1)1 REPORT — 1869. 



This curve is called the Sphinx. Capito is a studded Sphinx, and Cassis a studded 

 Cippus. In the fourth class X has three real unequal factors, and there is an outlying 

 oval. When X=(a — x)(b — x)(c — x) it is called Mountain and Moon. AVhen 

 X=(a — x)(b-\-x){c+x),it is Mountain and Tarn, provided that D>B''. But if 

 B' > D, we get the Clock and Pendulum, differing from Capito only in having an 

 oval for a stud. If D = B^, you have tlie Bell and Clapper, which differs similarly 

 from Helmet. 



The third group has also a small class of centric curves, x{m'^x--\-fi-i/'^) +2(Ex+Fy) 

 = ; but thej^ are most elegantly treated hj the polar equation of their chief variety. 

 Under p^=a-tan d+b" we include three curves, the third being obtained by b=0. 

 Each is a Twisted Bow ; they differ only in the amount of twist. The triple sys- 

 tem is here called Cornutus. All centric systems are of the form T, + T,=0. 



Under the fourth group we have an analogous centric system, reducible by alike 

 compact to p^ cos 26= a" tan d+b- ; but there are five species, as it is now important 

 whether a > Zi or « < 6. (We cannot have a = 6 without degeneracy.) These curves 

 are called the Buttei-fly. The Twisted Bow appears like two -wings. Two hyper- 

 bolas are added. There are three rectilinear asymptotes, as always in this fourth 

 group. 



It is here shown that for a single diameter it is essential ritJier that the curve be 

 of the first group, or else the general equation must admit the form (x— ey)U' +C=0, 

 where the equation U'=0 represents a conic curve ; and the equation of the dia- 

 meter whose chords are parallel to x — ey = is precisely the same in the Tertian as 

 with this conic. But further, when Tg has three real unequal factors, three diame- 

 ters become possible. If the diametral equation in this gi-oup be expressed by 

 fx-xy-=.i'^ — 2ax'^ + Cx-\-T), we find two additional diameters when C= a", which 

 makes ju^ary'":=a:(.T — o)^-(-U; such curves are called Trijuga or Triga. There are 

 three species. First, when a = 0, we have the Starry Triga, ii'xy-=x^ — P. The 

 other two species are conjugate and expressible by ix-xy- = .v(x — a)-+b^. The chief 

 Stariy Triga (when /i^=l) has a remarkable polar equation. 



When the three asymptotes make a triangle, this is called the Cloister. When 

 they all pass through one point, the figure is called Starry. The Cloister is here 

 generally drawn with its vertex upward. (Many drawings accompanied this 

 paper.) 



Taking ^^ir3/^=a;^ — 2f«--f-Ca:-l-D = X, when C is not =a' we may always suppose 

 n positive, except when a=0, which makes the system Stany. The height of the 

 Cloister is a. Now, first, if X=(,r — w)%we have an analogue to Pyramid; itishere 

 called the Crane. The height of the Cloister is ~n. Next, if X=(a:— «i)^(a:— «) 

 and »i> n, it is Crane and Sack, with some analogy to Festoon. Both are special 

 cases of X=(.r— m)(.r — n)(.T—p), which is called Swing and Chair, the oval oeing 

 conceived of as a chair. But ifX = {.r—7n){x+n){.r+p}, the curve is called Trophj^, 

 as it seems to show a Bow, with Shields and Spears. If in the last n=p, the curves 

 cross, and we have the Knotted Flower-pot. In fact if X has onlj' one real factor, 

 saj, fji"xy- = (.T — m)\{x-l-by' + c'-^ and r-=6'^+c-. If r-={^m+b]-, we get Triga. 

 But if P > i^m+by, the curve is the Swing without the Chair. But if (^m+&)= > r^, 

 the curve is the Flower-pot, if we conceive of the upper hyperbola as a calyx. The 

 two lower hyperbolas are so twisted as to exhibit a kind of urn. And whenc = 0, 

 we regain the Knotted Flower-pot. This completes the enumeration. 



It is observed that in certain cases the asymptotes have quadratic approach to 

 the curve ; and the paper contains various other details. 



Postscript. 



It is possible for the Swing and Chair to become a Triga, For this, in 

 ix'.vy-=(j.'—m){.x — n){.v—p)^e need as a condition {m-\-n+p)-=4{mn + mp-\-np), or 

 (7n—n~py=4np, which can evidently be fulfilled. Indeed cither side may be 

 the gi-eater. Conversely, a Triga of the form fi-y'= (r— a)" - 6lr~ ' (where a and b^ 

 have the same sign) may at the same time belong to the species Swing and Chair ; 

 that is, have an outlying oval within the Cloister. The condition is that .lix -a)- — b^ 

 shall be resolvable into three real unequal factors ; so then must {^+a)^- — P ; which 



