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It is to be observed that the equations in trilinear co-ordinates being homogeneous, we are not concerned with the actual lengths of the perpendiculars from any point on the lines of reference, but only with their mutual ratios. Thus the preceding equation is not altered if we write pa', pß', py', for a', B', y'. Accordingly if a point be given as the intersection of the lines B Y

a

ī

=

m n

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we may take l, m, n as the trilinear co-ordinates of that point. For let p be the common value of these fractions, and the actual lengths of the perpendiculars on a, ß, y are lp, mp, np where p is given by the equation alp+bmp+cnp=M, but, as has been just proved, we do not need to determine p. Thus, in applying the equation of this article, we may take for the co-ordinates of intersection of bisectors of sides, sin B sin C, sin C sin A, sin A sin B; of intersection of perpendiculars, cos B cos C, cos C cos A, cos A cos B; of centre of inscribed circle 1, 1, 1; of centre of circumscribing circle cos A, cos B, cos C, &c.

Ex. 1. Find the equation of the line joining intersections of perpendiculars, and of bisectors of sides.

Ans. a sina cosa sin(B-C)+ẞ sin B cos B sin(C-4)+ y sin C'cos Csin(A-B)=0. Ex. 2. Find equation of line joining centres of inscribed and circumscribing circles.

Ans. a (cos B - cos C) + B (cos C - cos A) + y (cos A - cos B) = 0.

66. It is proved, as in Art. 7, that the length of the perpendicular on a from the point which divides in the ratio l:m, the line joining two points whose perpendiculars are a', a" is la + ma" Consequently the co-ordinates of the point dividing

7 + m

in the ratio 7: m the line joining aß'y', a"B"y" are la' +ma", lB' + mB", ly' + my". It is otherwise evident that this point lies on the line joining the given points, for if a'B'y', a"B"y" both satisfy the equation of a line Aa+ BB+ Cy=0, so will also la' + ma", &c. It follows hence without difficulty that la-ma", &c. is the fourth harmonic to la+ma", a', a": that the anharmonic ratio of a-ka", a' - la", a' -ma", a'-na", is (n − 1) (m −k)

(n-m) (l-k); and also that given two systems of points on

two right lines, a' — ka", a' — la", &c., a"" – ka", a" — la"", &c., these systems are homographic, the anharmonic ratio of any four

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points on one line being equal to that of the four corresponding points on the other.

Ex. The intersection of perpendiculars, of bisectors of sides, and the centre of circumscribing circle lie on a right line. For the co-ordinates of these points are cos B cos C, &c., sin B sin C, &c., and cos 4, &c. But the last set of co-ordinates may be written sin B sin C - cos B cos C, &c.

The point whose co-ordinates are cos (B - C'), cos(C – A), cos (A – B) evidently lies on the same right line and is a fourth harmonic to the three preceding. It will be found hereafter that this is the centre of the circle through the middle points of the sides.

67. To examine what line is denoted by the equation

a sin A+B sin B+ y sin C=0.

This equation is included in the general form of an equation of a right line, but we have seen (Art. 63) that the left-hand member is constant, and never = 0. = 0. Let us return, however, to the general equation of the right line, Ax + By + C=0. We

saw that the intercepts cut off on the axes are

C C Α' Bi

consequently, the smaller A and B become, the greater will be the intercepts on the axes, and, therefore, the more remote the line represented. Let A and B be both = 0, then the intercepts become infinite, and the line is altogether situated at an infinite distance from the origin. Now it was proved (Art. 63) that the equation under consideration is equivalent to Ox+0y+ C=0, and though it cannot be satisfied by any finite values of the co-ordinates, it may by infinite values, since the product of nothing by infinity may be finite. It appears then that a sin A+B sin B+ y sin C denotes a right line situated altogether at an infinite distance from the origin; and that the equation of an infinitely distant right line, in Cartesian co-ordinates, is 0.x +0.y + C=0. We shall, for shortness, commonly cite the latter equation in the less accurate form C=0.

68. We saw (Art. 64) that a line parallel to the line a=0 has an equation of the form a+ C=0. Now the last Article shows that this is only an additional illustration of the principle of Art. 40. For, a parallel to a may be considered as intersecting it at an infinite distance, but (Art. 40) an equation of the form a+C=0 represents a line through the intersection of the lines

a=0, C=0, or (Art. 67) through the intersection of the line a with the line at infinity.

69. We have to add that Cartesian co-ordinates are only a particular case of trilinear. There appears, at first sight, to be an essential difference between them, since trilinear equations are always homogeneous, while we are accustomed to speak of Cartesian equations as containing an absolute term, terms of the first degree, terms of the second degree, &c. A little reflection, however, will show that this difference is only apparent, and that Cartesian equations must be equally homogeneous in reality, though not in form. The equation x=3, for example, must mean that the line x is equal to three feet or three inches, or, in short, to three times some linear unit; the equation xy = 9 must mean that the rectangle xy is equal to nine square feet or square inches, or to nine squares of some linear unit; and so on.

