as to bring in sucli a term, 



\Jq — Sq . V + KV^V, as before, second line of (4 A), 

 = Sq . V + (1-2V)V?V, because K = S-V = 1-2V, 

 = (S? + V</)V-2VV?V, identically, 

 = gV-2VV ? V, • (4B) 



and by substituting in (1 B), 



S7{qr)=\/q.r-rqVr-2VYqV.r, . . (IE) 



which is in the form we started to obtain. 



Third, suppose we Avish to obtain that form of expansion 

 which shall most clearly exhibit the quaternion equation as a 

 generalization for three dimensions of the theory of ordinary 

 complex quantities represented on a plane. We shall then 

 keep nabla always at the left, and, instead of interchanging 

 V and q, in the last term of (1 B) , shall invert the order of 

 the two quaternions q and r ; the steps are not unlike those 

 in(4B):— 



= SrSg + Yr . $q -f KYrYq, scalars being commutative, 



= Br . q + Yr . $q + (1 - 2 V) YrYq, because K = 1 - 2Y, 



=. Sr .q + Yr. (Sj + Yq) -2YYrVq, identically, 



= rq-2YYrY.q, (3 B) 



another formula for inverting two quaternions, giving by (IB) 



V(?r) = V^ . r + \Jr . q - 2VYYr'Yq, . (IF) 



which is in the form we started to obtain, for if q and r are 



* By putting two vectors for q and r, and by operating with S or V 

 either before or after such a substitution,we may obtain a number of special 

 cases very useful in practice. For another quite different method of 

 obtaining them see Phil. Mag. [6] No. 18, June 1902, p. 580. 



t See, for instance, the physical applications in Tait's ' Quaternions, * 

 Art. 497 et seq. 



