268 
stated, a number of points remain which must of necessity coincide 
in groups of 3. If we therefore call this number w, the number of 
triple points of the rest nodal curve is 4+ zv. These points lie in pairs 
symmetrical with regard to 3; the number of points in 8 therefore, 
which are centres of circles that cut k” thrice perpendicularly, is & «, 
and these points are triple pöints for the locus of the points of equal 
tangents. 
We find the following formula for w: 
v = (2u-+ 2v—4¢e—26 — 2) d—2 (p—2e — 0)’ —2 (5u — 3p + 3u— 86 —30) — 
— 16(u—se—2)(v —2¢ — 6) —24d — 18x —a —b —c — 46 —(Q9 -20) (v-0-2)— 
— 6x(2u-+ 2v—4e —26—5) . 
It is of course possible to express « exclusively in the funda- 
mental characters chosen by us, viz. u,v, €6,¢; the formula in that 
case, however, becomes very intricate, so we prefer to leave it in the 
form given here, a form which is not more circumstantial for 
the caleulation, and has the advantage that of the parts that must 
be subtracted, the meaning is easily recognized. 
If it is applied either to the general conic or the parabola, it 
gives «=O, which is correct, as with the conic no circles can 
the 
3 
appear that cut the curve thrice perpendicularly ; for the c,’ 
calculation “Is. -as\ follows?) gp d, Edd OF ee 
berg 120 bt he 56) 24. VCombeq ment Ly, jas 
= 10.36—18—18—18—48—1 20--24—42 —72; there are there- 
fore 12 points that are centres of circles that cut c,’ thrice perpen- 
dicularly and are therefore triple points of the locus of equal tangents 
at £# this locus has moreover 7 cusps on the cuspidal tangent of 
(“, and with that line itself as tangent; they are the centres of the 
circles that cut c,* perpendicularly in the cusp and moreover some- 
where’else. Por oc, we have: @ 3,10 4 2 =o — 0) dk 
ik LAS ii lel Ue p= A == 36, and so: 
w — 12.48 — 32 — 24 — 64 — 24 — 120 — 72 = 240 ; 
the locus of the points of equal tangents has therefore 40 triple points. 
To wind up this § we will now sum up what we have found 
of the locus of the points of equal tangents at /”. 
This curve is of order d* (§ 6); in each node of k” it has 2 cusps, 
while through each cusp of k’ pass 3 branches, which all touch at 
the cuspidal tangent. Further it passes through each vertex and through 
each focus of ke and it possesses nodes in the intersections of the 
asymptotes of hk’, while the nodal tangents bisect the asymptotal angles. 
Its. points at infinity are: 1. 26(6—1) simple points (§ 7), 2. 
ude do 20-fold points lying in the entersections of k* with |l, 
kenne 
