common radical axis, viz., the axis of similitude. If in the second determinant we change the sign either of r', r", or r'", we get the equations of the other three axes of similitude. Now it has been stated (Art. 118) that it is optional which of two supplemental angles we consider to be the angle at which two circles intersect; and if in any line of the first determinant of this article we substitute for its supplement, this is equivalent to changing the sign of the corresponding r. Hence it is evident that we may have four systems of circles cutting the given three at equal angles, each system having a different one of the axes of similitude for radical axis; calculating by the usual formula the radius of the circle whose equation has been written above, we get R in terms of λ, and then from the equation 2R cosλ we get a quadratic to determine the value of corresponding to any value of 0.

=

Ex. 1. To find the condition for the co-existence of the equations

ax + by + c = ax + b'y + c′ = a′′x + b′′y + c′′ = a""x + b'"'y+c"".

Let the common value of these quantities be λ; then eliminating x, y, λ from the four equations of the form ax + by + c = λ, we have the result in the form of a determinant

or A+ C = B+ D, where A, B, C, D are the four minors got by erasing in turn each column, and the top row in this determinant.

To find the condition that four lines should touch the same circle, is the same as to find the condition for the co-existence of the equations a = ẞ=y=d. In this case the determinants A, B, C, D geometrically represent the product of each side of the quadrilateral formed by the four lines, by the sines of the two adjacent angles.

Ex. 2. The expression, p. 129, for the distance between two points may be written 2 (a sin A+ẞ sinB + y sin C)2 =

0, 0,

0, 0, a
a'

β

• γ

cos C,
1

cos B

- cos A

Ex. 3. To find the relation connecting the mutual distances of four points on a circle. The investigation is Prof. Cayley's (see Lessons on Higher Algebra, p. 23). Multiply together according to the ordinary rule the determinants

which are only different ways of writing the condition of Art. 94; and we get the required relation

where (12)2 is the square of the distance between two points. expanded is equivalent to (12) (34) ± (13) (42) ± (14) (23) = 0.

This determinant

Ex. 4. To find the relation connecting the mutual distances of any four points in a plane. This investigation is also Prof. Cayley's (Lessons on Higher Algebra, p. 24). Prefix a unit and cyphers to each of the determinants in the last example; thus

We have then five rows and four columns, the determinant formed from which, according to the rules of multiplication, must vanish identically. But this is

(12)2 (34)2 {(12)2 + (34)2 — (13)2 — (14)2 — (23)2 — (24)2} + (13)2 (24)2 {(13)2 + (24)2 — (12)2 — (14)2 — (23)2 — (34)2} + (14)2 (23)2 {(14)2 + (23)2 — (12)2 — (13)2 — (24)2 — (34)?}

+ (23)2 (34)2 (42)2 + (31)2 (14)2 (43)2 + (12)2 (24)2 (41)2 + (23)2 (31)2 (12)2 = 0.

If we write in the above a, b, c for 23, 31, 12; and R+r, R + r', R + r" for 14, 24, 34, we get a quadratic in R, whose roots are the lengths of the radii of the circles touching either all externally or internally three circles, whose radii are r, r', r", and whose centres form a triangle whose sides are a, b, c.

Ex. 5. A relation connecting the lengths of the common tangents of any five circles may be obtained precisely as in the last example. Write down the two matrices 1

1,

0, 0, 0, 0 x2 + y22 - p12, - 2x', - 2y', 2r', x''2 + y''2 — p''2, — 2x", - 2y", 2r", 1 &c.

