and with the co-ordinates. These co-ordinates for each chord hold the places of x, y, z respectively in the general equation (A); therefore, on substituting them, that equation becomes, for the chord k',

A(Y′+uk')2 +B (d' + vk' )2 + c (e' + tk' )2 + D (y' + uk' )(8' + vk' ) + E(y' +uk')(e' + tk') + F (d' + vk') (e' + tk') + G (y' + uk' ) + H (d' + vk′) + K (e' + tk′ ) = L. (a)

Imagining this to be resolved as a quadratic equation, the two roots+k' and -k' being equal to one another and having contrary signs, the sum of the coefficients of the first power of k must be zero; therefore, collecting these coefficients, we have

2дuy' +2Bvd' + 2 cté + Dvy' + Dud' + Ety' + Eu é′ + Ft d′ + Fvé
+GU+HV + Kt=0;

or putting pe' for y', which may be done if a corresponding point be assumed as the origin of the co-ordinates,

2 ▲ pué′+2 в v d′+2 cte' + Dp vé' + Dud' + Epte' + Eué' + Ft d′+ F vé′ (b)

+GU+HU+Kt=0.

Putting first double, and then triple accents on ♪ and ɛ, corresponding equations will be obtained for the chords k", k""; therefore, subtracting successively the equation for k from those for k" and k"", and reducing,

(=

=

2 BV+Du+Ft

2 Apu +2 ct+Dpv + Ept+ Eu + F v°

On substituting qɛ' for d′, qe" for d", and proceeding in like

manner there would be obtained a value of Elle'

"

or its equal

Now, imagine a fourth chord to be parallel to the three former, and let either of its segments made by the plane which bisects the others be represented by k; also let y, d, e be the co-ordinates of the point in which it is intersected by that plane; then y+uk, d+vk, e+th will be co-ordinates of either of the points in which that chord cuts the curve surface; and substituting these co-ordinates for x, y, z in the equation (A), the result will be identical with (a) if the accents in the latter be effaced. But all the points in which the four chords are intersected by the given plane being, of course, in that plane, and the co-ordinate lines being parallel to one another, the differences between the like co-ordinates of the four points are proportional; therefore

Substituting pe for y in the equation said to be identical with (a) when the accents in the latter are omitted, and resolving the equation as a quadratic with respect to k, the sum of the coefficients of the first power of k will have for its numerator

2 Aрue +2в vd+2cte+Dp ve+Dud+Epte + Eu€+Ftd+FV€ (b')

+GU+HV + Kt,

the denominator being the sum of the coefficients of k2.

2 apu é' + 2cté' +Dpvé' +Epté' + EU é′ + FV é' + 2 BV d′ + D u d' + Ft d′
=2Apue+2cte + Dpve + Epte + Eue + Fv € + 2 B v d + Dud + Ftd.

(c) But the first nine terms in (b) constitute the second member of the equation (c); they are therefore equivalent to its first member; and, comparing (b') with (6), it will be seen that the former, that is the sum of the coefficients of the first power of k, is zero, or the two values of k in the quadratic equation are equal to one another. Thus the fourth chord is bisected by the plane which bisects the three others; and it follows that the same plane will bisect all chords which are parallel to these.

The equation for this plane becomes, on making (b') equal to zero and restoring y for pɛ,

(2 AU+DV + Et) y + (2 B v+Du+Ft)d+(2ct+Eu+Fv) e

191. DEF. 22. A plane bisecting all the chords which are parallel to one another is called a diametral plane. The intersection of any two diametral planes is evidently a diameter of the curve surface.

192. COR. If, in the equation (d'), u and y, v and §, t and * be respectively interchanged, we should have

this is the equation for a plane of which u, v, t represent the general co-ordinates, and which bisects all chords parallel to the plane whose general co-ordinates are y, d, ɛ. Thus the diametral plane which bisects chords parallel to another diametral plane, is parallel to the chords bisected by the latter.

