1451 
$ 2. We will now investigate the influence which have the 
singularities d, x. f, r‚£,5 provisionally supposed equal to zero in the 
preceding §; that it is necessary to consider them follows among 
others from this that already in the two simplest cases imaginable, 
viz. if £” is a straight line or a circle, the number 2u* appears to 
be incorrect for the order of £2; for the straight line, £2 is evidently 
the vertical plane passing through that line, so m = 1, and for the 
cirele {2 is, as is known, the hyperboloid of revolution of one sheet 
with that circle as gorge, so m= 2, whereas 2° would give 2 
and respectively 8. The differences are easy to explain in either 
case. The plane is apparently to be counted twice, as through each 
of its points two 45°-lines pass; for the hyperboloid of revolution 
the same holds good, but there the circle passes moreover through 
the two points /,,, /,,,, so that ¢ = 1, and consequently the influence 
of e must be investigated. 
Let us now suppose that a tangent ¢ has been drawn out of 
I,,, to kv, we then have to connect the point of contact P with /,, 
according to § 1; if, however, 4” itself passes through /,,, and if 
tis the tangent in this point, then the line P/,, becomes indefinite 
in the plane passing through ¢ and Z,, so that the peneil with 
vertex 7, lying in this plane branches off, and that twice, as the 
loo? 
tangent ¢ represents two coinciding tangents of 4”; every time 
therefore when 4” passes through one of the cyclic points, a pencil, 
counted twice, branches off from @. In our example mentioned 
above, we found e=—=1, consequently two planes, each counted 
twice, branch off from {2; the order of the complete surface was 
8, and is therefore reduced to 4, as the twice to be counted 
hyperboloid of revolution requires. 
Besides « the number o (the number of times that /” touches the 
straight line /, of 8) is also of influence on the order of the “true” 
surface £2, as easily appears from the following consideration. 
According to § 1 the base 3 can contain beside the nodal curve 
kv only isotropic generatrices of £2; through /,, pass only y—2e—o 
tangents, not having their point of contact on /,, so only 2(~~—2e—o) 
isotropic generatices lie in 8; if we are therefore able to prove that 
Ll, itself does not belong to the “true” surface, i is proved by this 
that the order of Q is: 
m = 2(u + vw — Ze — 6). 
As to / we may observe the following. We have to intersect 
each tangent ¢ of 4” with /,, to connect the intersection with Ls 
and to connect the two intersections of this connecting line and 
