( 3in ) 



3. Tlie osculating space belonging to tlie point L of C4 has for 

 equation 



3-0 A* — 4 3^1 A3 + 6 .7-2 A3 — 4 3:3 A + «-4 = . . . (3) 



So the coordinates .rg, .ci , xo,, xo, x^ are the coefficients of the form 



«■0 (- If + 4 .ri (- If + 6 .r^ (- If + 4.3 (- A) + -u 



of the fourth order in ( — A) , bare of the binomial factors. 



So the result (2) can be written as j = and represented symboli- 

 cally by {hcf {caf {ahf z= (i (see a. 0. Clebsch-Lindkmann's '^]^orle- 

 sumjen ilber Geometvié'\ I, page 229). From this ensues at the same 

 time that any point of the obtained locus is distinguished moreover 

 from any point taken at random in the space 'S* by the property that 

 the four osculating spaces of C4 passing through it belong to four 

 harmonic poiuts of the curve. We shall point this out more directly. 

 We suppose in the formulae (1) the (juantities Xi,'k2^P\iP2, to be 

 given ; then by substituting in (3) the values ensuing from this for 

 n. we shall find 



Pi{^-hf^P2Q^-hf = ^ • (4) 



as the equation which gives us the parametervalues of the four 

 points L of C4 , whose osculating spaces intersect in the point A 

 of the line L] L^ given by(l). If for convenience' sake we take the 

 points Li and Lj as base points with the parametervalues and 00, 

 this equation can be reduced to A*— 1=:0 and the roots 1,-1, 

 V — 1, — V — 1 show immediately that the pairs of points belonging 

 to (1,-1) and \y — 1,-1^ — 1) separate each other harmonically, 

 whilst each of these pairs behaves in tlie same way with reference 

 to the pair of base points L-^ Z3 belonging to (0, co). By this, not 

 only the harmonic position of the four points (4) has been indicated 

 but moreover the following theorem has been proved: 



"Any two points L^ , L^ on Ca determine on this curve a qua- 

 "dratic involution I^ of which they are the double points. If ot 

 "this 1 2 we join two pairs separating each other harmonically we 

 "get the biquadratic involution /} represented by the equation (4) 



work [Phil. Mag. 1. c.) first a binary form of odd degree is discussed and tlie 

 invariant belonging to forms of even degree are reached at the end by a round-about 

 way. In Salmon (1. c.) and otherwhere the determinant is regarded as an extension 

 of the covariant of Hesse built up out of second differentialquotients, here differen- 

 tialquotients of the éth order, etc. 



