TRANSACTIONS OF THE SECTIONS. REESE 193RARY An examination of the surfaces described by the motion of a plane parabola of F any order with its diameters parallel to a fixed right line showed that the con SITY dition of a pair of equal roots in the equation of the third order, CALIFORNIA considered as an equation in λ, was satisfied by all surfaces traced out by a plane parabola moving parallel to a fixed line and enveloping any curve in space whatever. As singular cases, he noticed the spindle made by causing a parabola (whether fixed or of variable size) to rotate round any diameter, the ruled surface with a director plane, and developable surfaces. He also showed that when three of the roots were equal, the surface necessarily reduced to a plane or a cylinder. These results are, however, restricted by the method of generating the surface. In fact, for the case of three equal roots, when the partial differentials of the third order are in continued proportion, Mr. Cayley has shown that the resulting equations can be integrated and that the integration gives a more general result. r= funct. s is integrated as the equation of a developable surface (p instead of z), viz. we have, say, p=ax+hy+g (a and g functions of h) and Observe that the constants have been taken so that =h, dq=h; but in order dy that h may in the two pairs of equations mean the same function of x, y, we must 1 g have a' = = that is The last equation gives h as a function of x, y; and the values of p, q are then such that dz=pdx+qdy is a complete differential, so that we obtain z by the integration of this equation. A simple example is On Doubly Diametral Quartan Curves. By F. W. NEWMAN, Emeritus Professor of University College, London. This paper aimed to detail the form of the curves, and point out the simplest modes of investigating their peculiarities. It distributed the general equation into three groups of five, five, and four families, and was accompanied by seventy-six diagrams. If we call Ax1+2Dx2y2+By1+2Ex2+2Fy2+C=0 the general equation, the first group of five families is when A or B or both vanish, the second group when D or F or both vanish, and these together nearly include all the forms. For in the third group, from which either no term of the original equation vanishes, or only C, three of the families are at once reducible to the second group by putting either y2+f2=y", else x2+e2=x2; then the proposed curve of (xy) is visibly at most a mere variety of the preceding, being either of the same species with, or of a lower species than, that of (xy) or of (xy). In the case of B=B, D=82, F=-f2, this reduction is impossible; but then by operating on a instead of y, it becomes possible unless also A=a2, E=-e'2; that is, the method fails only when A, B, D are of one sign, and E, F of the opposite. The analysis, thus limited, readily yields the same result, that the forms have nothing new. A cross division of the species is into Limited and Unlimited loci. All are Centric, the origin being the Centre. Finite forms are Monads, Duads, and Tetrads. Monads are:-1. A symmetrical oval (say, a Shield), as from x2y2+a2y2+b2x2 = m1. 2. An Oval with undulating sides (Viol or Dumb-bell), as a2y2=(m3-x2)(x2+n2), m2> n2. 3. A Lemniscate or Double Loop, as a2y2=(m2 —x2)x2. 4. A Scutcheon, with four sides undulating, as B2y2=f2±√(m2− x2)(x2+n2), when m2> n2, and f2 <mn. 5. Ovals in Contact, as By = (m2-x2)x2. 6. Pointed Hearts, crossing obliquely at the centre, as 2 2 B2y2=mn+82x2±√ {(m2 —x2)(x2+n2)}, when m2 —n2 > 2mn, and 82 > m2 —n 7. Hearts in Contact; the same equation as before, only with 82 —m2 —n2 = 2mn 2mn 8. Intersecting Hearts, as B2y2={(m2+n2)±√ {(m2 —x2) (x2+n2)}, when m2>n2. 9. Intersecting Ovals; the same equation as in 8, only with m2>n2. All curves are here deemed Monads which can be drawn without taking the pencil off the paper. Duads are:-1. Twin Ovals (rot singly symmetrical on opposite sides), as a2y2 = (m2 — x2)(x2 —n2). 2. Twin Beans (or Hearts, Dicuamos). 