508 
We control this result by using a second formula of ScuuBerr 
viz '): 
Op +29 + £8 = geh gE An 
where op indicates the number of pairs whose components without 
lying at infinitesimal distance intersect each other, whilst the point 
of intersection lies in a given plane, thus evidently in our case 
the order of the complete nodal curve, however taken twice, be- 
cause each ray can be a g as well as an 4, and therefore each 
pair of rays satisfying the condition op counts for two pairs; g, 
designates the number of pairs where the line y passes through 
a given point, a number which is evidently zero in our case, 
because all our rays belong to a surface and can therefore not 
pass through a point taken arbitrarily; for the same reason we 
find 4, zero. On the other hand gh designates the number of 
pairs where g intersects a given line 7, and A a given line J, 
a number which in our case evidently amounts to 24.24 = 576, 
because /, is intersected by twenty-four generatrices g, /, by twenty- 
four generatrices /, and each line of one group can be joined to 
each of the other. As eq=24, ¢8=46, op becomes 576—24—46—506, 
and as the order of the nodal curve is half of it, we find back the 
quantity 253. 
In the formula: 
Ge + 8&9 + B= ge + gh + he,’*) 
which is dualistically opposite to the last but one, oe indicates the 
number of pairs of rays whose components intersect each other 
and whose plane passes through a given point. Now, too, each pair 
we find is counted double, because each ray can be g as wellash; 
so soe is the class of the developable, enveloped by the double 
tangential planes of 2*'. The quantity g, indicates the number of 
pairs where the ray g lies in a given plane, and A, indicates the 
same for 4; both numbers are in our case evidently zero; and from 
this ensues oe = 6p = 506, so that the class of the doubly circum- 
scribed developable of 2°* amounts to 253. 
For the sake of completeness we shall discuss in short the sur- 
face formed by the chords of 4% resting on #*. Through any point of 
k* passes one, so that #* is a single curve: through any point of 4? 
on the other hand eight pass, because the quadratic cone projecting 
k* out of that point is intersected by £* in eight points; so &* is 
an eightfold curve. From this ensues again that an arbitrary chord 
bic pe GO, NE Be: 
=) die. ps 60, NOEDE 
