12. Two lettres of the Geometrical correspondence between M. Donkin and M. Spottiswoode. The following is partly a simplification and partly a development of a method which you once shewed me for determining two cyclie sections of a surface of the second order. suppose If o, u, l, m, n, l', m', n', P, Q, R, K, be any constants (I shall it is clear that the surface, whose equation is (1) 0(x2+y2+x3) + μ (lx +my+nz)(l'x +· m'y+n'z)+Px+Qy+R≈ = K, is cut circularly by the two systems of planes represented by since either of these assumptions reduces (1.) to the equation of a sphere. Now (1) will coïncide with the general equation (2) Ax2+By2+Cz2+2 Fyz+2 Gzx + 2 Hxy + Px + Qy+ Rz = provided we have K; A', B-0 B', C-6=C') the following Thus the six direction-cosines l, m, n, l', m', n' are completely determined as soon as (which is involved in A' etc.) is known. To find an equation for determining 0, we have only to substitute for m:n, n;l, l:m, their values in the identical equation But it is better to employ your original method. Reverting, namely to the equations (1 and 2), we obtain, by subtraction: A'x2 + B'y2+C'z2+2 Fyz +2 Gzx+2Hxy so that we have only to express the condition that the function on the lefthandside of this equation is capable of being resolved into linear factors, which, as is well known, is (4) A'B'C' A'F-B'G' - C'H2+2FGH = 0. It is to be remarked, that the expressions mn'-m'n, etc. are proportional to the direction-cosines of the line of intersection of the cyclic planes; and their values being proportional (3) to √(F2—B'C'), √(G2—C'A'), √(H2— A'B'), it follows that the line in question coïncides with one of the principal axes of the cone. In fact, the equation (4), multiplied by A', may be put in the form (G2 — C'A') (H2 — A'B') = (GH-FA')2 and in like manner, multiplying by B' and C', we set the two other forms by means of which expressions it is easy to deduce the above forms for the direction - cosines of the principal axes. But there are other important consequences to be drawn from the equation (4), when put in the three forms last written. In the first place we see that the three equations F, G, H, have necessarily the same sign; so that 1F, G, H are either all real, or all imaginary. Next taking any one of the three forms, for example (G2 — C'A') (H2 — A'B') — (G'H-FA') let B, B' be the two values of which satisfy the equation G2 - C'A' = 0 (which are necessarily real), and in like manner let 7, 7' be the two roots of H2- A'B' = 0. Let these four quantities, arranged in order of magnitude, be called [1], [2], [3], [4]. In the above equation, which we will write thus: SGHA2=0 [1], (which is of the fourth degree, with respect to 0, having been multiplied by A'). Let us examine the value of S, while varies continuously from - ∞ to +∞. We see that when ∞, G and H are both negative and S is positive. 0When 0 we have S negative; so that there is a root of S less than [1]. But since G and H are negative as long as [1], such a root gives imaginary values to the direction - cosines l, m, n; l', m', n'. Again no root of S can lie between [1] and [2], because it is plain that for values of between these limits, G and H have different signs. For the same reason no root can lie between [3] and [4]. When 0 = [4], we have S negative, and when +∞, S is positive. Therefore there is one root greater than [4]; but this again gives imaginary direction - cosines, because G and H are both negative, when > [4]. Of the two remaining roots, one is given by A'=0, or 0A, and is extraneous to the geometrical question, having been introduced by multiplication. us to shew that the fourth root lies between [2] and [3]. and 0 [3] both give S negative, both roots, or else between [2] and [3]. Now it is easy to shew that A does lie between these limits. For the quantities [1], [2], [3], [4], are the values of the expressions \(A+C)±√(}(A— C)2+G2), } (A + B) ± √(} (A — B)2 + H2), Crelle's Journal f. d. M. Bd. XLVII. Heft 3. But it will enable For since 0=[2] neither, must lie 31 which may obviously be written in the form A±C, A+C,, where C and C are positive. And it is plain that either A±C are the values of [1] and [4] and A+C, of [2] and [3]; or vice versa. In either case A lies between [2] and [3], and therefore also so does the fourth root of S. And since for values of between [2] and [3], G and H are both positive, the root in question gives real values of the direction-cosines. These conclusions may be recapitulated as follows. Let 0', 0" be two roots of that one of the three equations whose roots are closet together, and 0,, 0, the roots of that one that are farthest a part. Then of the three roots of the cubic [4], two lie respectively without the limits 0,, 0,,, and the third within the limits ', 0"; and this third root gives an unique system of real values to the direction - cosines l, m, n; l', m', n'; thus determining an unique pair of real cyclic planes. It would be easy to examine the peculiarities of the cases in which the cubic has equal or evanescent roots. But I will not pursue the subject farther. It should be added that the limits of the roots of the cubic [4] were originally tracted by M. Cauchy (see Exercices de Math. 1828. p. 9.), but in a manner less simple, as it appears to me, than the above. Oxford, March 24. 1853. W. F. Donkin. Your method in applicable to the case of four variables, and when so applied, gives rise to some interesting geometrical results. Let Ax2 + By2+ Cz2 + Ew2 +2(Fyz+Gzx + Hxy + Lwx+Mwy + Nw≈) (1) U = then, if U can be put in the form (2) 0(x2+y2+z2 + w2) + (lx+my+wz + kw) (l'x+m'y+n'z+k'w) = 0 we have, writing from which, combined with (5), we derive the following systems: 2/N m M+√M = H+√H G-VG L-VL M-VM N+√N' with a corresponding set of values for l': m': n': k', obtained from (8) by writing FF etc. instead of FF etc. troughout. obtained the equations of condition: etc. troughout. From these may be each of which, although apparently two distinct equations, is equivalent to only one, on account of the identities |