( 772 ) 



curve, were (f and if? possess but real coefficients. For that curve J 

 has of course the same vahie as for tlie curve <f — i if' =r 0, e. g. 

 (he vahie J^. 



For the curve ^- -j- ip^ = 0, consisting of the two tirst-nientioued 

 curves, ./ has thus the value '2J^, as J consists only of terms, which 

 are formed additively for a degenerated curve out of the corresponding 

 terms of the partial curves. The equation of that curve is however 

 real, so that the relation (2) is applicable to it. Fr«nn this ensues 

 J=1J,^0, so J,=0. 



This proves, that J has the value zero for the curve </ -|- / if ? = ( ) 

 too, and that for this the equation (2) also holds good. 



For this deduction we have tacitly made the supposition that q) 

 and If? have no common divisor, as otherwise ihe curve y^-["^^ = ^ 

 would possess a part counting double and thus an infinite number 

 of singularities. If (p and if? have such a common divisor, that is if 

 the curve degenerates into a curve with real equation and into one 

 with an imaginary one, the relation (2) still holds good. For, as we 

 have seen, this is the case for the two partial curves, from which 

 ensues b}^ addition the corresponding equation for the total curve. 



Klein (1. c, p. 207) linds by applying his e(|uation to the curve 

 (f"^ -[- if;"^ = 0, for a curve with imaginary equation, of which the /) 

 real points and the /• real tangents are not singular, the relation 



II ^ )' ^z. k -\- p (8) 



This equation can be immediately deduced from (2). 



Farthei-more ensues from (2) : 



The equation (8) of Klein for imagiiKiri/ curee.'^ holds rjood aUo 

 if that carve possesses real singular points or real singular tangents 

 if only they count for as many real points or tangents as is indicated 

 hy the order resp. the class. 



§ 5. (Jther forms for the equation (2). 



The equation (2) can be reduced to still other forms. The Plücker- 

 equation o {k — n) i:^ ,■? — y. becomes namely for a cui've wqth higher 

 singularities 8 (/.: — //) = ,i — y- + ^ d^* — y-,% or according to (4) ') 

 3 (A — /<) z= (i — X 4- 2£ {i\ — tj, where the JS'-sign must be extended 

 to the higher singularities, the real ones as well as the imaginary 

 ones. By including the PLüCKER-singularities, and if one likes also 

 the ordinary points of the curve, under the JS'-sigu the equation 

 becomes 



^) See note p. 767 — 708. 



