( 318 ) 

 Tbc first member of tbis equation is again an invariant of tbe sextic 



a; (- A)G + G ri (- If+lb X, (- Xf + . . . + 6 .r, (_ A) + .r^, 



whicb made equal to nougbt indicates tlio osculating space belonging 

 to tbe point L. 



Now we find in the ordinary manner by passing to the use of 

 symbolic coefficients and by noticing their mutual equivalence tbat 

 tbe indicated invariant may be represented by (a&)2(rtc)2(a(Pj(6c)2(6c?)2(crf)-. 

 Naturally if tbe invariant i) vanisbes there is a connection between 

 tbe six points of Cg whose osculating spaces pass through the point 

 A of tbe plane L\ L^ L^ indicated by the formulae (5), for substi- 

 tution of the values following out of (5) for tbe seven coordinates 

 •n in the equation of tbe osculating space of the point L gives 



Pia-^if+p2i>^->^2f+P3i^-h)' = ^ ... (6) 



So we have tbe following theorem : 



"Any tbree points L-^, L^, L^ on Cg determine on tbis curve an 

 "involution -?o" of tbe second dimension and tbe sixth order of which 

 "eacb of tbe three points counted six times represents a sextuple. 

 "Tbe osculating spaces belonging to the points of any sextuple inter- 

 "sect in a point A of tbe plane L^ L^ L-^ ; if tbe sextuple describes 

 "7e^, the point A generates in tbe plane L-^ L^ L^ a plane system in 

 "projective correspondence with /e*-" 



The considered invariant of a^^ is indicated by Sylvester as 

 "catalecticant" because its vanishing is the condition under which 

 aj^ can be represented as the sum of three sextic powers; in con- 

 nection with this an aj^ allowing tbis reduction is called a "meio- 

 catalectic" sextic [Pliil. Mofj. I. c. page 408). 



5. Finally we examine in the space S^" tbe locus of the linear 

 space S"-i baving n points Lj, L^, ■ ■ . L„ in common with the 

 normal curve C2n represented by 



1) If according to tlie common notation / = US'' and /• = ( ƒ, jy — (« 0}' lu- bz^ = 

 tlie t'ourlh transvectant of f with itself, tlien the indicated invariant is the fourtli 

 transvectaut (/.-, /••)'' of k with itself (see Gordan-Kirschensteiner " ForlumiiffeH über 

 Itwaricmtmitheorie", Vol 2, page 3S6). 



For the following case ƒ=«/ we have got to deal with an iuvariaut of the fifth 

 order in the coeflicients. There being (see a.o. von Galj/s two papers in the "Math. 

 Aim." Vol 17, p. 31—51 and 139—153, 1S90 on "Das volhtamlit/e lormuimyüem mier 

 biniiren Form adder Ordnnwf) hul one invariant of this kind, our invariant must in 

 this case correspond to the one indicated by the sign /i, t. 



