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and determine the mean point F of this system, there will in general be a set of fifteen lines, of the kind above considered, all passing through this sixth point F: and these will be arranged generally in fifty-five distinct planes, whereof twenty-five will be what we have called triple, the thirty others being of the non-triple kind.
97. More generally, if a, .. a, be, as before, a system of m given and co-initial vectors, and if a1, am be any system of m given scalars (17), then that new co-initial vector ẞ, or OB, which is deduced from these by the formula,
may be said to be the Complex Mean of those m given vectors a, or OA, considered as affected (or combined) with that system of given scalars, a, as coefficients, or as multipliers (12, 14). It may also be said that the derived point B, of which (comp. 96) the position is independent of that of the origin o, is the Barycentre (or centre of gravity) of the given system of points A considered as loaded with the given weights a1...; and theorems of intersections of lines and planes arise, from the comparison of these complex means, or barycentres, of partial and total systems, which are entirely analogous to those lately considered (96), for simple means of vectors and of points.
(1.) As an Example, in the case of Art. 24, the point c is the barycentre of the system of the two points, A and B, with the weights a and b; while, under the conditions of 27, the origin o is the barycentre of the three points A, B, C, with the three weights a, b, c; and if we use the formula for p, assigned in 34 or 36, the same three given points A, B, C, when loaded with ra, yb, zc as weights, have the point P in their plane for their barycentre. Again, with the equations of 65, E is the barycentre of the system of the four given points, A, B, C, D, with the weights a, b, c, d; and if the expression of 79 for the vector or be adopted, then xa, yb, zc, wd are equal (or proportional) to the weights with which the same four points A.. D must be loaded, in order that the point P of space may be their barycentre. In all these cases, the weights are thus proportional (by 69) to certain segments, or areas, or volumes, of kinds which have been already considered; and what we have called the anharmonic co-ordinates of a variable point P, in a plane (36), or in space (79), may be said, on the same plan, to be quotients of quotients of weights.
(2.) The circumstance that the position of a barycentre (97), like that of a simple mean point (96), is independent of the position of the assumed origin of vectors, might induce us (comp. 69) to suppress the symbol o of that arbitrary and foreign point; and therefore to write" simply, under the lately supposed conditions,
* We should thus have some of the principal notations of the Barycentric Calculus: but used mainly with a reference to vectors. Compare the Note to page 56.
CHAP. III.] BARYCENTRES OF systems of points.
It is easy to prove (comp. 96), by principles already established, that the ordimate of the barycentre of any given system of points is the complex mean (in the sense above defined, and with the same system of weights), of the ordinates of the points of that system, with reference to any given plane: and that the projection of the barycentre, on any such plane, is the barycentre of the projected system.
(3.) Without any reference to ordinates, or to any foreign origin, the barycentric ΣΠΑ
notation B= may be interpreted, by means of our fundamental convention
(Art. 1) respecting the geometrical signification of the symbol B-A, considered as denoting the rector from A to B: together with the rules for multiplying such vectors by scalars (14, 17), and for taking the sums (6, 7, 8, 9) of those (generally new) vectors, which are (15) the products of such multiplications. For we have only to write the formula as follows,
Σα (Α - Β) = 0,
in order to perceive that it may be considered as signifying, that the system of the vectors from the barycentre B, to the system of the given points A1, A2, . . when multiplied respectively by the scalars (or coefficients) of the given system a1, a2,.. becomes (generally) a new system of vectors with a null sum: in such a manner that these last vectors, α1. BA1, α2. BA2... can be made (10) the successive sides of a closed polygon, by transports without rotation.
(4.) Thus if we meet the formula,
B = (A1 + A2),
we may indeed interpret it as an abridged form of the equation,
OB = (OA1 + OA2);
which implies that if o be any arbitrary point, and if o' be the point which completes (comp. 6) the parallelogram 110^20', then в is the point which bisects the diagonal oo', and therefore also the given line A1A2, which is here the other diagonal. But we may also regard the formula as a mere symbolical transformation of the equation,
(A2-B) + (A1 − B) = 0;
which (by the earliest principles of the present Book) expresses that the two vectors, from B to the two given points A1 and A2, have a null sum; or that they are equal in length, but opposite in direction: which can only be, by B bisecting A1A2, as before. (5.) Again, the formula, B1=(A1 + A2 + A3), may be interpreted as an abridgment of the equation,
OB1 = (OA1+0A2 + OA3),
which expresses that the point B trisects the diagonal oo' of the parallelepiped (comp. 62), which has OA1, OA2, OA3 for three co-initial edges. But the same formula may also be considered to express, in full consistency with the foregoing interpretation, that the sum of the three vectors, from в to the three points A1, A2, A3, vanishes: which is the characteristic property (30) of the mean point of the triangle A1A2A3. And similarly in more complex cases: the legitimacy of such transformations being here regarded as a consequence of the original interpretation (1) of the symbol B – A, and of the rules for operations on vectors, so far as as they have been hitherto established.
SECTION 6.-On Anharmonic Equations, and Vector-Expres
sions, of Surfaces and Curves in Space.
