942 
X, different from the tangent to u, in X,; then the C, describes a 
pencil, one curve of which passes through any point of &. The coin- 
cidence of A with X, then furnishes a C, having in X, three points 
in common with u, and two points with 4; so this C, has a node in 
X, and one of its branches touches w,. If now we allow £ to move 
along LX, to X, and afterwards M along MX, to X,, we generate 
a C, having still threefold points in D,, D,....D,, now admitting 
a ninth threefold point in X, and passing moreover through an 
arbitrarily chosen point .V (compare § 3). So the point X, isa point 
of the curve j, under discussion. Therefore : 
The curve jr; cuts any curve of (3’) besides in the base points in 8 
points more. It is at the same time the locus of the points forming 
with B, on the curves of (B) a Steinerian pair of the third order. 
§ 8. In order to determine the curve j, more closely it is neees- 
sary to know the order of multiplicity of the points D,, D,,..., Ds 
on it, i. e. how many times each of these points happens to form 
with B, a Steinerian pair of order three on a curve of (8’). Let u, 
(fig. 2) be once more an arbitrary curve of (8/); then we project 
B, out of D, on u, (i. e. we determine the third point A, common 
to D,B, and u,), from this point A, we project D, on uw, into A,, 
from A, we once more project B, on u, into A, and so on, alter- 
nately projecting B, and D,. Then we allow wu, to describe the 
pencil (3’) and determine the loci of the points Ay; Age sy Aap ten 
every coincidence of A, with J, points to a curve out of (8) on 
which 2, and D, form a Steinerian pair of order three. 
So we find for the locus of 
A, > theshne. Dr B 
A,: a C, with a double point in D,, not passing through D, and 
B, ibut cointainine B De „4D 
A,: a C, with an ordinary point in D,, 
a threefold point in D,, 
double poim#ts.an’ 0. Dad 
a fourfold point in &, ; 
8? 
A: a C., with a sixfold point in 2,, 
a threefold point in D,, 
fourtold: pons an de eI a One 
a double point in JB, ; 
A,; a C,, with a fourfold point in D,, 
a sevenfold point in D,, 
sixfold -pemis-in: De Di DE 
a ninefold point in JB, ; 