3. Pair of Sandals (Disandalon): this has always two double tangents parallel, yet the disposition of the four points of contact is not the same in all cases. (They form a rectangle when D=0; they are in lines diverging from the centre when F-0.) 4. Pair of unsymmetrical Lemniscates, which I call Four Kites. Tetrads can only consist of unsymmetrical Ovals, symmetrically disposed. Of the infinite curves, one very limited class may be called Parabolic, those in which B=0 and D=0, which reduces the equation to the form a2y2=x++2hx2+k, when the locus is infinite. It has as curvilinear asymptotes the Proximate twin Parabolas, a22(x2+h). The Species are:-1, Twin Goblets; 2 (when their vertices unite), Pointed Goblets or Knotted Parabolic Hour-glass; 3, Parabolic Hour-glass; 4, Perforated Hour-glass (with disk in centre); 5, Hollow-bottomed Goblet. When A=0, we may have asymptotes parallel to the axis of x, and when B=0, to the axis of y. Such curves must be treated apart. When a Quartan Hyperbola is confined between parallel asymptotes, I call it an Arch, Round-headed or Hollowheaded, as the case may be; they are found, of course, in pairs. A Quartan Hyperbola which is confined within diverging asymptotes like the Conic Hyperbola, I call a Basin; when it crosses or otherwise envelopes its diverging asymptotes, I call it a Cup. Cups and basins may be Round-bottomed or Hollow-bottomed. Again, a Quartan Hyperbola may lie between an oblique and a vertical asymptote; I then call the Hyperbola itself Oblique, equally when it lies between two asymptotes of different systems. Such an Hyperbola may cut one, and only one, asymptote; then I call it Paratomous: if it cut both, it is an Oblique Cup. Cups may be Pointed at bottom and unite; they may be also in Contact at bottom, or they may intersect. Vertical and horizontal asymptotes develope other and simpler forms. Conchoids, grouped in pairs, generate one class, and Arches another. Arches may intersect, Basins also may intersect sideways; I call this Paratomy. Such are the elements (adding only Studs or Conjugate Points) of which all the loci are composed. Four Hyperbolas, of whatever class, are the utmost that can arise as locus of a Quartan equation; whether in square, each in one quadrant, or as Cross Arches, or as Oblique, or Oblique and Paratomous, or as around and crossing the axes, or between unsymmetrical asymptotes, or it may be Cups instead of Basins. In the midst of these infinite curves, some one of the Monad or Twin Ovals are often found as Satellites. It must be added that when AB-D2 and the locus is infinite, we find Oblique Parallel Asymptotes, and even, related to them, Oblique Paratomous Arches. Such is the general description of the forms. The investigation is simple. We know that a straight line can cut a Quartan at most in four points. This often shows what forms are impossible. ADE Put V D B F then by Conics we know that if V=0, our general equaEF C tion will degenerate into the product of two quadratic factors. Besides, if A, B, C are positive and E, F negative, and E2=AC, F2=BC, the equation degenerates into two ellipses. If F2=BC, and B, F have opposite signs, the curve crosses itself (in a Knot) where x=0 and By+F=0; but if B, F have the same sign, the Knots become Studs. Thus if E2=AC, and F2-BC, but E, F have opposite signs, there are two Knots on one axis and two Studs on the other. We find where the curve crosses its axis, by putting Ar+2Ex+C=0_when y=0, and By+2Fy3+C=0 when x2=0. Then if AC is positive, E must be negative, if there be any vertex in OX. If AC is negative, there are two vertices. Put T Ar+2Dx2y2+By+; .. T+(2E2x2+2F2y2+C)=0 is the equation to the curve. When T is essentially positive, the curve is finite. This happens when A, D, B are all of one sign, or when AB-D2 0. When A=0 and B=0, the curve is finite only when D, E, F are of one sign. If B=0, -y2= Ax+2Ex2+C 2Da2+F and the curve is finite when A, D, F are of one sign. If D and F are of opposite signs, there are asymptotes parallel to y, viz. 2Dx2+F=0. Indeed now T=(Ax2+2Dy2)x2; thus, if A and D have opposite signs, Ax2+2Dy2=0 are oblique asymptotes. When