79. Let the axes be rectangular, then the formulas in Art. 77 and 78, become

80. By supposing the angle which the radius vector makes with the axis, to pass through all degrees of magnitude between O and 360°, the signs of the trigonometrical functions of that angle will determine the position of the point P. Hence, in the application of the foregoing formulas, the radius vector (r) must always be considered as positive; and those values of (r), which are negative, must be rejected. See Scholium,

p. 35.

ON THE DISCUSSION AND PROPERTIES OF LINES OF

THE SECOND ORDER.

CHAP. I.

ON LINES OF THE SECOND ORDER IN GENERAL.

81. A CURVE, whose equation is of the nth degree, is called a line of the nth order.

Agreeably to this definition, the straight line, since it is characterized by an equation of the first degree, is a line of the first order, and the circle, for a similar reason, is a line of the second order.

82. PROP. 1. To find the locus of the equation of the second degree between two variables.

The general form of this equation is

ay2+ bxy+cx2+dy+ex+f=0,

in which a, b, c... are constant quantities that may be positive or negative, fractional or integral.

The equation being resolved in terms of x and y successively, we have

√{(b2-4ac) x+2(bd−2ae) x+d2-4af},

√ { (b2—4 ac) y2 + 2 (be−2cd) y+e2—4cf}.

Let the known coefficients in the irrational part of these equations be represented by single lettters, and let

1st. To construct the values of (y).

This value consists of two parts, of which the rational part, mx+n, is equation to a straight line, and may therefore be constructed by (51). Let BR, (fig. 42.) be this line, the axes AX, AY, being inclined at any angle whatever. In AX, take any point M, and draw MP parallel to AY, meeting BR in N.

Let AM=x, then MN will = mx + n.

Again, take NP, Np, each =

√ { A x2 + 2 Bx + C},

2 a

then MP, Mp, will be the values of (y) required.

2dly. To construct the values of x.

Let DS (fig. 43.) represent the line m'y + n', take AM' =y,

draw M'Q parallel to AX, meeting DS in N', then, assume N'Q, N'q, each

and it may be shewn, as before, that M'Q, M'q are the two values of r.

The curves, which are the loci of the above equation, are termed lines of the second order.

83. PROP. 2. A straight line cannot cut a line of the second order in more than two points.

Let the curve be cut by a straight line, whose equation is y=mx+n; then, at the points of intersection, the co-ordinates of the line and of the curve will be identical. Substituting, therefore, the value of (y) in the general equation, we have

a (mx+n)2+ bx (mx + n) + cx2+d(mx+n) ̈, x+f=0, which, on reduction, becomes

{am2+bm+c} x2+{(2am+b) n+dm+e} x+an2 +dn+f=0.

This equation, being of the second degree, can have only two roots, which, when real, represent the abscissas of the points of intersection. Hence, the truth of the proposition.

84. PROP. 3. To find the locus of the middle points of any number of parallel chords.

Let y = mx............(a)

be the equation to any line APp drawn through the origin, and cutting the curve in the points P, p (fig. 44): take 0 (x', y'), the middle point of any chord Qq, parallel to APP; then, the question is to find the relation between the co-ordinates of 0.

If the origin be transferred to O, the equation to the curve will become

a(y+y')2+b(x+x') (y+y)+c(x+x′)2+d(y+y')+e(x+x')+f=0,

and the equation to Qq, will become y = mx.

Now the points in which Qq intersects the curve will be determined by supposing the co-ordinates in the last two equations identical, whence

a(mx+y')2+b(x+x') (mx+y')+c(x+x')2+d(mx+y')+e(x+x′)+f=0.

But Qq being bisected in O, the roots of this equation will be equal, and the coefficient of the second term will, therefore, vanish; whence, collecting the terms involving x, we have

2 amy +by+bmx' +2cx'+dm+e=0,

or, arranging, and suppressing the accents,

(2 am+b) y+(bm +2 c) x + dm+e=0....................(1),

the equation to a straight line, which is, therefore, the locus required.

The straight line which has thus been shewn to bisect a system of parallel chords is called a diameter; and the points in which it meets the curve are called the vertices.

COR. The diameter corresponding to any other chord

y = m'x.

will have for its equation

...

..(B),

(2 am'+b) x+(bm' + 2c) y + dm' + e=0.........(2).

85. PROP. 4. If either of these diameters be parallel to the corresponding chords of the other, then, reciprocally, the latter will be parallel to the corresponding chords of the former.

Let the diameter (1), for instance, be parallel to the chord (B); we are to prove that the diameter (2) will be parallel to the chord (a).