( 771 ) 



n f /r -f- 2 t" 4- ^'v^ = ^ ^ ;c' + 2 d" + r ^", ... (5) 

 where the two siijninations must also be extended to the imaginary 

 sing'uhirities with real points and tangents. 



§ 3. Proof of i'([iuillon (2) for d carve with real equation. 



The considerations of the two preceding paragraphs all lead up to 

 the equation (5) thus holding good for every curve with real equation 

 and with higher singuhxrities. The summations must be extended 

 only to the hhjlier singularities, namely '^'t^ to those with real points, 

 '21't\ to those with real tangents. 



The equation (5) can be considerably simplified by including also 

 the PLÜCKEU-singnlarities among the i^'-signs. 



Inflexion. For an intlexion we have ^ = 1, y ^ 2. If we omit 

 /i' but extend '^'t^ and '^' i\ to the real inflexions then in (5) due 

 consideration is taken of the presence of those inflexions. 



Cusp. For this / = 2, i? = 1, so that for the cusps the same holds 

 good as for the inflexions. 



Isolated point. An isolated point is formed hy two conjugate 

 imaginary elements, of which the points are real, thus coinciding, 

 the tangents imaginary, thus ditïering. For each of those elements 

 ^=r = l. If we extend the summations to the isolated points, this 

 has no influence on 2'i\ (the tangents being imaginary), Avhilst on 

 the contrary ^'t^ increases with 2ff". If now in (5) we omit the 

 term 2d", but extend 2't^ to the isolated points, the equation remains 

 true. 



Isolated bitangent. This is formed by two elements {t = v = 1) 

 with real tangents and imaginary points of contact. For this holds 

 good the correlative of what was obsei'ved for an isolated point. 



So if the summations are extended also to the PLücKER-singularities 

 the equation (5) becomes 



n^^'c,=k^:£\, (2) 



where, if one pleases, every other element of the curve may i)e 

 included arnona- the 2£"-si2ns. 



'to 



§ 4. Proof of equation (2) for a curve with unafumry equation. 



To prove the relation (2) for a cur\^e with imaginary equation, 

 we write it in the form : 



./ = /• - /' + ^'t, — ::£'i\ — 0. 



So we must show, Ihal also for a curve witli imaginary equation 

 ./ has Ihe value zero. liCl <f -\- i i]' = {) be the cquatioji of that 



51* 



