Hence, the ratio of Pm to Pn is constant, whatever point on the locus P may represent.

that locus is a straight line.

This can only be true when

9. To find the co-ordinates of the point of intersection of two given straight lines.

Let the equations of the two straight lines be

give the values of a, B, y, at the point of intersection.

10. To find the equation of the straight line, passing through two given points.

Let f, g, h; f, g, h, be the co-ordinates of the two given points, and suppose the equation of the required straight line to be

=

M

N

gh' - g'h hf" - h'ƒ ̄ ̄ƒg' —ƒ'g'

giving, as the equation of the required line,

(gh' — g'h) a + (hf" — h' ƒ) ẞ + ( ƒg' — ƒ'g) y = 0.

11. To find the general equation of a straight line, passing through the point of intersection of two given straight lines.

If the equations of the straight lines be

la + mB+ny = 0,

l'a + m'B + n'y = 0,

every straight line, passing through their point of intersection, may be represented by an equation of the form

la + mB + ny = k (l'a + m'B + n'y),

where k is an arbitrary constant. For this equation is satisfied when the equations of the given straight lines are both satisfied, and, being of the first degree, it represents a straight line. It is therefore the equation of a straight line passing through their point of intersection.

12. To find the condition that three points may lie in the same straight line.

Let a1, B1, Y1; d2, B2, Y 2 i α g, B3, Y,, be the co-ordinates of the three given points, then, if these points lie in the same straight line, suppose the equation of that line to be

λα + β + γ = 0.

Then λ, μ, v must satisfy the following equations :

λα + μβ + vy, = 0,

λα + β + γ = 0,

λα + β + γ = 0,

whence, eliminating λ, μ, v, by cross multiplication,

the required condition.

=

13. To find the condition that three straight lines may intersect in a point.

Let the equations of the straight lines be

l1a+m1ß + n2y=0,

la+m2ß + n2y= 0,

l ̧α+m ̧ß +n ̧y=0.

-If these three straight lines intersect in a point, the above three equations must be satisfied by the same values of α, β, γ. This gives, eliminating a, B, y by cross-multiplication,

1mn − 1 ̧m ̧n+l ̧m ̧n ̧ − Im‚n ̧ + l ̧m ̧n − 1 ̧m ̧n ̧ = 0, the required condition.

The identity of form between the conditions that three straight lines should intersect in a point, and that three points should lie in a straight line, is worthy of notice. Its full geometrical meaning will be seen hereafter.

We shall sometimes, in future investigations, speak of the straight line represented by the equation la + mẞ+ny =0, as the straight line (l, m, n). Adopting this phraseology, it will be seen that the condition that the three points (11, m1, n1) (11⁄2 më në) (1, m, n) should lie in the same straight line, is the same as the condition that the three straight lines (11, m1, n1) (l2, m, n) (1з, m, n,) should intersect in a point.

14. To find the condition that two straight lines may be parallel to one another.

Let the equations of the two straight lines be

Let (f, g, h) (a, B, y) be the co-ordinates of any two points

in (1),

(f', g', h') (a', B', y') be the co-ordinates of any two points in (2).

Also, recurring to the investigation of Art. (8), fig. 7,

These two equations are equivalent to one only, since they may be written in the form

where it will be seen that the equality of any two members implies that the third is equal to either of them.

Multiplying the numerators and denominators of the several members of (3) by l', m', n' and adding, we obtain the condition under the form

(mn' — m'n) a + (nl′ — n'l) b + (lm' — l'm) c = 0......(4).

This is the necessary condition of parallelism, and is generally the most convenient form which can be employed. It is equivalent to

(mn' — m'n) sin A + (nl′ — n'l) sin B + (lm' — l'm) sin C = 0,

a form which we shall occasionally use.

It will be observed that this condition is the same in form as that which results from the elimination of a, ß, y between the equations

The last of these is, as we know, an equation which cannot be satisfied by any values of a, B, y, since, as we have already proved (Art. 2), aa+bB+cy = 24. Hence the equation (4) may be looked upon as an expression of the fact that the two equations

cannot be simultaneously satisfied by any values of a, ß, y, or, in other words, that the two straight lines represented by them do not intersect, which is known to be a necessary condition of their parallelism, and also a sufficient condition, since the two straight lines are in the same plane.

Although, however, no values of a, B, y exist which will satisfy the equation aa + bß +cy = 0, yet we can always satisfy the equation la + mß + ny = 0, where the ratios l m n approach as nearly as we please to the ratios a: b: c.

:

By referring to the investigation of Art. (7) it will be seen that, q, r, denoting the distances from A, of the points in which the straight line (l, m, n) cuts AC, AB respectively,

It hence appears, that by making the ratios : m : n sufficiently nearly equal to the ratios a:bc, the values of q and r may be made as great as we please, in other words, that the straight line (l, m, n) may be removed as far as we please from the triangle of reference. The limiting position,