(30) Hence the planes of constant quadratic moment Mk2 will envelop the quadric a2+b2+c2~~ (31)
The diameter of a quadric surface is a line at the extremities of which the tangent planes are parallel.
The physical properties of a heterogeneous body (provided they vary continuously from point to point) are known to depend, in the neighbourhood of any one point of the body, on a quadric function of the co-ordinates with reference to that point.
Same is true of physical quantities such as potential, temperature, &c., throughout small regions in which their variations are continuous; and also, without restriction of dimensions, of moments of inertia, &c. Hence, in addition to its geometrical applications to surfaces of the second order, the theory of quadric functions of position is of fundamental importance in physics.
And therefore varies as the square of the perpendicular drawn from 0 to a tangent plane of a certain quadric surface, the tangent plane in question being parallel to (22).
If the co-ordinate axes coincide with the principal axes of this quadric, we shall have ~(myz) =0, ~(mzx) =0, Z(mxy) = 0~ (24) and if we write ~(mx) = Ma, ~(my1) = Mb, ~(mz) =Mc2, (25) where M=~(m), the quadratic moment becomes M(aiX2+bI,s2+ cv), or Mp, where p is the distance of the origin from that tangent plane of the ellipsoid ~-,+~1+~,=I, (26)
If we construct the quadric Axi+By2+Czi 2Fyz2Gzx 2HXy = M~4, (3c~) where e is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction X, u, v is found by putting x, y, z=Ar, ur, Pr.
The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius.
When referred to its principal axes, the equation of the quadric takes the form Axi+By2+Czi=M.
Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at 0.
A curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax+by+c = o, or say (1 x, y, 1) = o; or by an equation of the second order, ax 2 +2hxy+by e +2fy+2gx+c=o, say (*I x, y, 1) 2 =o; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.
A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve.
It is to be remarked that an equation may break up; thus a quadric equation may be (ax+by+c) (a'x.+b'y+c') = o, breaking up into the two equations ax+by+c = o, a'x+b'y+c' = o, viz.
The pair of lines is considered as a quadric curve.
But it is an improper quadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a.
It is a leading point in the theory that a curve in space cannot in general be represented by means of two equations U= o, V = o; the two equations represent surfaces, intersecting in a curve; but there are curves which are not the complete intersection of any two surfaces; thus we have the cubic in space, or skew cubic, which is the residual intersection of two quadric surfaces which have a line in common; the equations U= o, V= o of the two quadric surfaces represent the cubic curve, not by itself, but together with the line.
He next gives by aid of these projective rows and pencils a new generation of conics and ruled quadric surfaces, "which leads quicker and more directly than former methods into the inner nature of conics and reveals to us the organic connexion of their innumerable properties and mysteries."