CHAP. II.] POINTS AND LINES IN A GIVEN PLANE.

And the three equations, of which the following is one,

(b − c) a" − (2b + c + a) ẞ” + (2c + a + b) y′′ = 0,

23

with the relations between their coefficients which are evident on inspection, show (by 23) that we have the three additional collineations, A"B"C", B′′C"A", C"A"B", as indicated by three of the dotted lines in the figure. Also, because we have the two expressions,

we see (by 26) that the two points A", A" are harmonically conjugate with respect to B' and c'; and similarly for the two other pairs of points, B′′, B′′, and c", c", compared with c', a', and with a', B': so that, in a notation already employed (25, 31), we may write,

(B ́A′′C ́A′′) = (C ́′B′′A′B′′) = (A'c′′B'C') =

== 1.

34. If we begin, as above, with any four complanar points, o, A, B, C, of which no three are collinear, we can (as in Fig. 18), by what may be called a First Construction, derive from them six lines, connecting them two by two, and intersecting each other in three new points, A', B', c'; and then by a Second Construction (represented in Fig. 21), we may connect these by three new lines, which will give, by their intersections with the former lines, six new points, A", . . c". We might proceed to connect these with each other, and with the given points, by sixteen new lines, or lines of a Third Construction, namely, the four dotted lines of Fig. 21, and twelve other lines, whereof three should be drawn from each of the four given points and these would be found to determine eightyfour new points of intersection, of which some may be seen, although they are not marked, in the figure.

But however far these processes of linear construction may be continued, so as to form what has been called* a plane

By Prof. A. F. MÖBIUS, in page 274 of his Barycentric Calculus (der barycentrische Calcul, Leipzig, 1827).

geometrical net, the vectors of the points thus determined have all one common property: namely, that each can be represented by an expression of the form,

xaa+ybß + zcy ρ xa + yb+zc

where the coefficients x, y, z are some whole numbers. In fact we see (by 27, 31, 33) that such expressions can be assigned for the nine derived vectors, a', . . . y", which alone have been hitherto considered; and it is not difficult to perceive, from the nature of the calculations employed, that a similar result must hold good, for every vector subsequently deduced. But this and other connected results will become more completely evident, and their geometrical signification will be better understood, after a somewhat closer consideration of anharmonic quotients, and the introduction of a certain system of anharmonic co-ordinates, for points and lines in one plane, to which we shall next proceed: reserving, for a subsequent Chapter, any applications of the same theory to space.

SECTION 4.-On Anharmonic Co-ordinates and Equations of Points and Lines in one Plane.

35. If we compare the last equations of Art. 33 with the corresponding equations of Art. 31, we see that the harmonic group BACA", on the side BC of the triangle ABC in Fig. 21, has been simply reflected into another such group, B'A"c'a", on the line B'C', by a harmonic pencil of four rays, all passing through the point o; and similarly for the other groups. More generally, let OA, OB, OC, OD, or briefly O. ABCD, be any pencil, with the point o for vertex; and let the new ray OD be cut, as in Fig. 22, by the three sides of the triangle ABC, in the three points A1, B1, C1; let also

so that (by 25) we shall have the anharmonic quotients,

CHAP. II.] POINTS AND LINES IN A GIVEN PLANE.

25

and let us seek to express the two other vectors of intersection, B, and y, with a view to determining the anharmonic ratios of the groups on the two other sides. The given equation

(27),

whence we derive (by 25) these two other anharmonics,

y

(CB′AB;) = ' + 2;

y

so that we have the relations,

(BC ́AC1) = 2-y;

(CB'AB1) + (CA'BA1) = (BC'AC1) + (BA'CA1) = 1.

But in general, for any four collinear points A, B, C, D, it is not difficult to prove that

whence by the definition (25) of the signification of the symbol (ABCD), the following identity is derived,

(ABCD) + (ACBD) = 1.

Comparing this, then, with the recently found relations, we have, for Fig. 22, the following anharmonic equations:

and we see that (as was to be expected from known princi

E

ples) the anharmonic of the group does not change, when we pass from one side of the triangle, considered as a transversal of the pencil, to another such side, or transversal. We may therefore speak (as usual) of such an anharmonic of a group, as being at the same time the Anharmonic of a Pencil; and, with attention to the order of the rays, and to the definition (25), may denote the two last anharmonics by the two following reciprocal expressions:

with other resulting values, when the order of the rays is changed; it being understood that

(0. CABD) = (C`A`B`D`),

if the rays oc, OA, OB, OD be cut, in the points c', a', b', d`, by any one right line.

may represent the vector of any point P in the given plane, by a suitable choice of the coefficients x, y, x, or simply of their ratios. For since (by 22) the three complanar vectors PA, PB, PC must be connected by some linear equation, of the form a'. PA + b. PB + c'. PC = 0,

and the proposed expression for p will be obtained. Hence it is easy to infer, on principles already explained, that if we write (compare the annexed Fig. 23),

CHAP. 11.] POINTS AND LINES IN A GIVEN PLANE.

27

we shall have, with the same coefficients xyz, the following expressions for the vectors OP, OP2,

OPs, or P1, P2, P3, of these three points

of intersection, P1, P2, P3:

which give at once the following anharmonics of pencils, or of groups,

whereof we see that the product is unity. Any two of these three pencils suffice to determine the position of the point P, when the triangle ABC, and the origin o are given; and therefore it appears that the three coefficients x, y, z, or any scalars proportional to them, of which the quotients thus represent the anharmonics of those pencils, may be conveniently called the ANHARMONIC CO-ORDINATES of that point, P, with respect to the given triangle and origin: while the point P itself may be denoted by the Symbol,

P = (x, y, z).

With this notation, the thirteen points of Fig. 21 come to be thus symbolized: