Page images



1. We shall now give a concise account of a system of co-ordinates which differs from that which has been the subject of the preceding chapters in assigning a slightly different interpretation to the co-ordinates.

In the system which we are about to explain, the position of a point P is considered as determined by the ratios of the areas of the triangles PBC, PCA, PAB, to the triangle of reference ABC. If these quantities be denoted by the letters x, y, z, they will be connected by the identical relation

2 + y +2=1. 2. In this method, as in that of trilinear co-ordinates, an equation of the first degree represents a straight line, and one of the second degree a conic.

Again, since x: aa :: Y: bB :: 2 : cy, it follows that if the same straight line be represented in the two systems by the equations

la + mp + ny = 0,

l'x + m'y + n'z=0;

.:.l : l'a :: m : m'b :: n : n'c. Hence we may pass from any relation among the coefficients in the trilinear system to that in the present one, by writing la, mb, nc, for" l, m, n, respectively. Similarly, in conics, we may pass from any such formula to the corresponding one, by writing

ua, vb?, wc', u'bc, v'ca, w'ab, for u, v, w, u', u', w

And, since


= vw u", we must write for U, b*c*U, and similarly for V and W, c'a' V, a'b' W. Also, since

U' = v'r' uu', we must write for U', abcU', and similarly for V' and W', b'ca V', c'ab W'.

Hence we obtain the following synopsis of formulæ :

The straight lines drawn through the angular points of a triangle, bisecting the opposite sides, are represented by

y-z=0, 2 - x=0, x-y=0. The internal bisectors of the angles, by


0, = 0, -3=0.
The perpendiculars, by
y cot B -% cot C = 0, % cot C – x cot A = 0,

x cot A-y cot B=0. The distance between two points, by {(y y') (z' — 2) + b’ (z – z") (ac' — «) + c* (x − x) (y' y)}4, or by +/(+c9-a) (x– x')+(c*+a?—8)(y-y')+(a’+b?—c*)(2–2')?


[ocr errors]
[ocr errors]



The condition of parallelism of the straight lines
lx + my + nz=0, l'x + m'y + n'z = 0, is

1, l, l'
1, m, m'
1, n, n'

[ocr errors]

or mn' m'n + nl-nl + Im' - I'm = 0,

The condition of perpendicularity,
2ll'a* + 2mm'b' + 2nn'c - (mn' + m'n) (+ * - a)

- (nl' + n'?) (c + a62)

- (2m + 2 m) (a + bo c) = 0, or {(l m) (l' n') + (l n) (l' – n')} a?

+{(m n) (m' 1') + (m 1) (m' n')} 62

+{(n 1) (n' m') + (n – m) (n' – 1')} c? = 0. The perpendicular distance from the point (x, y, z) to the line lac + my + nz =

(lx + my + nz) 2A {(– m) (1 – n) a? + (m – n) (m 1) 8+ (n 1) (n – m) c?}" The line at infinity will be represented by x+y+z= 0.

0, is

or if

3. Again, in conics we have the following formulæ :
The conic will be a parabola, if

u, w', u', 1 = 0,
w', v, u', 1
0, au ,

W, 1
1, 1, 1, 0

U+ V + W +2U' +2V' + 2W'= 0.
A rectangular hyperbola, if
ua vb? wc? Ú (24 + c* — aʻ) — v' (c + a' 62)

w' (a? + 3— c) = 0,
(u + u' – o' – w') ao +(0+v' – w' – U') 6?

+ (w + w'- u – v') cʻ = 0. A circle, if v + w - 2u w+U-20'

u tv - 2w a



[merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small]

The equation of the asymptotes is u, w', u', 1$ (x, y, z) + u, w', u' | (x + y +z)' = 0. 2', v, w, 1

w', v, u o', u', w, 1

v', u', w 1, 1, 1, 0

Other formulæ may be adapted in a similar manner.



1. The theory of Reciprocal Polars, which will be treated of in this chapter, discusses the relations which exist between systems of points and straight lines which are the poles and polars of each other with regard to any conic; and shews how from the properties of a curve, regarded as the locus of a moving point, may be deduced those of another curve which is always touched by the polar of this moving point with regard to a fixed conic. The theory is especially valuable when the conic, with respect to which the poles and polars are taken, is a circle.

2. The polar of the point of intersection of two given straight lines is the straight line which joins the poles of those straight lines. This will readily be seen to follow geometrically from the definitions of a pole and polar; or it may be analytically proved thus. Let the two straight lines be represented by the equations

la + m,B+n,y= 0............(1),

1,4 + m,ß +n y= 0....... (2). At their point of intersection, we have


min, - min, nl, nelaim, -1,m,

« PreviousContinue »