CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. B 13 a plane AOB, in which the (now actual) vector, represented by the sum aa + bẞ, is situated. For if, for the sake of symmetry, we denote this sum by the symbol-cy, where c is some third scalar, and y = oc is some third vector, so that the three co-initial vectors, a, ẞ, y, are connected by the linear equation, aa + bB + cy = 0; B Α' Fig. 15. C then the two auxiliary points, a' and B', will be situated (by 19) on the two indefinite right lines, OA, OB, respectively: and we shall have the equation, so that the figure A'Oв'c is (by 6) a parallelogram, and consequently plane. 22. Conversely, if c be any point in the plane AOB, we can draw from it the ordinates, CA' and CB', to the lines OA and OB, and can determine the ratios of the three scalars, a, b, c, so as to satisfy the two equations, = after which we shall have the recent expressions for oA, OB', with the relation oc OA'+ OB' as before; and shall thus be brought back to the linear equation aa+b+cy = 0, which equation may therefore be said to express the condition of complanarity of the four points, 0, A, B, C. And if we write it under the form, xa + yẞ + zy = 0, and consider the vectors a and ẞ as given, but γ as a variable vector, while x, y, z are variable scalars, the locus of the variable point c will then be the given plane, oAB. 23. It may happen that the point c is situated on the right line AB, which is here considered as a given one. In that AC case (comp. Art. 17, Fig. 13), the quotient must be equal AB to some scalar, suppose t; so that we shall have an equation of the form, t, or y = a+ t (B-a), or (1−t) a + tẞ − y = 0 ; and under this last form it expresses a geometrical relation, which is otherwise known to exist. 24. When we have thus the two equations, aa+bB+cy = 0, and so that the three co-initial vectors a, ß, y terminate on one right line, and may on that account be said to be termino-collinear, if we eliminate, successively and separately, each of the three scalars a, b, c, we are conducted to these three other equations, expressing certain ratios of segments: or a (a − y) + b(ẞ − y) = 0 ; 0 b. ABC. AC c. BC + a. BA = a.cA+b.cв. = Hence follows this proportion, between coefficients and seg ments, a:b:c= C = BC CA: AB. CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 15 We might also have observed that the proposed equations give, a= whence I 25. If we still treat a and ẞ as given, but regard y and will express that the variable point c is situated somewhere on the indefinite right line AB, or that it has this line for its locus: while it divides the finite line AB into segments, of which the variable quotient is, AC y = CB x Let c' be another point on the same line, and let its vector be, then, in like manner, we shall have this other ratio of seg If, then, we agree to employ, generally, for any group of four collinear points, the notation, may be said to denote the anharmonic function, or anharmonic quotient, or simply the anharmonic of the group, A, B, C, D: we shall have, in the present case, the equation, 26. When the anharmonic quotient becomes equal to negative unity, the group becomes (as is well known) harmonic. If then we have the two equations, the two points c and c' are harmonically conjugate to each other, with respect to the two given points, A and B ; and when they vary together, in consequence of the variation of the value of y they form (in a well-known sense), on the indefinite right line AB, divisions in involution; the double points (or foci) of this involution, namely, the points of which each is its own conjugate, being the points A and B themselves. As a verification, if we denote by u the vector of the middle point м of the given interval AB, so that μ so that the rectangle under the distances MC, Mc', of the two variable but conjugate points, c, c', from the centre м of the involution, is equal to the constant square of half the interval between the two double points, A, B. More generally, if we then in another known and modern phraseology, the points c and c' will form, on the indefinite line AB, two homographic divisions, of which A and B are still the double points. More generally still, if we establish the two equations, * See the Géométrie Supérieure of M. Chasles, p. 107. (Paris, 1852.) CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 17 y being still constant, but variable, while a' = oa', B' = OB', and y' = oc', the two given lines, AB and A'B', are then homographically divided, by the two variable points, c and c', not now supposed to move along one common line. 27. When the linear equation aa + bẞ+ cy=0 subsists, without the relation a + b + c = 0 between its coefficients, then the three co-initial vectors a, ẞ, y are still complanar, but they no longer terminate on one right line; their term-points A, B, C being now the corners of a triangle. In this more general case, we may propose to find the vectors a', B', y' of the three points, C respectively opposite sides. The three collineations OAA', &c., give (by 19) three expressions of the forms, where x, y, z are three scalars, which it is required to determine by means of the three other collineations, A'BC, &c., with the help of relations derived from the principle of Art. 23. Substituting therefore for a its value a1a', in the given linear equation, and equating to zero the sum of the coefficients of the new linear equation which results, namely, and eliminating similarly ß, y, each in its turn, from the original equation; we find the values, |