261 
coincide with Z,, the first polar surface therefore cuts that 
straight line as well in 2u — de — 25 points coinciding with Z,, 
and further in the 2r—1 harmonic poles of 7, with regard to the 
2p points of 2 not coinciding with 7. Now the first polar surface 
of Z must of course be symmetrical with regard to 3, as 2 is so 
too; but the number 2p 
can lie in infinity; consequently one must lie in 3, and this holds 
good for any straight line passing through Z,, 
Analytically too it is easy to see. The 2r points referred to higher 
up, may be represented by an equation of the SA 
(NRE (aye 
and the harmonical centres for the pole BG are ed from it by 
differentiation with regard to z; it appears then at once that each 
term contains the factor z. 
The first polar surface of Z, consequently breaks up into the 
plane 3 and a surface z,, which only reaches the order 2u-+-2r— 
4e —20—2, and only contains /” as a simple curve. By the way we 
will observe that @ contains still other torsal lines with vertical 
tangent planes, viz. the tangents out of the two absolute circle- 
points at kv ($ 2), and that these lines therefore also belong to the 
intersection of 2 with the first polar surface ; as they lie, however, 
in @, and are only to be counted once, they do not lie on 2,. 
The intersection of 2 with a, now consists of the following parts: 
J. The curve 4’. We saw already that 2, touches the two sheets 
of 2 passing through /» in £” itself, but those two sheets osculate 
each other, that is to say, in each intersection with a plane passing 
through Z, they have not 2 but 3 points in common; 2, must 
also pass through this third point, that is to say, osculates each of 
those two sheets, and consequently has the curve 4” six times in 
| is odd, while of these points not one 
common with 2. 
2. The curve 4. It is for 2 (v—o)-fold ($ 2), consequently for 
mn, (v )-fold: in the intersection of 2 and z, this conic counts 
therefore (v—-c) (»—o—1) times. 
3. The uw—2e—2o double torsal lines of 2 arising from the simple 
intersections of k# with /,; these lines show for a, the same char- 
acter as for @, that is to say, through each of these lines, which 
are to be considered twice as torsal lines, pass 2 sheets of 2 and 
2 of ‚so that such a line counts & times in the intersection. 
4. The » double torsal lines of 2, arising from the points of 
contact of kv with /; for them the same holds good as for those 
of the preceding group, each of these 5 lines therefore counts 5 
times in the intersection. 
