About the Book
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1906 edition. Excerpt: ...L is at infinity, and is equivalent to a plane-vector, say to t; hence, substituting in (84), PA = nP, -vP, =?AA-v =-PJv (86) Product of Three Plane-Sects.--By (85) and (83) this must be the square of a volume times the common point of the three planes; or, if A, A, A, A De taken in such manner that P, = AAA. P, =PtP, Pl. P, = AAA. then /VP, = 023.031. 012 = 023.0123.01 = (AAAA)3-A; (87) the suffixes being used instead of the corresponding points. If A be at infinity, the three planes are parallel to a single line, and may be written p, =, eAA, etc-, and then treated as above. Product of Four Plane-Sects.--Let the planes be P ... P, and let A A be the four common points of the planes taken three by three. 0...;?3 may be so taken that P0 = njlp, p3, tc.; then P.P.P.P, = n.nln, n, . 123. 230. 301.012 = 'WWfAAAA)'-(83) Grassmann (1S62), Art. 300. Product of Two Plane-Vectors.--Let 7, and 7, be two planevectors or lines at infinity; let e be parallel to each of them, and e, and e, so taken that 7, = ee, 7, = ee, then 7,7, = ee, . ee, = ee, e, . e =--7,7, (89) because 7, and 7, determine a common direction e, and a parallelepiped of which three conterminous edges are equal to e, e e respectively. Product of Three Plane-Vectors.--Take e e, e, so that ViV.V, = e, e, . e, e, . e, e, = (, )'. (90) The directions e, ... e, are common to the plane-vectors 7, ... 7, taken two by two. Several conditions are given here together which follow from the results of this article. AA =. PA = o, (90 Two points coincide. Two planes coincide. AAA = o, PlPJ, = o, (92) Three points collinear. Three planes collinear. AAAA = AA AA /VW. = Pf, -L, L, = 0, = L, L, -o, (93) Four points coplanar; two Four planes confluent; two lines...
