Now, the three quantities B-A A-B A-C are SO related to one another, that only one of them can be positive. For the A BNo2 the product of all three being (4-5)2, , a quantity neces sarily positive, it follows, that one, at least, of the factors, must B-A be supposed positive; also be positive. Let the first, then, B-A and A-C, and, therefore, their sum B-C, must have the same sign; consequently, B-C and A~B; C-B and AC must have different signs; that is, the second and third must be negative. Similarly it may be shewn, that if the second or third be supposed positive, the remaining two will, in each case, be negative. Hence, of the three quanties (1), (2) and (3), one only is real, and this, having a double sign, furnishes two values of the corresponding tangent. II. Let the surface be supposed not to have a centre. Then, as before, the section will be a circle, when, in equation (e'), M cos cos + M' sin2 = M' sin and 2 M sin cos cos 0=0.. Equation (n) will be satisfied, by supposing .... (m'), (n'). But, if be supposed =0, the coefficient of x2, in equation (e'), would =0, which, when the section is a circle, cannot be Whence, we can only admit the two last hypotheses. the case. 1 In the elliptic paraboloid, one of these quantities will be real, and the other imaginary, according as M is greater or less than M'; the real quantity will furnish, as before, two values of the corresponding sine. In the case of the hyperbolic paraboloid, M and M' having different signs, the values of sin 0, and sin , will be imaginary; whence, no plane can be drawn, which shall intersect this surface in a circle. See Art. 393. We conclude, therefore, that in all surfaces of the second order, except the hyperbolic paraboloid, two planes may be drawn, such, that their intersection with the surface shall be a circle. The sub-contrary section of the cone (346), is to be regarded as a particular case of this Proposition. 399. COR. 1. In finding the nature of the section, Art. 396, suppose that the origin, as well as the direction of the axes is changed, and let a, ß, y be the co-ordinates of the new origin. Then, since the equations of condition (m), (n), and (m'), (n'), which determine the section to be a circle, are independent of the quantities a, ß, y, the Proposition just proved will hold true, whatever be the point through which the intersecting plane is drawn. Whence it follows, that the pairs of planes which intersect the surface in circles, will be infinite in number, and parallel to one another. 400. COR. 2. If we replace the indeterminate quantities give to a2, b3, c2 their proper signs, according as the surface belongs to one or other of the three species (362), we shall have no difficulty in proving, that the sections are circles in each of the following cases: (1) In the ellipsoid. When the intersecting plane passes through the mean axis 2b, and is inclined to the plane of xy, at angles When the intersecting plane passes through the greatest axis 2a, and forms with the plane of xy angles (3) In the hyperboloid of two sheets. When the intersecting plane is parallel to that which passes through the mean axis 26, and forms with the plane of xy angles 401. PROP. 3. Surfaces of the second order, with the exception of the hyperbolic paraboloid, may be generated in two different ways, by the motion of a circle parallel to itself, and of variable radius. Having already shewn, that an infinite number of parallel planes may be drawn, intersecting the surfaces in circles, it only remains to prove, that the centres of these circles will be situated on the same straight line. For this purpose, let the origin be transferred to the centre of any one of the circular sections: the terms involving x' and y', in equations (e), and (e), Art. 396, will, therefore, vanish (91), and we shall have Cy sin 0 – BB cos 0 cos +Aa cos 0 sin =0, BB sin +Aa cos p.. =0. .... Each of these equations being linear, it follows, that the locus of the centre is a straight line. Whence the truth of the Proposition. E CHAP. X. ON THE CONDITIONS WHICH DETERMINE A SURFACE OF THE SECOND ORDER TO BE A SURFACE OF REVOLUTION. 402. PROP. I. To find what must be the relation among the coefficients of the general equation, in order that it may represent a surface of revolution. The principle of the transformation of co-ordinates, leads, in a very simple manner, to the solution of the proposed question. The surface being referred to rectangular axes, originating at the centre, its equation will be Ax2+By+Cz2 + 2 A y z + 2 B x z +2 Cry=D.... (1). Let x, y, z' be the co-ordinates of another rectangular system, which has the same origin as the first. Then, supposing the surface to be one of revolution about the axis z', its equation will be of the form or, since the centre is the origin, (350) x22 + y2 = az12 +b ........ (2). a and b being constant quantities. If we now transform the co-ordinates from this, to the primitive system, the transformed equation will be identical with equation (1). Replacing, therefore, r', y' and ', by their values in (316), S s |