124 
33. We have further: 
(a). A.pq=A:V.pq=A-.Vp-.Vq—V: Vp Vq=A. eee 
(b). A. pKq=—V.pKq= —2A.Sp.q—A. pq. 
(ce): (Se poe — F. Kp.q=2A.Sp.q—A.pq. 
Note.—These and similar formulas are useful in computing alternates of 
higher order. Thus a factor pq of a product may be replaced by A. pg [th 4] or 
any of its equivalent values in (a) with or without the partial A. 
34. Resolve q into q/ + q” respectively parallel and perpendicular to p. 
Then A.pKg—A.pKo’ =p. KY [th6, art 31,]. Its tensor is therefore base X 
altitude of parallelogram on q, p, as sides. We call A. p K q the vector area of the 
parallelogram (q, p). It gives plane, direction and tensor, by the plane, direction 
of turn, and tensor of the vector. Observe that A.pq—V. Vp Vq is perpen- 
dicular to the three-space (1, Vp, })’q) —(J, p, ¢). 
35. The alternates of third order are: 
(a). A.Vp.Vq.Vr=A.A. Vp. Vq. Vr=A. 8. V p. Vg Vr— 
SVp.Vq.Vr. We call this scalar — a. 
(b). 3A.Sp.qr=38A.Sp.Vqr=—bi+e7-+ dk. 
The four independent scalar alternates of third order are a, b, ¢, d, respec- 
tively the coefficients of ¢ (i, j, k), @ (1, j, k), 0 (1, k, 7) © (1, 7, j) in the expansion 
of any linear alternate ¢ (p, q, r). 
36. We have further: 
(a) A.pqr=A.Vp.qr+A.Sp.qr 
(b). S.A. pqr=8.Vp.Vq.Vr=S.pAqr—SA.pAgqr— 
4S(pAqr+qA rp+rApq)ete. 
(c). V.A.nqr=A. 8.9. qr=—<A. pe 8ger des 
4V.(p.Aqr+qArp+rA pq) ete. 
(d) A.p.Kq.r=—S.Apqr=3.V.Apqr 
(e). A.Kp.g.Kr=—KA.p.Kq.r=SA.pqr—3V.Apqr 
=S.pAqr—3A.Sp.Aqr. 
Note.—This alternate is Shaw’s A. pqr, and his formulas hold in the present 
notation with this value of his A4.pqr. In the present notation a function that 
is used as a variable must be enclosed in brackets. Thus A [S p] q —0, where the 
S follows the p, but 4. Sp.q isnotzero. Similarly, Shaw’s value of A.pq.Arst 
becomes A. Kp.q. K[A. Kr.S.Kt]=6A.r.Sps. Sqt. 
37. Resolve r into 1’ +r” respectively parallel and perpendicular to the 
plane of p, g, and then A. p. Kq.r—=(A.p. Kq).r”, whose tensor is base X alti- 
tude of the parallelopiped on p, q, 7 as edges. We call this the quaternion volume 
of parallelopiped. It will be shown that this is a line perpendicular to the edges 
