(b − g) {(b2 — ac) (b +g) + 2 (af2 + ch2 — 2bƒh)} — 20 (abc — aƒ2 — bg3 — ch2 +2fgh) + 0* (4fh−2b3 — ac−g3) + 0ʻ=0, (0+b−g) {03 − (b −g) 02 + (4fh — ac - 2bg-b*) 0 + (b2 − ac) (b+g) + 2 (af2 + ch3 — 2bfh)} = 0. NOTE ON RHIZIC CURVES. By WILLIAM WALTON, M.A., Fellow of Trinity Hall. LET f(x) represent the expression where P P P P are all possible constants. Write u+v√(-1) for x, where u and v are possible quantities: then f(x) = P+ Q √(− 1), P and Q being possible functions of u and v. The two curves P=0, Q=0, referred to rectangular axes of coordinates u and v, I propose to call rhizic curves, in virtue of their relation to the roots of the equation f(x)=0, and I proceed to investigate some of the properties of these curves. Since the equation f(x)=0 has m roots, the rhizic curves intersect each other at m points. Let 0, 0, be the inclinations of the two curves, at a point of intersection, to the axis of u. dQ dP But (I refer to my article on a Theorem in Maxima and Minima, Quarterly Journal, Vol. X., p. 253) we know that dQ dP du dv dv du hence the equation (2) becomes dP dv dP cos + sin = 0......................... (3). du From (1) and (3) we see that cos(0-4)=0, and consequently that, when there is no multiplicity at an intersection, I. "The rhizic curves intersect each other always at right angles." Imagine that (u, v) is a multiple point, of th multiplicity, of the curve P=0; then will determine the directions of all its tangents at this point. Suppose the multiplicity to be even, and put v=2r. Then, as may be seen by reference to the article quoted above, Then the directions of the tangents at the multiple point are determined by the 2r values of 0, viz., In like manner, if the multiplicity be even, the equation for multiple tangency for the curve Q=0 will be, by the said article, and therefore dar Q dP sin 2rp = 0, (+α), tan 2ro= == cota = tan the directions of the tangents being therefore determined by 2r values of o, viz., II. "When a rhizic curve has even multiplicity at any point, the multiplicity is equiangular.” III. "When the two curves intersect at a point of even multiplicity, the rotation, in the plane of the curves, of the tangents to either of them at the point, through an angle π 4r will bring these tangents into coincidence with the tangents to the other at the point." Next, let us suppose that there is odd multiplicity at any point (u, v) of the curve P=0, and put accordingly v=2r+1. Then the tangency is given by the equation u+h, v+k, being the coordinates of a point of the curve indefinitely near to (u, v). It is easily seen, by reference to the quoted article, that If the multiplicity of one curve at an intersection of the two be even, that of the other is also even and of the same degree. which, if h, k, be replaced by p cose, p sine, respectively, the attitudes of the tangents at the point are given by the following 2r+ 1 values of 0, viz., IV. "When a rhizic curve has a point of odd multiplicity, the multiplicity is equiangular." Let us now consider odd multiplicity in the curve Q=0, the equation of tangency being |