Let the two planes be designated A and B, and let a third plane c which bisects chords parallel to в be so situated that these bisected chords are also parallel to A; then c will bisect the intersection of A and B. Now, since B will bisect chords parallel to c, and it already bisects chords parallel to A, it will bisect the intersection of A and c. In like manner it may be shown that A will bisect the intersection of B and c. Thus each of the three diametral planes will bisect the chords which are parallel to the intersection of the two others.

193. DEF. 23. Three planes so situated are called conjugate diametral planes; and it is manifest that, for a curve surface of the second order, there may be an infinite number of such conjugate diametral planes. The common intersection of all the diametral planes is the centre of the curve surface.

194. The equation for any diametral plane being of the form expressed in (d'); when one of the chords which it bisects coincides with the axis of x in the general equation (A), the angle in the denominator of the fraction which expresses the value of u becomes zero, in which case u or u becomes infinite: again, when one of the chords is parallel to the axis of y, the angle in the denominator of v vanishes; whence v or becomes infinite: lastly, when one of the chords

t

ย

t

บ

is parallel to the axis of z the angles in the numerators both of u and v vanish; whence both=0 and = 0. Making therefore infinite in (d) and dividing every term

и

t

by it; that equation is reduced to

2 Ay+DO+E+G=0:

t

t

y,

= 0 and = 0, the equation becomes

v t

Ey+F+2 Ce+K=0.

These are the equations of three diametral planes which bisect, respectively, the chords which are parallel to the axes of x, y, z, or of 7, 8, e. Now, let the co-ordinate planes be represented by ZOX, ZOY, XOY, and let them be parallel to three conjugate diametral planes; then the diametral plane represented by the first of these equations being parallel to ZOY, y is constant, while & and are variable; the second being parallel to zox, is constant while y and ɛ are variable; and the third being parallel to xOY, is constant while y and are variable. But every equation in which some terms are constant and others variable is absurd; and to render such equation consistent with itself, the coefficients of the variable quantities must be separately zero. Therefore, in the above equations, D=0, E=0, F=0. It follows that, when the co-ordinate planes are parallel to a system of conjugate diametral planes, the general equation (A) must

have the form

Ax2+By2+Cz2 + Gx+Hy+Kz=L.

If the co-ordinate planes be co-incident with any system of conjugate diametral planes, the constant terms G, II, K in the three equations above must be separately zero; therefore, in this case, the general equation (A) must have the form

The general equation (4) might have been reduced to the form at (A') by transformations of the co-ordinates of a point in the curve surface, and the establishment of equations of condition among the arbitrary quantities. (La Croix, Traité du Calcul Differentiel et Integral, No. 301.) In that work the values of the arbitrary quantities are proved to be real, and it is shown that there may be one system of conjugate diametral planes at right angles to one another; it is also shown that, when the curve surface is not one of revolution, there can be only one such system: the co-ordinates of the centre of the curve surface are investigated, and the conditions under which that centre may be infinitely distant from the surface are thence inferred.

195. DEF. 24. The intersections of three conjugate dia

metral planes at right angles to one another are called the principal axes of the curve surface.

196. The equation (A') may be expressed in terms of the conjugate diameters formed by the intersections of any system of conjugate diametral planes, rectangular or oblique : -thus, let a, b, c, be the conjugate semi-diameters; then, in that equation, on making y=0 and z=0, when x=a,

again, making x=0, y=0; when z=c,

It follows that the above equation may be put in the form

The curve surfaces to which the equation just given relates will be of different kinds, according to the signs of the terms in its first member.

PROPOSITION II.

197. To find the nature of the curve resulting from the intersection of the figure by a plane parallel to one of the co-ordinate planes; all the terms in the equation for the surface being positive.

Note. In the following propositions of this section, the co-ordinate axes are supposed to coincide with the principal axes of the curve surface.

Let the co-ordinate plane be XOY; then z being a given or constant quantity, if it be represented by m the equation of the surface will be

which, while m is less than c, is manifestly [(b) Art. 56.] the equation for an ellipse, and it follows that a section parallel to, or (supposing z or m to be zero) coincident with xoy, is an ellipse. In like manner it may be shown, that a section pa

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