Now we can prove, that the three points L, M, N are all in one right line, whose equation is la + mẞ + ny = 0,

for this line passes through the

and (mẞ+ny - la, a) or L.

The equation of CN is

points (la + mß – ny, y) or N, (la – mß + ny, ß) or M,

la + mẞ = 0,

for this is evidently a line through (a, ß) or C, and it also passes through N, since it = (la + mß + ny) — ny.

=

are,

Hence BN is cut harmonically, for the equations of the four lines CN, CA, CF, CB

α = 0, B = 0, la - mß = 0, la + mß = 0.

We shall often afterwards meet with equations of the form discussed in this example.

Ex. 3. If two triangles be such that the intersections of the corresponding sides lie on the same right line, the lines joining the corresponding vertices meet in a point.

Let the sides of the first triangle be a, ß, y; and let the line on which the corresponding sides meet be la + mß + ny: then the equation of a line through the intersection of this with a must be of the form l'a + mß + ny = 0, and similarly those of the other two sides of the second triangle are

la + m'ẞ+ny = 0, la + mẞ + n'y = 0.

But subtracting successively each of the last three equations from another, we get for the equations of the lines joining corresponding vertices

(1 − 1') a = (m − m')ß,

which obviously meet in a point.

(m— m')ß = (n − n') y, (n − n') y = (1 − 1') a.

61. We have seen that having assumed any three right lines, we can express the equation of any right line in the form

Aa + BB + Cy = 0,

and so solve any problem by a set of equations expressed in terms of a, ẞ, y, without any direct mention of x and y. This suggests a new way of looking at the principle laid down in Art. 58, &c. Instead of regarding a as a mere abbreviation for the quantity x cos ay sina - p, we may look upon it as simply denoting the length of the perpendicular from a point on the line a. We may imagine a system of trilinear co-ordinates in which the position of a point is defined by its distances from three fixed lines, and in which the position of any right line is defined by a homogeneous equation between these distances of the form

Aa + BB + Cy = 0.

The advantage of trilinear co-ordinates is, that whereas in Cartesian (or x and y) co-ordinates the utmost simplification we can introduce is by choosing two of the most remarkable lines in the figure for axes of co-ordinates, we can in trilinear co-ordinates obtain still more simple expressions by choosing three of the most remarkable lines for the lines of reference a, ß, y. The reader will compare the brevity of the expressions in Art. 56 with those corresponding in Chap. 11.

62. To reduce a non-homogeneous equation (for example, a = 3) to the homogeneous form la + mẞ + ny = 0.

Let a, b, c be the lengths of the sides of the triangle formed by the three lines of reference; then since a denotes the length of the perpendicular from any point O on a, aa is double the area of the triangle OBC; in like manner bß is double OAC; and cy double OAB; therefore, no matter where the point O be taken, the quantity aa + bẞ + cy is always constant, and equal double the area of the triangle ABC. The reader may suppose that this is only true if the point O be taken within the triangle; but he is to remember that if the point O were on the other side of any of the lines of reference (a), we must give a negative sign to that perpendicular, and the quantity aa + bẞ + cy would then

=

that is, still twice the area of the triangle. If, then, we call the double area M, the equation a = 3 may be written

Ma = 3(aa + bß + y),

which is the required form. If A, B, C be the angles (opposite a, ẞ, y respectively) of the triangle formed by the lines of reference, it is plain that a sin A + B sin B + y sin C is also constant, M sin A

being

=

α

63. To express in trilinear co-ordinates the equation of the rallel to a given line Aa + Bß + Cy = 0.

pa

In Cartesian co-ordinates two lines are parallel if their equations Ax + By + C = 0, Ax + By + C' = 0 differ only by a constant. It follows, then, that the equation

Aa + BB + Cy+k(a sin A + B sin + y sin C) = 0

B'sin+

denotes a line parallel to Aa + Bẞ+ Cy = 0, since the two equations differ only by a constant.

Ex. 1. To find the equation of a parallel to the base of a triangle drawn through the Ans. a sin A+B sin B = 0. For this, obviously, is a line through aß, and writing the equation in the form

vertex.

y sin C (a sin A + ẞ sin B + y sin C) = 0,

it appears that it differs only by a constant from

=

0.

We see, also, that the parallel a sin A + ẞ sin B, and the bisector of the base a sin A - ß sin B form a harmonic pencil with a, ẞ (Art. 55).

Ex. 2. The line joining the middle points of sides of a triangle is parallel to the base. Form (Art. 59) the equation of the line joining (3 sin B

y sin C, B), when we get a sin A + ẞ sin B parallel to y.

y sin C

=

y sin C', a), (a sin A

0, which, by this article, is

=

Ex. 3. To find the equation of a perpendicular to any side at its middle point. This is to draw a parallel to the line a cos A B cos B 0 through the point (a sin A - ẞ sin B, y). Ans. a sin A - ẞ sin B + y sin (A – B) = 0.

Ex. 4. The three such perpendiculars meet in a point. Their equations vanish when multiplied, respectively, by sin 2C, sin 2B, sin 2A, and added together. The equations of

the lines joining their intersection to the vertices will be found to be

α В

=

&c.

cos A cos B Ex. 5. Verify that this point lies on the line whose equation is given Art. 59, Ex. 2. Ex. 6. Find the length of the perpendicular from a point a'ß'y' on Aa + Bß + Cy = 0. Aa' + BB'+CY'

Ans.

√(A2 + B2 + C2 — 2AB cos C2BC cos A - 2CA cos B)

64. 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 that the co-ordinates of any finite point render the quantity a sin A + B sin B + y sin C = a certain constant, and never = 0. Let us return, however, to the general equation of the right line, Ax + By + C = 0. We saw that the

intercepts which this line cuts off on the axes are

с с

A' B

; con

sequently, the smaller A and B become, the greater will be the intercepts on the axes, and, therefore, the more remote the line represented by Ax+ By + C = 0. 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. Hence we arrive at the conclusion, that the paradoxical equation C = 0, a constant = 0,

(and therefore, likewise, a sin A + B sin B + y sin C = 0), represents a right line situated altogether at an infinite distance from the origin.

65. We saw (Art. 63) 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. 36. For, a parallel to a may be considered as intersecting it at an infinite distance, but (Art. 36) an equation of the form a + C = 0 represents a line through the intersection of the lines a = 0, C=0, or (Art. 64) through the intersection of the line a with the line at infinity.

66. We have to add, in conclusion, 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. 64) 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.

CHAPTER V.

EQUATIONS ABOVE THE FIRST Degree rePRESENTING RIGHT

LINES.

67. 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.

If we take any number of equations, L = 0, M = 0, N = 0, &c., 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 degrees, 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.

68. A homogeneous equation, of the nth degree between the variables, denotes n right lines passing through the origin. Let the equation be

n-1

aп - px2-1y + qx2-2 y2 - &c. . . . + ty" = 0.

Divide by y", and we get

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

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