504 
bof #*. If it cats £* in B,, B,, we must bear in mind that these 
points according to § 2 are for the surface {2° single points only, from 
which ensues that through those points only one plane a passes 
which comes in consideration if we make, as is done here, a point 
P to describe the surface and if we ask after the surface to be 
enveloped by the planes zr; this one plane however does not pass 
in general through 5. Besides b,, B, 6 has with 2° four more 
points S in common; through each of these evidently passes a plane 
x containing 6. 
However, there are of course now again planes a through 5, 
whilst point P lies outside 6. A plane 8 through 6 contains three 
chords a and these describe when 2 rotates round 5 a scroll of 
order four with 4? as a nodal curve and 5 as a single directrix ($ 3). 
The plane 8 contains moreover 6 chords of £4, of which however 
one coincides with 6, so that one diagonal point lies on 6 and two 
outside 5. These describe when 2 rotates round / a twisted curve 
of order four, resting in 6,, 6, on 6; if namely p touches 4* in 
B, or B,, it is easy to see that one of the two diagonal points lying 
in general outside h coincides with the point of contact. This curve 
of order four intersects fhe just mentioned scroll of order four in 
sixteen points, to which however belong B, and 4, as these lie in 
6 and therefore on the scroll too; if we set these aside, because 
they do not satisfy the question, fourteen are left, and these added 
to the four points on 4, which do satisfy the question, give us again 
the number 18. 
We can also determine by the way followed here the eighteen 
tangential planes of @,, through an entirely arbitrary line /. The 
chords of #? resting on / lie again on a surface of order four, and 
the diagonal points of the complete quadrangles in the planes 4 through 
/ lie on a curve of order five resting in two points on /; for, the 
chord a of 4° which we discussed above is for /* an arbitrary line, 
so it contains as many diagonal points as in the general case. The 
curve and the surface intersect each other now in twenty points, 
but to these belong the two points of intersection of the curve 
with /, which do not satisfy the question; so there are again 
eighteen left. 
8. An arbitrary plane through one of the twenty common chords 
of 4° and #* contains beside this chord, representing an a as well 
as a 4, one chord / more, cutting the other outside /*, and therefore 
it is a plane a to be counted once; so through each of the twenty 
chords pass an infinite number of tangential planes of 2,,, from 
