TRANSACTIONS OF THE SECTIONS. 21 



■which B=0 and D=0, which reduces the equation to the iovm.a'^y'^=.x*+2hx'+k, 

 when the locus is infinite. It has as curvilinear asymptotes the Proximate twin 

 Parabolas, a-y'^=(x-+h)-. The Species are : — 1, Twin Goblets ; 2 (when their 

 vertices imite), Pointed Gohlets or Knotted Parabolic Hour-glass; S, Parabolic 

 Hour-glass ; 4, Perforated Hom'-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 ciu-ves must be treated apart. When a Quartan Hyperbola is 

 confined between parallel asymptotes, I call it an Arch, Roimd-headed or Hollow- 

 headed, as the case may be ; they are foimd, of coiu'se, in pairs. A Quartan Hy- 

 ferbola which is confined within diverging asj-mptotes like the Conic Hyperbola, 

 call a Basin ; when it crosses or otherwise envelopes its diverging asymptotes, I call 

 it a Cup. Cups and basins may be Eoimd-bottomed or HoUow-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 

 dilferent 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 foi-ms. Conchoids, 



S'ouped in pairs, generate one class, and Arches another. Arches may intersect, 

 asins also may intersect sideways ; I call this Paratomy. Such are the elements 

 (adding only Studs or Conjugate Points) of which aU 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 quadi-ant, or as Cross Arches, 

 or as Oblique, or Oblique and Paratomous, or as ai-oimd and crossing the axes, or 

 between unsymmetrical asymptotes, or it may be Cups instead of Basins. 



In the midst of these infinite cui-ves, some one of the Monad or Twin Ovals 

 are often found as Satellites. It must be added that when AB=D- and the 

 locus is infinite, we find Oblique Parallel Asymptotes, and even, related to them, 

 Oblique Paratomous Arches. 



Such is the general description oi\hQ forms. The investigation is simple. 



We know that a straight line can cut a Quartan at most in four pomts. This 

 often shows what forms are impossible. 

 A D E 



Put V= DBF , then by Conies we know that if V=0, our general equa- 

 E F C 

 tion will degenerate into the product of two quadratic factors. Besides, if A, B, C 

 are positive and E, F negative, and E-=AC, F^=BC, the equation degenerates 

 into two ellipses. 



If F^=BC, and B, F have opposite signs, the curve crosses itself (in a Knot) 

 where .r^O and By--|-F=:0 ; but if B, F have the same sign, the Knots become 

 Studs. Thus if E-=AC, and F-=BC, but E, F have opposite signs, there are two 

 Knots on one axis and two Studs on the other. 



We find where the cm've crosses its axis, by putting Aa'*4-2E.i'--|-C=0 when 

 «/^=0, and Bif+2Yif+C=0 when .r^=0. Then if AC is positive, E must be ne- 

 gative, if there be any vertex in OX. If AC is negative, there are two vertices. 

 Put T=Aari-|-2D.ry+B«/S 

 .-. T+(2E-^x2-j-2Fy + 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 — D-> 0. 



When A=0 and B=0, the curve is finite only when D, E, F are of one sim. If 

 B=:0, J » > 



_,.2_ A.r*+2Ea:^4-C 

 •^ 2D.i--'-i-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. 2Dx-+F=0. Indeed now 



T=(Aa;H2Dy^).r2; 



thus, if A and D have opposite signs, A.x'^+2J)y^=.0 are oblique asymptotes. 



