TRANSACTIONS OF THE SECTIONS. H 



T3 mav be (1) an algebraic cube, (2) may bave two equal factors, (3) may bare 

 only one real factor, (4) may bavo three' real unequal factors. Accordingly the 

 curves fall under four distinct groups. 



The first group, if T„(or 'Ax'^+2D'xy+B'y-) contains in square the same factor 

 which in T3 is in cube, is reducible to the single species mj- = x^, the Twisted 

 Parabola, here called the Whip-snake. If T^ contains, once only, the factor 

 which is cubical in T,„ the curve is known as the Trident, and is reducible to 

 aij = x-—¥x~^ ; but this is neither centric nor diametral, but scalene. If T, has 

 no factor common to T^, then (in this group) the equation is reducible to y- = X, 

 and the curve is here called Calyx. When further reduced to ay-—x^+Cx-\-I> 

 the new origin is here called the Pole. The curve is of at least six species, here 

 named Lily, when C = 0, D = 0; Tulip, especially when C and 1) are both positive; 

 Hyacinth, when 0=0 and D is ])0>iitu-e ; perhaps Convolvulus when C = and D is 

 negatioe ; Pink, wlien C = - W, D = 2c^ if <■ be positive and > & ; but if c = - 36- and 

 'D-2b\ it is Fuchsia (or Fucia), the hiwtted Calyx. If c= - Sb'-, and D = —-2b>, it 

 is the studded Calyx, or Anti-fucia. If C= -36= and T)—-2c', and c<b, the curve is 

 here called Bulbus ; being a calyx with a bulbous poot below it. 



The curves of the second group here treated fall into two classes. First, the Palm- 

 stems, xy- = a" ; the Archer's Bow, xy- = '3b-(a - x); the Twisted Bow, x(ij- + b-) = aby ; 

 the Pilaster (?) x(y--h-) = aby ; the Archway or Tunnel, x(y- — b'') = ab'\ A second 

 (diametral) class is called Vas, xy-=X=7nx'^ -f nx+p, in which m is always posi- 

 tive. The parabola, y'--mx-\-n, is asymptotic to it, and has elegant geometrical 

 relations witli it. Wlien X has no real factor, the curve is called the Urn, if 

 X-m\(x-^by-tc-}; but the Goblet ii X = m{(x-b)2+c-}. AVhen c=0, the 

 latter becomes the Knotted Goblet; but the case of X = m(-v+bf is called 

 the Studded Goblet. An outlying conjugate point is in this paper entitled 

 a Stud (clavus), and the curve which has one. Studded (clavatus)." But X 

 Ttmj have two real factors, under which division there are three species. For 

 a:y'-=a(a;+6)(a?+c) is an Urn with an outlying oval, and is called the Dripping 

 Urn. But xy^ = a(x — b)(x—c) is a Goblet -uith broken stem ; xy'- = a{x + b){x—c) 

 further has the foot reversed. 



The diametral curves of the third group are reducible to the form 



/x^a;/=D-|-3Cx+3Ba;^ -a;' = X, 

 where D is positive. They naturally fall into four classes, in strict analogy to 

 the four groups of Tertians. When X= (a — x)'', there is but one species, here 

 called Pj-ramid ; but when ju- = l, it is the Kissoid of Diodes. Pyramid is the 

 analogue of Lily. If X = (a—x)(b—x)-, and a^6, we get the Festoon (Cirrus), 

 which is the analogue of Fucia, being knotted. But if a <b, it is the Overstudded 

 Hillock; and when X = (a — a;)(.j;+6)^ the curve is in general the Uuderstudded 

 Hillock. Yet if a > 8i, the head overhangs the sides, and it is called Capito 

 (Great Head) ; and when a = Sb, the sides appear at one point perpendicular. The 

 last is called Cassis (Helmet). This makes five species in second class. But in fact 

 four of them are only degenerate forms of third or fourth class. 



In the third class, X=(a — a')|(a' + i)'--f c-|. That a curve may be a Conchoid, 

 generally, when X is a function of x of anj' degree, but essentially positive, we 

 must have xy-=(a— x)X. The simplest case is X = constant, which makes the 

 Archer's Bow in our second group. But in the third group, tor a Conchoid (here 



regarded as a hill, whether Tumidus or Mons), -^ —.0 must be impossible within 



ax 



the limits of the curve, .•. we need D>B''. Thus, \i ij.-xy-={a—x')[(x-^b)-+c-^^ 



and r^ = b^-\-c', we need ar^:^ ( — ^ 1 . But the c/ut/ Tertian Conchoid is xy-= 



a' — x^, with polar equation p-'cos6—a^. To this T3 + C = is reducible in this. 



group. But wlien D = B'', -^ =0 has two e^ual roots, and a tangent parallel to x 



cuts the curve in its point of flexure. This is the Tombstone, Cippus. And when 



B-' >• D, ^ =0 has three real roots, and t/ has both a maximum and a minimum. 

 ax 



