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CHAP. 111.]



the point a", to which that line tends, is of the type P2, 1, or belongs to the first group of points of second construction. A second inscribed triangle, A”B”C””, for which Fig. 21 may be consulted, is only indicated by the number 2 placed at the middle of the side B'c', to suggest that this bisecting point a” belongs to the second group of points P2. The same number 2, but with an accent, 2', is placed near the corner Ao of the exscribed triangle ABCo, to remind us that this corner also belongs (by a syntypical relation in space) to the group P2, 2. The point A", which is now infinitely distant, is indicated by the number 3, on the dotted line at the top; while the same number with an accent, lower down, marks the position of the point A”. Finally, the ten other numbers, unaccented or accented, 4, 4, 5, 5′, 6, 6′, 7, 7′, 8, 8', denote the places of the ten points, ▲ˇ, A1ˇ, A”, A1, A, A1, AV, A 4, 41. And the principal harmonic relations, and relations of involution, above mentioned, may be verified by inspection of this Diagram.

95. However far the series of construction of the net in space may be continued, we may now regard it as evident, at least on comparison with the analogous property (42) of the plane net, that every point, line, or plane, to which such constructions can conduct, must necessarily be rational (77); or that it must be rationally related to the system of the five given points: because the anharmonic co-ordinates (79, 80) of every net-point, and of every net-plane, are equal or proportional to whole numbers. Conversely (comp. 43) every point, line, or plane, in space, which is thus rationally related to the system of points ABCDE, is a point, line, or plane of the net, which those five points determine. Hence (comp. again 43), every irrational point, line, or plane (77), is indeed incapable of being rigorously constructed, by any processes of the kind above described; but it admits of being indefinitely approximated to, by points, lines, or planes of the net. Every anharmonic ratio, whether of a group of net-points, or of a pencil of net-lines, or of net-planes, has a rational value (comp. 44), which depends only on the processes of linear construction employed, in the generation of that group or pencil, and is entirely independent of the arrangement, or configuration, of the five given points in space. Also, all relations of collineation, and of complanarity, are preserved, in the passage from one net to another, by a change of the given system of points: so that it may be briefly said (comp. again 44) that all geometrical nets in space are homographic figures. Finally, any five points of such a net, of which no four are in one plane, are sufficient (comp.

* These general properties (95) of the space-net are in substance taken from Möbius, although (as has been remarked before) the analysis here employed appears to be new: as do also most of the theorems above given, respecting the points of second construction (92), at least after we pass beyond the first group P2,1 of ten such points, which (as already stated) have been known comparatively long.

45) for the determination of the whole net: or for the linear construction of all its points, including the five given ones.

(1.) As an Example, let the five points A1B1C1D and E be now supposed to be given; and let it be required to derive the four points ABCD, by linear constructions, from these new data. In other words, we are now required to exscribe a pyramid ABCD to a given pyramid A¡B ̧C1D1, so that it may be homologous thereto, with the point E for their given centre of homology. An obvious process is (comp. 45) to in. scribe another homologous pyramid, A3B3C3D3,, so as to have Ag= E^1*B1C1D1, &c. ; and then to determine the intersections of corresponding faces, such as ABC1 and A3B3C3; for these four lines of intersection will be in the common plane [E], of homology of the three pyramids, and will be the traces on that plane of the four sought planes, ABC, &c., drawn through the four given points D1, &c. If it were only required to construct one corner a of the exscribed pyramid, we might find the point above called A as the common intersection of three planes, as follows,

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Or the point A might be determined by the anharmonic equation,

(EAA1A3) = 3,

which for a regular pyramid is easily verified.

(2.) As regards the general passage from one net in space to another, let the symbols P1 = (1.. v1), . . P5 = (X5. . v5) denote any five given points, whereof no four are complanar; and let a'b'c'd'e' and u' be six coefficients, of which the five ratios are such as to satisfy the symbolical equation (comp. 71, 72),

a' (P1) + b′(P2) + c′ (P3) + ď′ (P4) + e' (P5) = — u'(U) :

or the five ordinary equations which it includes, namely,

a'x1+..+exs=..= av1 + ..+évs: == -u'.

Let P' be any sixth point of space, of which the quinary symbol satisfies the equation,

(P) = xa' (P1)+yb' (P2) + zc' (P3) + wď’(P1) + ve' (P5)+u( U ) ;

then it will be found that this last point P' can be derived from the five points P1.. Ps by precisely the same constructions, as those by which the point P = (ryzwv) is derived from the five points ABCDE. As an example, if v′ = x+y+z+w-3v, then the point (xyzwv) is derived from A1B]C1DE, by the same constructions as (xyzwv) from ABCDE; thus ▲ itself may be constructed from A1.. E, as the point P= (30001) is from A.. E; which would conduct anew to the anharmonic equation of the last sub-article.

(3.) It may be briefly added here, that instead of anharmonic ratios, as connected with a net in space, or indeed generally in relation to spatial problems, we are permitted (comp. 68) to substitute products (or quotients) of quotients of volumes of pyramids; as a specimen of which substitution, it may be remarked, that the anharmonic relation, just referred to, admits of being replaced by the following equation, involving one such quotient of pyramids, but introducing no auxiliary point:

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In general, if zyzw be (as in 79, 83) the anharmonic co-ordinates of a point p in space, we may write,

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with other equations of the same type, on which we cannot here delay.

