( 630 ) 
the point agr, whilst it touches Ba in its points of intersection with 
the curve 6," for which the corresponding ray passes through afz. 
The curve of the complex of the plane x has the right line Ba 
for n-fold tangent, so it is rational. 
If the curve 5,* touches the intersection Ba, then the multiple 
tangent is at the same time inflectional tangent. 
We now pay attention to the tangents 7 out of the point S= a’ 
to the curve 4* corresponding to a. The envelope of these tangents 
has the right line «8 as multiple tangent; its points of contact are 
the 2(n—1) coincidences of the involution, determined by the 
pencil (6") on eg. As S evidently sends out n(n — 1) right lines r 
the indicated envelope is of class (n — 1) (mn + 2). 
The planes containing a curve of the complex of which the n-fold 
tangent is at the same time injlectional tangent envelop a plane 
curve of class (n— 1) (n + 2). 
$ 7. The curve (a) can break up in three different ways. 
First the point egt may correspond to itself, so that (a) breaks 
up into a pencil and into a curve of class n. This evidently takes 
place when ar passes through one of the principal points Cp. 
Secondly the involution on gr may break up, so that all its groups 
contain a fixed point; then also a pencil of rays of the complex 
separates itself. This will take place, when 2 passes through one 
of the principal points Ay. 
Thirdly the curve a may contain the principal point A. Then the 
curve 4 corresponding to the ray a — aa determines on ga the 
vertices of 7 pencils, whilst also A is the vertex of a pencil. The 
curve a is then replaced by (+ 1) pencils. 
In a plane through ag, thus through all principal points Cy, the 
curve (a) consists of course also of (n + 1) pencils. 
A break up into two pencils with a curve of class (n — 1) takes 
place when the plane a contains two principal points B, or a point 
B, and a point Cy. 
§ 8. To obtain an analytical representation of the complex we 
can start from the equations 
Da) 5 wv, J Aw, = 0; 
Dai) 5 a + ab” ==" (5 
Here at and & are homogeneous functions of #2, 2, a, of 
order 7. 
For the points of intersection Y and Y of a ray of the complex 