A, D, B are finite, solve the equation for y2, regarding B as positive. Put g=D-AB, h=DF-EB, k=F-CB; .. By2+Dx2+F=±√(gx1+2hx2+k). For D2-AB>0, we may assume g=1; then for the upper sign we get, as Proximate Conic Hyperbola, if D be <l, By+Dr2+F=x2+h. If D is negative, we have a second Proximate Conic for the lower sign, By,2+D+F=-x2-h. Of course the asymptotes are By2+Dx2+2, or only By2+Dx2=x2. If D2-AB=0, g=0, and the curve is infinite only when h is positive: then if D is negative, By2+Dx2+F=±√(2h)x is two Proximate Conic Hyperbolas, and the asymptotes are oblique and parallel in pairs; they do not pass through the centre, but are equidistant from it. Evidently if C=O and E, F have opposite signs, the curve crosses itself in the centre; but if C=0 and E, F have the same sign, the centre is a mere Stud. An undulation of the curve implies a double tangent. Such double tangents are always parallel to one axis. I desire a general proof.] There can be only two pairs parallel to one axis. To ascertain whether there is undulation across OY, put x=0, and try whether y2 is there a maximum or a minimum. Making a2 infinitesimal and k positive (which is implied), h x4 2k Nk' h whence y2 is a minimum at x2=0 if -D>0, a maximum if -D<0. Yet Nk h when =D, y2 is a minimum at x=0 if gk-h> 0, or a maximum if gk—h2 <0. Jk Now gk-h2-BV, and we cannot have V=0 without degeneracy. Hence this test is final. Also h=D √k is equivalent to BE2-2DEF+CD2=0. If the branch we are investigating is infinite and y2 is a minimum, there is no undulation; but if y2 is a maximum, it begins to decrease, yet must afterwards increase; hence there is undulation. On the contrary, if the upper branch be finite and y2 be a maximum, there is no undulation; but there is undulation if y2 be a minimum. In general, for tangents parallel to the axis of x, putting vious solution r=0, when there is a vertex on the axis of y. have a double tangent where D√(gx1+2hx2+k)=gx2+h, dy dx =0, we have the obBesides this, we may which yields gx2+h D /h2-gk (= √(gx1+2hx2+k)=By2+Dx2+F. D2-g Hence at the points of contact gx2+h=±D√ − (~~), gy2+h'=I√(−AV), if h'-DE-FA. When T(x2-p2y2) (p2x2+o1y2), the curve has oblique asymptotes, X'a2=μ3y 3· To try whether it ever cut its asymptotes, put y=y; then at the common point A2x2=μ3y, and 2Ex2+2Fy2+C=0. If the x, y hence determined is within the limits of the curve, it does thus cut; if not, it does not. IfT=(x2-μy) (p2x2-σ2y2), there are two pair of oblique asymptotes, X2x2=μ3y2, p2x2=σ22; and by combining either of them with the second equation, we decide on Paratomy. 2Ex2+2Fy2+C=0, When the general equation is given to us in this form and we desire to find the Proximate Conics, the most direct method is to assume p2=1, σ2 = 1, and whence or (y2-λ2x2-M)(y2-p2x2-N)=T+2(Ex2+F2y2)+MN; These are closer indications of the infinite branches than the asymptotes. Put MNC', ADE But in general it is expedient to put X=gx1+2hx2+k, and study the variations of X in the equation By2+Dx2+F=+X. In many cases the lower sign is inadmissible; in most it is more restricted than the upper. When we have only the upper, evidently there is no undulation across the axis of x; for y2 has then but one positive value for any given value of x. The forms of X are as follows: Remarks on Napier's original Method of Logarithms. By Professor PURSER. On Linear Differential Equations. By W. H. L. RUSSELL, F.R.S. The object of this paper was to explain the progress the author is making in his theory for the solution of Linear Differential Equations, especially when the complete integral involves logarithmic functions. On MacCullagh's Theorem. By W. H. L. RUSSELL, F.R.S. This paper was intended to simplify the process given by Dr. Salmon to prove MacCullagh's theorem relative to the focal properties of surfaces of the second order. Note on the Theory of a Point in Partitions. By J. J. SYLVESTER, F.R.S. In writing down all the solutions in positive integers of the indefinite Equation of Weight, x+2y+32+...=n, or, in other words, in exhibiting all the partitions |