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

since, as we have Hence the equaof the fact that

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

la + mß + ny = 0,

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

cannot be simultaneously satisfied by any values of a, B, 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 ab: 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 l: m : n sufficiently nearly equal to the ratios ab: c, the values of q and 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,

therefore, to which the straight line (1, m, n) continually approaches, and with which it ultimately coincides, when the ratios 1:m:n continually approach to, and ultimately coincide with, the ratios a:b:c, is a straight line altogether at an infinite distance.

This is often expressed by saying that the equation

aa+bB+cy=0,

or the equivalent equation

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

represents the straight line at infinity.

This phraseology is very convenient, and free from objection, if the conventions on which it is adopted be clearly understood. It is, however, desirable that attention should be called to the fact, that the equation

aa+bB+cy=0

is, in itself, impossible, in fact, a contradiction in terms,and can only be admitted as a limiting form to which possible equations may continually tend.

15. To find the equation of a straight line, drawn through a given point, parallel to a given straight line.

Let (l, m, n) be the given straight line, (f, g, h) the given point, then the equation of the required straight line will be

For this straight line passes through the point (f, g, h), and does not intersect the straight line (l, m, n); since, if it did, we should have aa+bB+cy = 0.

Since af+bg+ch = 2A, this equation may also be written

la + mB+ny =

=

If + mg + nh

24

(aa+bB+cy).

COR. The general equation of a straight line parallel to (l, m, n) is

la + mẞ+ny = k (aa+bB+cy),

where k is an arbitrary constant.

16. To find the inclinations of a straight line, drawn through one of the angular points of the triangle of reference, to the sides which intersect in that point.

Let the equation of the straight line AP be

and let be its inclination to AD, the internal bisector of

Hence, the inclinations of the given straight line to AB, AC, are determined.

17. To find the condition that two given straight lines may be perpendicular to one another.

Let (l, m, n), (l', m', n') be the two given straight lines. Through A draw two straight lines parallel to them. These will be represented by the equations

And these straight lines must be at right angles to one another.

If 0, ' be the respective inclinations of these straight lines to the internal bisector of the angle A, then, by the result of the last article,

And, if these be at right angles to one another,

(lcna) (l'cn'a) + (ma - lb) (m'a- l'b)

+ {(lcna) (m'a - l'b) + (ma - lb) (l'cn'a)} cos A=0;