to express that the five coefficients, x'... v', of the one symbol, are separately equal to the corresponding coefficients of the other, x' = x, v = v.

72. Writing also, generally,

(tx, ty, tz, tw, tv) = t (x, y, z, w, v),

(x' + x, . . v' + v) = (x',.. v') + (x,. .v), &c.,

and abridging the particular symbol (1, 1, 1, 1, 1) to (U), while (Q), (Q'),.. may briefly denote the quinary symbols (x,.. v), (x', .. v),.. we may thus establish the congruence (71),

(Q'′)=(Q), if (Q) = t (Q') + u (U);

in which t and u are arbitrary coefficients. For example,

(0, 0, 0, 0, 1) = (1, 1, 1, 1, 0), and (0, 0, 0, 1, 1) = (1, 1, 1, 0, 0); each symbol of the first pair denoting (65) the given point E; and each symbol of the second pair denoting (66) the derived point D1. When the coefficients are so simple as in these last expressions, we may occasionally omit the commas, and thus write, still more briefly, (00001) (11110); (00011) (11100).

73. If three vectors, p, p', p', expressed each under the first form (70), be termino-collinear (24) and if we denote their denomitors, xa + x'a + . ., x'a + . ., by m, m', m", they must then (23) be connected by a linear equation, with a null sum of coefficients, which may be written thus:

tmp + t'm'p' + t''m''p'' = 0;

tm + t'm' + t'm" +0.

We have, therefore, the two equations of condition,

t(xaa+. + vee) + t' (x'aa + + v'ee) +t'' (x''aa + . . + v''ee) = 0;

..

t(xa +. +ve) + t' (x'a + . . + v'e) + t'' (x'' a + .. + v'e) = 0 ;

where t, t', t' are three new scalars, while the five vectors a.. e, and the five scalars a.. e, are subject only to the two equations (65): but these equations of condition are satisfied by supposing that

tx + t'x' +t''x'' = . . = tv + t'v' + t''v'' = — u,

where u is some new scalar, and they cannot be satisfied otherwise. Hence the condition of collinearity of the three points P, P′, P", in which the three vectors p, p', p" terminate, and of which the quinary symbols are (Q), (Q'), (Q"), may briefly be expressed by the equation,

* This quinary symbol (U) denotes no determined point, since it corresponds

(by 70, 71) to the indeterminate vector with other quinary symbols, as above.

0

=

; but it admits of useful combinations

0

CHAP. III.]

QUINARY SYMBOLS OF PLANES.

t (Q) + t′ (Q') + t'' (Q'') = − u ( U);

59

so that if any four scalars, t, t, t, u, can be found, which satisfy this last symbolic equation, then, but not in any other case, those three points PP'P" are ranged on one right line. For example, the three points D, E, D1, which are denoted (72) by the quinary symbols, (00010), (00001), (11100), are collinear; because the sum of these three symbols is (U). And if we have the equation,

(Q'') = t (Q) + t' (Q') + u (U),

where t, t', u are any three scalars, then (Q") is a symbol for a point P'', on the right line PP'. For example, the symbol (0, 0, 0, t, t') may denote any point on the line DE.

74. By reasonings precisely similar it may be proved, that if (Q) (Q) (Q′′) (Q"") be quinary symbols for any four points PP'P''p''' in any common plane, so that the four vectors pp'p"p"" are terminocomplanar (64), then an equation, of the form

t(Q) + t' (Q') + t'' (Q'') + t''' ( Q''') = − u(U),

must hold good; and conversely, that if the fourth symbol can be expressed as follows,

(Q''') = t(Q) +t' ( Q') + t'' ( Q'') + u(U),

with any scalar values of t, t', t", u, then the fourth point p'"' is situated in the plane PP'P" of the other three. For example, the four points,

(10000),

(01000),

(00100),

(11100),

or A, B, C, D (66), are complanar; and the symbol (t, t', t", 0, 0) may represent any point in the plane ABC.

75. When a point P is thus complanar with three given points, Po, P1, P2, we have therefore expressions of the following forms, for the five coefficients x,.. v of its quinary symbol, in terms of the fif teen given coefficients of their symbols, and of four new and arbitrary

scalars:

=

...

v = tovo + t¿v1 + t2Vq + U.

And hence, by elimination of these four scalars, to..u, we are conducted to a linear equation of the form

1(x-v)+m(y− v) + n ( z − v) + r (w-v) = 0,

which may be called the Quinary Equation of the Plane PPP, or of the supposed locus of the point : because it expresses a common property of all the points of that locus; and because the three ratios of the four new coefficients l, m, n, r, determine the position of the plane

in space. It is, however, more symmetrical, to write the quinary equation of a plane II as follows,

lx + my + nz + rw + sv=0,

where the fifth coefficient, s, is connected with the others by the relation,

and then we may say that [l, m, n, r, s] is (comp. 37) the Quinary Symbol of the Plane II, and may write the equation,

II = [l, m, n, r, 8].

For example, the coefficients of the symbol for a point P in the plane ABC may be thus expressed (comp. 74):

w = u,

v=u;

x = to + u, y = t1 + u, z = t2+u, between which the only relation, independent of the four arbitrary scalars to.. u, is w v=0; this therefore is the equation of the plane ABC, and the symbol of that plane is [0, 0, 0, 1, -1]; which may (comp. 72) be sometimes written more briefly, without commas, as [00011]. It is evident that, in any such symbol, the coefficients may all be multiplied by any common factor.