If we wish to have our equation homogeneous in form as well as in reality, we may denote our linear unit by z, and write the equation of the right line

Ax+ By + Cz = 0.

Comparing this with the equation

Aa + BB + Cy = 0;

and remembering (Art. 67) that when a line is at an infinite distance its equation takes the form z=0, we learn that equations in Cartesian co-ordinates are only the particular form assumed by trilinear equations when two of the lines of reference are what are called the co-ordinate axes, while the third is at an infinite distance.

70. We wish in conclusion to give a brief account of what is meant by systems of tangential co-ordinates, in which the position of a right line is expressed by co-ordinates, and that of a point by an equation. In this volume we limit ourselves to what is not so much a new system of co-ordinates as a new way of speaking of the equations already in use. If the equation (Cartesian or trilinear) of any line be λx+μy + vz = 0, then evidently, if λ, μ, v be known, the position of the line is known: and we may call these three quantities (or rather their mutual ratios with which only we are concerned) the co-ordinates of the right line. If the line pass through a fixed point x'y'z', the relation

must be fulfilled x'λ+y'μ + z'v = 0; if therefore we are given any equation connecting the co-ordinates of a line, of the form aλ+bμ+cv=0, this denotes that the line passes through the fixed point (a, b, c), (see Art. 51), and the given equation may be called the equation of that point. Further, we may use abbreviations for the equations of points, and may denote by a, the quantities x'λ+y'μ+z'v, x"λ+y′′μ+z"v; then it is evident that la+mẞ=0 is the equation of a point dividing in a given ratio the line joining the points a, B; that la=mß, mß=ny, ny = la, are the equations of three points which lie on a right line; that a + kß, a − kß denote two points harmonically conjugate with regard to a, B, &c. We content ourselves here with indicating analogies which we shall hereafter develope more fully; for we shall have occasion to show that theorems concerning points are so connected with theorems concerning lines, that when either is known the other can be inferred, and often that the same equations differently interpreted will prove either theorem. Theorems so connected are called reciprocal theorems.

Ex. Interpret in tangential co-ordinates the equations used Art. 60. Ex. 2. Let a, ẞ, y denote the points A, B, C; mẞ – ny, ny - la, la - mß, the points L, M, N; then mẞny – la, ny + la – mß, la + mß - ny denote the vertices of the triangle formed by LA, MB, NC; and la + mß + ny denotes a point O in which meet the lines joining the vertices of this new triangle to the corresponding vertices of the original: mẞny, my la, la + mẞ denote D, E, F. It is easy hence to see the points in the figure which are harmonically conjugate.

CHAPTER V.

EQUATIONS ABOVE THE FIRST DEGREE REPRESENTING
RIGHT LINES.

71. BEFORE proceeding to speak of the curves represented by equations above the first degree, we shall examine some cases where these equations represent right lines.

&c.

If we take any number of equations, L=0, M=0, N=0, and multiply them together, the compound equation LMN, &c.=0

will represent the aggregate of all the lines represented by its factors; for it will be satisfied by the values of the co-ordinates which make any of its factors = 0. Conversely, if an equation of any degree can be resolved into others of lower degres, it will represent the aggregate of all the loci represented by its different factors. If, then, an equation of the nth degree can be resolved into n factors of the first degree, it will represent n right lines.

72. A homogeneous equation, of the nth degree in x and y denotes n right lines passing through the origin.

Let the equation be

x” − px” ̄1y + qx”-2y2 - &c. ...+ ty" = 0.

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Let a, b, c, &c. be the n roots of this equation, then it is resolvable into the factors

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(-a) C-4) - c) &c. = 0,

and the original equation is therefore resolvable into the factors

(x — ay) (x — by) (x − cy) &c. = 0.

It accordingly represents the n right lines x-ay = 0, &c., all of which pass through the origin. Thus, then, in particular, the homogeneous equation

x2 - pxy + qy2 = 0

represents the two right lines x-ay=0, x-by=0, where a and b are the two roots of the quadratic

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It is proved, in like manner, that the equation

(x − a)” − p (x − a)" ~1 ( y − b) + q (x − a)"~" ( y − t)"...+ t ( y − b)" = 0

denotes n right lines passing through the point (a, b).

Ex. 1. What locus is represented by the equation xy = 0 ?

Ans. The two axes, since the equation is satisfied by either of the suppositions

=

0, y = 0.

Ex. 2. What locus is represented by x2 - y' = 0?

Ans. The bisectors of the angles between the axes, r±y = 0 (see Art. 35).

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