98. When, in the expression 79 for the vector P of a variable point P of space, the four variable scalars, or anharmonic co-ordinates, xyzw, are connected (comp. 46) by a given algebraic equation, f(x, y, z, w)=0, or briefly f=0,
supposed to be rational and integral, and homogeneous of the ph dimension, then the point P has for its locus a surface of the pth order, whereof f=0 may be said (comp. 56) to be the local equation. For if we substitute instead of the co-ordinates x.. forms,
w, expressions of the
to indicate (82) that P is collinear with two given points, P1, P1, the resulting algebraic equation in t: u is of the pth degree; so that (according to a received modern mode of speaking), the surface may be said to be cut in p points (distinct or coincident, and real or imaginary), by any arbitrary right line, PP. And in like manner, when the four anharmonic co-ordinates lmnr of a variable plane II (80) are connected by an algebraical equation, of the form,
F(l, m, n, r) = 0, or briefly F= 0,
where F denotes a rational and integral function, supposed to be homogeneous of the qth dimension, then this plane II has for its envelope (comp. 56) a surface of the qh class, with F= 0 for its tangential equation: because if we make
l=tlo + ul,... r = tr1 + uri,
to express (comp. 82) that the variable plane II passes through a given right line II, II, we are conducted to an algebraical equation of the qth degree, which gives q (real or imaginary) values for the ratio t:u, and thereby assigns q (real or imaginary†) tangent planes to the sur
* It is to be observed, that no interpretation is here proposed, for imaginary intersections of this kind, such as those of a sphere with a right line, which is wholly external thereto. The language of modern geometry requires that such imaginary intersections should be spoken of, and even that they should be enumerated: exactly as the language of algebra requires that we should count what are called the imaginary roots of an equation. But it would be an error to confound geometrical imaginaries, of this sort, with those square roots of negatives, for which it will soon be seen that the Calculus of Quaternions supplies, from the outset, a definite and real interpretation.
+ As regards the uninterpreted character of such imaginary contacts in geometry, the preceding Note to the present Article, respecting imaginary intersections, may be consulted.
CHAP. III.] ANHARMONIC EQUATIONS OF SURFACES.
face, drawn through any such given but arbitrary right line. We may add (comp. 51, 56), that if the functions ƒ and F be only homogeneous (without necessarily being rational and integral), then
[D1f, Dyf, D-f, Dwf]
is the anharmonic symbol (80) of the tangent plane to the surface ƒ=0, at the point (xyzw); and that
(DF, DF, DF, D,F)
is in like manner, a symbol for the point of contact of the plane [Imnr], with its enveloped surface, F = 0; D,, . . D¿, .. being characteristics of partial derivation.
(1.) As an Example, the surface of the second order, which passes through the nine points called lately
is a symbol for the tangent plane, at the point (x, y, z, w).
(2.) In fact, the surface here considered is the ruled (or hyperbolic) hyperboloid, on which the gauche quadrilateral ABCD is superscribed, and which passes also through the point E. And if we write
R= = (Oyz0),
P = (xyzw), Q = (xy00), then qs and RT (see the annexed Figure 31), namely, the lines drawn through P to intersect the two pairs, AB, CD, and BC, DA, of opposite sides of that quadrilateral ABCD, are the two generating lines, or generatrices, through that point; so that their plane, QRST, is the tangent plane to the surface, at the point P. If, then, we denote that tangent plane by the symbol [Imnr], we have the equations of condition,
0 = lx + my = my + nz = nz + rw=rw+ lx;
whence follows the proportion,
1:m:n: r=1:-y-1: 2-1:- w 1;
or, because xz = yw,
1:m:n: r=z: -w:x:- Y,
so that the variable generatrix qs divides (as is known) the two fixed generatrices AB and DC homographically*; AD, BC, and c'c2 being three of its positions. Conversely, if it were proposed to find the locus of the right line qs, which thus divides homographically (comp. 26) two given right lines in space, we might take AB and DC for those two given lines, and AD, BC, c'c2 (with the recent meanings of the letters) for three given positions of the variable line; and then should have, for the two variable but corresponding (or homologous) points Q, s themselves, and for any arbitrary point P collinear with them, anharmonic symbols of the forms,
Q = (s, u, 0, 0),
because, by 82, we should have,
between these three symbols, a relation of the form, (P)=t(Q) + v(s):
if then we write P= (x, y, z, w), we have the anharmonic equation xz=yw, as before; so that the locus, whether of the line qs, or of the point P, is (as is known) a ruled surface of the second order.
(4.) As regards the known double generation of that surface, it may suffice to observe that if we write, in like manner,
(P) = u(R)+8(T),
we shall have again the expression,
P= (st, tu, uv, vs), giving xz=yw,
as before so that the same hyperboloid is also the locus of that other line RT, which divides the other pair of opposite sides BC, AD of the same gauche quadrilateral ABCD homographically; BA, CD, and A'A2 being three of its positions; and the lines A'a2, c'c2 being still supposed to intersect each other in the given point E.
(5.) The symbol of an arbitrary point on the variable line RT is (by sub-art. 2) of the form, t(0, y, z, 0) + u(x, 0, 0, w), or (ux, ty, tz, uw); while the symbol of an arbitrary point on the given line c'c2 is (t', t', u', u'). And these two symbols represent one common point (comp. Fig. 31),
Hence the known theorem results, that a variable generatrix, RT, of one system, intersects three fixed lines, BC, AD, c'c2, which are generatrices of the other system. Conversely, by the same comparison of symbols, for points on the two lines RT and c'C2, we should be conducted to the equation rz = yw, as the condition for their intersection; and thus should obtain this other known theorem, that the locus of a right line, which intersects three given right lines in space, is generally an hyperboloid with those three lines for generatrices. A similar analysis shows that os intersects A'A2, in a point (comp. again Fig. 31) which may be thus denoted :
P"=QS A'A2 = (xyyx).
(6.) As another example of the treatment of surfaces by their anharmonic and local equations, we may remark that the recent symbols for P' and p", combined with
* Compare p. 298 of the Géométrie Supérieure.