( 54 ) 



4. The foi-nmla (D) and hence also the formula (A), which is now 

 proved for the case that the two surfaces are developables holds still 

 good when >Si and S^ are arbitrarj^ algebraic surfaces. Let S,^ and v^ 

 represent the degree of the total nodal curve and total cuspidal curve 

 of >Si, likewise §2 and v^ for S^. One of the formulae of Plücker 

 applied to an arbitrary plane section of S^ gives 



m^ = 11^^ — jij — 2 ^^ — 3 Vj , 



or 



= n^"^ — n^ — m^ — 2 §j — 3 r^. 



In like manner an arbitrary plane section of /S'2 gi\es 

 = 7^2' — '^ — ni^ — 2 ^2 — 3 i'2 

 hence 



= ^2 (»,^ — n^ — m^ — 2 §1 — 3 I'j) -f- n^ {n^^ — n^ - m^ — 2 §2 — Si'j) 

 or 



combining the formulae (Z)^) and {B) we get the formula {A). 



If ^§2 is ^ plane, n^ becomes equal to unity and 111^ equal to nought, 

 whilst the curve s becomes a plane section and the rank r of s 

 passes into the class of the plane section. So formula [A] gives for 

 that class 



r = mj — 2 (f — 3 /, 



which is indeed the class of a section of >S'i with a plane, having 

 with >S'i in (S points an ordinary contact and in •/ points a stationary 

 contact. 



5. If S^ is of the second degree and S^ of the degree n and 

 of the class m, the formula {A) gives for the rank of the curve 

 of intersection 



r =: 2 {m + w) — 2 d — 3 x- 



If S^ is a quadratic cone K^ this formula will be proved directly 

 once more as follows for the sake of verification. 



The rank of the curve of intersection s is the number of its 

 tangents meeting an arbitrary right line, e.g. a generator / of K^. 

 Each tangent of .v, meeting the generator / has three poijits in 

 common with the cone IC", in fact the two consecutive points it 

 has in common with .s' and its point of intersection with /, unless 

 the latter coincides with the point of contact to s. Each right line 

 having three points in common with A'^ lies entirely on lO. The 

 only tangents oï s meeting / are thus the generating lines of /v"'' which 

 are at the same time tangents of s and the tangents to ,s at its 

 points of intersection with /. The generator / of K^ meets >S\ and 

 therefore s too n times; through each of these points of intersection 



