502 
ji? +774, the section of 2° and 77,, by JF, then every point of inter- 
seètion with 4? counts for two, with r?* for one; therefore each of 
the twelve points under discussion counts for three. As the complete 
number of points of intersection of the three surfaces is 6.5.4 = 120, 
outside 4 there are 120—3.12— 84. It is wellknown that the tan- 
gents in these points to 7** pass through QO: thus the apparent circuit 
of 2° possesses eighty-four cusps. 
To determine the class of 2° and with it of the circumscribed cone, 
resp. the apparent circuit, we assume a second point O’, and we 
coustruct the first polar surface /7,’; this, too, passes through 4° and 
intersects the curve 7?! just found in 420 points of which twelve 
however lie on /*, and count singly, because 7** is a single section 
of 2* and 77, and # is again a single curve of 7/,; so outside 4? 
the three surfaces have 120 — 12 —108 points in common, so that 
the class of 2° amounts to 108. 
By applying the Pröcker formula v = u (u—l1) — 2d — 3x to the 
apparent circuit, we find 
2d = uw (u—1) -— vp — 84% = 24.23 — 108 — 3.84 
or 
d= 96. 
The projecting cone out of O contains therefore 96 double edges, 
the apparent circuit 96 nodal points. 
The PLickrErR equation dualistically related: 
u = v (p—1) — 21 — 381, 
applied to the apparent circuit furnishes us with 
27 + St =— vp (rl) —v = 108. 107 — 24 = 11532, 
whilst the third formula: «—x = 5 (»y—un) furnishes for « 
c— 84 + 3(108—24) — 336; 
go owe find: 27 = 11532 on 306 1052466 a — 262: 
Now however we have to remember that the planes through O 
and the twenty lines of 2° are threefold tangential planes of the 
cone, that their traces are therefore threefold tangents of the apparent 
circuit and that therefore they count together for sixty double tangents. 
If we subtract these from the entire number 5262, then for the uppa- 
rent circuit remain 5202 real double tangents completed by 20 three- 
fold ones. 
A cusp in the apparent circuit is generated by a principal tangent 
(a tangent with eontact in three points) of the surface passing through 
0; these principal tangents form a congruence, of which according 
to the above mentioned the first characteristic (number of rays through 
a point) is eighty-four. The second characteristic indicates the number 