76. The symbol of the plane PP,P2 having been thus determined, we may next propose to find a symbol for the point, P, in which that plane is intersected by a given line PP: or to determine the coefficients x.. v, or at least the ratios of their differences (70), in the quinary symbol of that point,

(x, y, z, w, v) = P = PP¿P2 * P3P4• Combining, for this purpose, the expressions,

X = t3x3 + t¿X4 + U', • • v = 13v3 +t4v4 + U',

(which are included in the symbolical equation (73),

(Q)=ts (Q3)+ts (Qs) + u' (U),

and express the collinearity PPP,) with the equations (75),

(which express the complanarity PPPP,) we are conducted to the formula,

t3 (lx3+. + 8v3) + ts (lxs + .. + sv1) = 0;

which determines the ratio t:t, and contains the solution of the problem. For example, if r be a point on the line DE, then (comp.

73),

P

CHAP. III.] QUINARY TYPES OF POINTS AND PLANES.

but if it be also a point in the plane ABC, then w therefore t-t1=0; hence

61

v=0 (75), and

(Q) = ts(00011)+u' (11111), or or (Q) = (00011); which last symbol had accordingly been found (72) to represent the intersection (66), d1 = abcde.

77. When the five coefficients, xyzwv, of any given quinary symbol (Q) for a point P, or those of any congruent symbol (71), are any whole numbers (positive or negative, or zero), we shall say (comp. 42) that the point P is rationally related to the five given points, A.. E; or briefly, that it is a Rational Point of the System, which those five points determine. And in like manner, when the five coefficients, Imnrs, of the quinary symbol (75) of a plane II are either equal or proportional to integers, we shall say that the plane is a Rational Plane of the same System; or that it is rationally related to the same five points. On the contrary, when the quinary symbol of a point, or of a plane, has not thus already whole coefficients, and cannot be transformed (comp. 72) so as to have them, we shall say that the point or plane is irrationally related to the given points; or briefly, that it is irrational. A right line which connects two rational points, or is the intersection of two rational planes, may be called, on the same plan, a Rational Line; and lines which cannot in either of these two ways be constructed, may be said by contrast to be Irrational Lines. It is evident from the nature of the eliminations employed (comp. again 42), that a plane, which is determined as containing three rational points, is necessarily a rational plane; and in like manner, that a point, which is determined as the common intersection of three rational planes, is always a rational point: as is also every point which is obtained by the intersection of a rational line with a rational plane; or of two rational lines with each other (when they happen to be complanar).

78. Finally, when two points, or two planes, differ only by the arrangement (or order) of the coefficients in their quinary symbols, those points or planes may be said to have one common type; or briefly to be syntypical. For example, the five given points, A, .. E, are thus syntypical, as being represented by the quinary symbols (10000),.. (00001); and the ten planes, obtained by taking all the ternary combinations of those five points, have in like manner one common type. Thus, the quinary symbol of the plane ABC has been seen (75) to be [00011]; and the analogous symbol [11000] represents the plane CDE, &c. Other examples will present themselves, in a

shortly subsequent Section, on the subject of Nets in Space. But it seems proper to say here a few words, respecting those Anharmonic Co-ordinates, Equations, Symbols, and Types, for Space, which are obtained from the theory and expressions of the present Section, by reducing (as we are allowed to do) the number of the coefficients, in each symbol or equation, from five to four.

SECTION 3.-On Anharmonic Co-ordinates in Space.

79. When we adopt the second form (70) for p, or suppose (as we may) that the fifth coefficient in the first form vanishes, we get this other general expression (comp. 34, 36), for the vector of a point in space:

xaa + ybẞ+zcy + wdd

and may then write the symbolic equation (comp. 36, 71),

P= (x, y, z, w),

and call this last the Quaternary Symbol of the Point P: although we shall soon see cause for calling it also the Anharmonic Symbol of that point. Meanwhile we may remark, that the only congruent symbols (71), of this last form, are those which differ merely by the introduction of a common factor: the three ratios of the four coefficients, x.. w, being all required, in order to determine the position of the point; whereof those four coefficients may accordingly be said (comp. 36) to be the Anharmonic Coordinates in Space.

80. When we thus suppose that v=0, in the quinary symbol of the point P, we may suppress the fifth term sv, in the quinary equation of a plane II, lx + +sv=0 (75); and therefore may suppress also (as here unnecessary) the fifth coefficient, s, in the quinary symbol of that plane, which is thus reduced to the quaternary form,

II=[l, m, n, r].

This last may also be said (37, 79), to be the Anharmonic Symbol of the Plane, of which the Anharmonic Equation is

lx + my + nz + rw=0;

the four coefficients, lmnr, which we shall call also (comp. again 37) the Anharmonic Co-ordinates of that Plane II, being not connected among themselves by any general relation (such as l+..+s=0): since their three ratios (comp. 79) are all in general necessary, in order to determine the position of the plane in space.

81. If we suppose that the fourth coefficient, w, also vanishes, in