0, 0, 0, 0,

1

where there are six rows and five columns, and the determinant formed according to the rules of multiplication must vanish. But this is

1, 0 (12), (13)2, (14)2, (15)2
1, (12)2, 0, (23)2, (24)2, (25)2
1, (13)2, (23)2, 0, (34)2, (35)2

1, (14)2, (24)2, (34)2, 0, (45)2

1, (15), (25), (35)2, (45)2, 0 = 0,

where (12), &c. denote the lengths of the common tangents to each pair of circles. If we suppose the circle 5 to touch all the others, then (15), (25), (35), (45), all vanish, and we get, as a particular case of the above, Dr. Casey's relation between the common tangents of four circles touched by a fifth, in the form

Ex. 6. Relation between the angles at which four circles whose radii are r, r', r", r”” intersect. If the circler have its centre at the point 1 in Ex. 4, r' at 2, &c. we may put for 122 = r2 + p12 — 2rr' cos 12, &c. in the determinant of that example which becomes then

subtracting from each row and column the first multiplied by corresponding square

If in this we let cos 21 = cos 31 = cos 41 = cos 0, we have the quadratic in A mentioned at the end of Art. 132 e.

CHAPTER X.

PROPERTIES COMMON TO ALL CURVES OF THE SECOND DEGREE, DEDUCED FROM THE GENERAL EQUATION.

133. THE most general form of the equation of the second degree is

ax2 + 2hxy + by* + 2gx+2fy + c = 0,

where a, b, c, f, g, h are all constants.

It is our object in this chapter to classify the different curves which can be represented by equations of the general form just written, and to obtain some of the properties which are common to them all.*

Five relations between the coefficients are sufficient to determine a curve of the second degree. For though the general equation contains six constants, the nature of the curve depends. not on the absolute magnitude, but on the mutual ratios of these coefficients; since, if we multiply or divide the equation by any constant, it will still represent the same curve. We may, therefore, divide the equation by c, so as to make the absolute term = 1, and there will then remain but five constants to be determined.

Thus, for example, a conic section can be described through five points. Substituting in the equation (as in Art. 93) the coordinates of each point (x'y') through which the curve must pass, we obtain five relations between the coefficients, which will enable us to determine the five quantities,,

a

&c.

134. We shall in this chapter often have occasion to use the method of transformation of coordinates; and it will be useful

*We shall prove hereafter, that the section made by any plane in a cone standing on a circular base is a curve of the second degree, and, conversely, that there is no curve of the second degree which may not be considered as a conic section. It was in this point of view that these curves were first examined by geometers. We mention the property here, because we shall often find it convenient to use the terms "conic section," or "conic," instead of the longer appellation, "curve of the second degree."

to find what the general equation becomes when transformed to parallel axes through a new origin (x'y'). We form the new equation by substituting x+x for x, and y+y' for y (Art. 8), and we get

a (x+x')2+2h(x+x') (y+y')+b(y+y′)*+2g (x+x')+2ƒf(y+y')+c=0. Arranging this equation according to the powers of the variables, we find that the coefficients of x, xy, and y3, will be, as before, a, 2h, b; that

the new c,

c' = ax" + 2hx'y' +by+2gx+2fy' +c. Hence, if the equation of a curve of the second degree be transformed to parallel axes through a new origin, the coefficients of the highest powers of the variables will remain unchanged, while the new absolute term will be the result of substituting in the original equation the coordinates of the new origin.*

135. Every right line meets a curve of the second degree in two real, coincident, or imaginary points.

This is inferred, as in Art. 82, from the fact that we get a quadratic equation to determine the points where any line y = mx + n meets the curve. Thus, substituting this value of y in the equation of the second degree, we get a quadratic to determine the x of the points of intersection. In particular (see Art. 84) the points where the curve meets the axes are determined by the quadratics

ax2+2gx+c=0, by2 + 2fy + c = 0.

An apparent exception, however, may arise which does not present itself in the case of the circle. The quadratic may reduce to a simple equation in consequence of the vanishing of the coefficient which multiplies the square of the variable. Thus

xy+2y2+x+5y + 3 = 0

is an equation of the second degree; but if we make y=0, we get only a simple equation to determine the point of meeting of the axis of a with the locus represented. Suppose, however, that in any quadratic Ax+2Bx + C=0, the coefficient C

This is equally true for equations of any degree, as can be proved in like manner.

T