( 502 ) 



(ai , a'i ), (6 , h') {ai , a'i ), (c , c') {a , a', ), (d , c?') 



and tlie corresponding surfaces 0J°, 0^^, (>ƒ, in order to propose 

 the question liow many of the 2336 points of intersection of OT 

 and o !-*^ satisfy the conditions of the problem. 



Here we must fix our attention upon the following groups of points 

 w^hich are to be discarded : 



a. the twenty common nodes D of 0^^, 0^^, O^f , 



b. the hundred nodes Ei, of 0]^ and the hundred nodes ^^ of 0""', 



c 



c. the sixteen points F common to qJ'*^ and any of the ten lines a,-, 



d. the twelve points G common to (>'•*<' and each of the twenty 



transversals of two of the five pairs {(ii,a'i), 



e. the forty points H common to any of the five surfaces 0* 



and the corresponding curve q]^. 



We consider each of these five groups separately. 



a. The 20 points JJ coiuit thrice among the points common to 

 oi-*'"' and O^^, for the curve touches in the node of the surface the 



^ b,c d 



tangential cone of the surface. 



b. P^ach of the 200 points E^j , E.: counts once. 



c. A point F common to a, and ^j^'*^ counts for four points of 



intersection ; for the cone with vertex F corresponding to the six 

 pairs {n^ , a\) , . . . , {a^ , n\), {b , b'), (c , c') cuts a\ twice and F lies 

 on a double line of 0^^. 



c 



(I. A point G common to ^1,2 and q^^^^ counts oiice. 



e. A point H common to 0* . and o'o lies on o'^^^ as it emits 



three complanar transversals to {a^,a\), {b,b'), {c,c') and four com- 

 planar transversals to the other pairs {a^,a\), ..,{a^,a\). As the tan- 

 gential plane in H to OJ" cuts the common tangential plane of (9^^ 

 and 0'^ in H according to the tangent in i7 to ()[^ the point ^counts 



for one point of intersection. 



As all these groups of points admit the property that the cone 

 for 0\^ differs from the cone corresponding to 0\^ and 0'^, they 



must be discarded. So the required number is 



2336 — 3 . 20 — 200 — 4 . 10 . 16 — 20 . 12 — 5 . 40 = 996. 



We can check easily the obtained result. In the special case of 

 eight pairs of intersecting lines, where (Ai , oi), f ^ 1, 2, . . . , 8 indi- 

 cate point of intersection and connecting plane for each pair, we 

 find the following solutions: 



