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Hence the line ww' is represented by the equation

ww' B+y=(w+w') ka.

24. To find the equation of the tangent at w.

This is obtained at once, from the result of the preceding article, by simply putting w' = w. It will then be seen to be

w2ß+y=2w.ka.

25. To find the pole of ww'.

The pole of ww' is the point of intersection of the tangents at w, w'.

It is therefore given by the equations

2w.ka-w3ß-y=0,

2w'. ka-w2ß-y=0,

whence

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26. To find the condition that a given straight line may

touch the conic.

Let the equation of the given straight line be

la+mB+ny=0.

From Art. 22, it appears that if this straight line touch the conic, it must admit of being put in the form

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The co-ordinates of the point of contact of this line will be determined by the equations

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COR. By writing a, b, c, respectively for l, m, n in the condition of tangency just investigated, we see that the necessary condition in order that the conic may be a parabola is

a2 = 4k2bc.

Or the equation of the parabola touching AB, AC in B, C, is

a2a2 = 4bc By.

27. To find the centre of the conic.

Since the conic touches AB, AC in BC, it follows that the straight line drawn through A, and the middle point of BC, will pass through the centre. The equation of this straight line is

19

bB-cy=0.

If f12 912 h1, f2, 92, h, be the co-ordinates of the points in which the straight line meets the conic, those of the centre will be

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2

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Now f f 92, 91, h,, h, are the respective values of a, B, y, obtained from the equations

k2a2- By = 0,

bB-cy=0,

aa+bB+cy = 2A.

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These are the values B, y at the centre. The corresponding value of a may be ascertained by substitution in the equation

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These values all become infinite when 4k bca", as manifestly ought to be the case, since, as has been shewn in Art. 24, the conic is then a parabola.

EXAMPLES.

1. A triangle is inscribed in a conic; prove that the points, in which each side intersects the tangent at the opposite angle, lie in a straight line.

2. A triangle is described about a conic; prove that the straight lines, joining each angular point with the point of contact of the opposite side, intersect in a point.

3. Find the equations of the normals to the conic Aẞy+μya + vaẞ= 0, drawn at the angular points of the triangle of reference; and prove that they will intersect in a point if

= (x2-) + " ("-") + " (" — μ9) = 0.

4. Three conics are drawn, touching respectively each pair of the sides of a triangle at the angular points where they meet the third side, and all intersecting in a point. Prove that the three tangents at their common point meet the sides of the triangle which intersect their respective conics in three points lying in a straight line; and that the other common tangents to each pair of conics intersect the sides of the triangle which touch the several pairs of conics in the same three points.

5. Prove that the points of intersection of the opposite sides of any quadrangle, and the point of intersection of the diagonals, form a conjugate triad with respect to any conic described about the quadrangle.

6. If R be the radius of the circle described about the triangle of reference, p that of the circle with respect to which the triangle of reference is self-conjugate, prove that

p+4R* cos A cos B cos C = 0.

7. If BC, CA, AB be three given tangents to a conic, P, Q, R three points on the curve, and if the areas of the triangles PBC, PCA, PAB be denoted by P1, P P respectively, and three of the triangles obtained by successively writing and R in place of P by I, I, I rrr, prove that

(P1L‚ˆ‚)*~ (P ̧¥‚†)*+ (P‚2‚ˆ‚)*— (P ̧?,r)2+ {P ̧¥‚ˆ‚)* — (P ̧£‚¿‚)2= 0.

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8. Prove that the diagonals of any quadrilateral described about a conic, and the lines joining the points of contact of opposite sides, all intersect in a point.

9. A system of conics is described touching three straight lines; prove that, if one of the foci move along a given straight line, the other will describe a conic about the triangle.

Hence prove that the circle, which passes through the points of intersection of three tangents to a parabola, passes also through the focus.

CHAPTER III.

ON ELIMINATION BETWEEN LINEAR EQUATIONS.

1. BEFORE entering upon the discussion of the conic represented by the general equation of the second degree, it will be necessary to devote a few pages to the subject of elimination between homogeneous linear equations, and to explain some of the terms recently introduced in connection with this branch of analysis.

We shall, however, only state and prove such elementary theorems as will be necessary in our future investigations; referring the reader who may be desirous of fuller information to Salmon's Lessons on the Higher Algebra; Spottiswoode, On Determinants (the second edition of which will be found in Crelle's Journal, t. 51, pp. 209, 328), and to the original memoirs communicated to various scientific Journals by Messrs Boole, Sylvester, Cayley, and others.

2. If we have given n homogeneous linear equations, connecting n unknown quantities x1, x...x, such as

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the quantities x,, x,... x, can be eliminated between them, and the result of the elimination may be expressed by omitting x,, x,... x, and writing the coefficients only in the order in which they appear in the given equations, thus

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