SECTION 5.-On Barycentres of Systems of Points; and on Simple and Complex Means of Vectors.

96. In general, when the sum Σa of any number of co-initial vectors,

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is divided (16) by their number, m, the resulting vector,

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is said to be the Simple Mean of those m vectors; and the point м, in which this mean vector terminates, and of which the position (comp. 18) is easily seen to be independent of the position of the common origin o, is said to be the Mean Point (comp. 29), of the system of the m points, A1, Am It is evident that we have the equation,

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or that the sum of the m vectors, drawn from the mean point м, to the points A of the system, is equal to zero. And hence (comp. 10, 11, 30), it follows, Ist., that these m vectors are equal to the m successive sides of a closed polygon; IInd., that if the system and its mean point be projected, by any parallel ordinates, on any assumed plane (or line), the projection M', of the mean point M, is the mean point of the projected system : and IIIrd., that the ordinate мM', of the mean point, is the mean of all the other ordinates, A1A'1, · · AmA'm• It follows, also, that if N be the mean point of another system, B1, and if s be the mean point of the total system, A1.. B,, of the m+n = 8 points obtained by combining the two former, considered as partial systems; while v and σ may denote the vectors, ON and os, of these two last mean points: then we shall have the equations, Σα+ Σβ=mμ + ην, m. MS n.8N;

ημ= Σα,

ην Ξ · Σβ,


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so that the general mean point, s, is situated on the right line MN, which connects the two partial mean points, м and N; and divides

that line (internally), into two segments Ms and SN, which are inversely proportional to the two whole numbers, m and n.

(1.) As an Example, let ABCD be a gauche quadrilateral, and let E be its mean point; or more fully, let

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that is to say, let a=b=c=d, in the equations of Art. 65. Then, with notations lately used, for certain derived points D1, &c., if we write the vector formulæ,

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we shall have seven different expressions for the mean vector, ε; namely, the following:

e=(a+3a1) = . . = 1 (8+381)

= (a + a2) = . . = { (y' + y2).

And these conduct to the seven equations between segments,

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which prove (what is otherwise known) that the four right lines, here denoted by AA1,.. DD1, whereof each connects a corner of the pyramid ABCD with the mean point of the opposite face, intersect and quadrisect each other, in one common point, E; and that the three common bisectors A′A2, B′B2, C'C2, of pairs of opposite edges, such as BC and DA, intersect and bisect each other, in the same mean point: so that the four middle points, c', A', C2, A2, of the four successive sides AB, &c., of the gauche quadrilateral ABCD, are situated in one common plane, which bisects also the common bisector, B'B2, of the two diagonals, AC and BD.

(2.) In this example, the numbers of the points A.. D being four, the number of the derived lines, which thus cross each other in their general mean point E is seen to be seven; and the number of the derived planes through that point is nike : namely, in the notation lately used for the net in space, four lines A1, three lines A2, 1, six planes II, and three planes II2, 1. Of these nine planes, the six former may (in the present connexion) be called triple planes, because each contains three lines (as the plane ABE, for instance, contains the lines AA1, BB1, C'C2), all passing through the mean point E; and the three latter may be said, by contrast, to be non-triple planes, because each contains only two lines through that point, determined on the foregoing principles.

(3.) In general, let ø (8) denote the number of the lines, through the general mean point s of a total system of s given points, which is thus, in all possible ways, decomposed into partial systems; let ƒ(s) denote the number of the triple planes, obtained by grouping the given points into three such partial systems; let (s) denote the number of non-triple planes, each determined by grouping those s points in two different ways into two partial systems; and let F(s) = ƒ(s) + ¥ (s) represent the entire number of distinct planes through the point s: so that

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Then it is easy to perceive that if we introduce a new point c, each old line MN furnishes two new lines, according as we group the new point with one or other of the two old partial systems, (M) aud (N); and that there is, besides, one other new line, namely cs: we have, therefore, the equation in finite differences,

$ (8 + 1) = 29 (s) + 1;

which, with the particular value above assigned for (4), or even with the simpler and more obvious value, ø (2) = 1, conducts to the general expression,

$(s) = 2s-1 −1.

(4.) Again, if (M) (N) (P) be any three partial systems, which jointly make up the old or given total system (8); and if, by grouping a new point c with each of these in turn, we form three new partial systems, (M') (N')(P'); then each old triple plane such as MNP, will furnish three new triple planes,


while each old line, KL, will give one new triple plane, CKL: nor can any new triple plane be obtained in any other way. We have, therefore, this new equation in differences:

But we have seen that

f(s+1)=3ƒ (s) + $ (8).

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(5.) Finally, it is clear that we have the relation,

3ƒ(3)+4(3)=3(3). (p(s)-1)=(211) (2-1);

because the triple planes, each treated as three, and the non-triple planes, each treated as one, must jointly represent all the binary combinations of the lines, drawn through the mean point s of the whole system. Hence,

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which last equation in finite differences admits of an independent geometrical interpretation.

(6.) For instance, these general expressions give,

$(5) = 15; ƒ(5)=25; (5)=30; F(5) = 55;

so that if we assume a gauche pentagon, or a system of five points in space, ▲ . . E,

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