y 
TRANSACTIONS OF SECTION A. 413 
Stapete-points, which are the dual forms of points of undulation, are discrimi- 
nated by a second curve (D,), the locus of («,A), when 
2. To determine curve (D,) when the quartics are bitangential. 
If the tests of singularity be applied to the quartic (#), 
K(1 + dé) =u, (& +97) + (EP + 0°)? 
xd = 2Ew,v, + [2mpE + (np + mq)n—(p+m)) (EF +n°) + ANE(E’ +n”) 
(A) 0 =2nu,v, + (2ngn + (mp + mg)E—(q+n)] (E +n") + 4\n(&* + n°) 
(B) «(4+ 8d) =[2—(mtp)E—(n+9)n] (E +7") 
By eliminating « and X, it appears that all the bitangents must touch the para- 
bolic cubic 
(C)  dn{(m+ p)E+ (n+ 9-2} = 
(4 + 3dE) { (mp + mg)(E? — n°) — 2(mp — nq) Ey + (m+ pn — (2 + QE} 
If &,n be eliminated from (A), (B), (C), the eliminant is the required curve 
(D,) referred to «, \ as ordinates. 
The curve (D,) is conveniently drawn by points, after drawing (C) by line —, 
or point —, co-ordinates: to each line or point in (C) a single value of A and « 
corresponds in (A) and (B). 
The curve (D,) has five asymptotes. 
To the values 
£=+n=o there correspond c=, 4X + mp + 2np + 2mq + 8ng =0. 
s= pel A=0,x=0. 
n=, A(n+q)+(4+8dé) (mptmg)=0, K=0,rA+ng=0. 
n=0, 4+8dé=0 corresponds «(1 + d&) = (m§ —1) (pE-1)& + rE". 
3. In order to obtain a!l possible varieties of curve (D,), the five cases of the 
cubic (C) must be examined. When 5S, the quartic, and 7’, the sextic, invariants of 
(C) have been found, the conditions are known for a nodal cubic (7° = 64"), and 
for a cusped cubic (S=0, 7=0). For slight alterations in the constants near these 
critical values the non-singular companion-cubics may be obtained. 
If two of the foci P, Q, R, S coincide, this cubic (C) degenerates into a conic 
and point. If R, S be at infinity, the line at infinity is a bitangent. 
4. To determine curve (D,) when the quartics of this group have stapete- 
oints. 
: If the equations to the quartic (¢ =0) be differentiated thrice, and the conditions 
introduced for a stapete-point 
the elimination of x, A gives the following equations: (¢ = =) 
{h(E +9?) (140) ~(E + nt)?} {mt nt) (pE + 91-1) + (pt gt) (mE +m —1)5 
=(mé + mn —1) (p+ 9n—-1) (E+nt) A+) 
I 2 plmtnt)(ptgt)Etnty  .« » (A) 
A (ede) (E+ nt) (E40) +0)-2 + 06) 
7 =P arysEaM Gem. . 6. B 
