262 
5. The 2x cuspidal edges of 2, the 45°-lines passing through 
the cusps of 4” ($ 3). Tbe euspidal tangent planes of these cuspidal 
edges always pass through 7, so that a line Z, P connecting Z, 
with a point P of such a cuspidal edge has in P with 23, and 
consequently with a, 2 points in common. It is, however, easy to 
see that in the intersection of 2 with 2,, each cuspidal edge is to 
be counted 4 times, and this is only to be brought into con- 
formity with the rest if we accept that a cuspidal edge of 2 is a 
nodal edge of zr, 
_ That this must be so in fact is most easily seen in the example 
of the plane cubic with cusp. It is of class 3, and the polar conic of 
an arbitrary pole P? passes through the cusp, touches at the cuspidal 
tangent here and consequently bas 3 points in common here’ with 
the curve; the remaining 3 intersections are the points of contact 
of the 3 tangents out of £. If, however, P lies on the cuspidal 
tangent there touch at the curve but 2 tangents besides this one; 
the polar conic of / must therefore have now in the cusp 4 points 
in common with the curve, but the cuspidal tangent has in the cusp 
only 3 points in common with the curve, and consequently only 
2 points in common with the polar conic. These various conditions 
are only satisfied at the same time if the polar conic degenerates 
into a pair of lines whose node lies in the cusp. By applying this 
argument to the euspidal edges of £2 we easily find that they are 
nodal edges of a,, and consequently count + times in the inter- 
section with 2; and as through each cusp of 4’ pass two of those 
cuspidal edges, the share contributed by all those cuspidal edges to 
the intersection is of order 8z. 
6. The 2 torsal lines of @ passing through the points of inflee- 
tion of k* (vide supra). 
7. The rest nodal curve. It lies on 2, as a simple curve, and 
consequently is counted twice in the intersection. The calculation 
of order d of the nodal curve in this way is as follows. The sur- 
faces 2 and a, are respectively of the orders 2u-}+-2r—4e—2o and 
4+s:—2o—2; the order of their complete intersection is 
therefore the product of these two numbers. In order to find 2d 
now this product is to be diminished by 6u, 2(r—o)(r—o—1),8(u— 
Ye—45); 80, 8x, 20. 
2u--2p 
$ 10. The second polar surface of 7, with regard to 2, which 
we shall call z,, is of order 2u-+2r—te—2o0—2, consequently as 2 
of even order, and therefore need not break up as the former. In 
fact, if we represent the intersections of a line passing through Z,, 
