Mr. J. J. Sylvester on a new Theory of Multiplicity. 463 



of first polar contact or of evection, a cusp situated or having its 

 nidus, so to say, at a point of inflexion. In other words, x=0, 

 y = will be a point whose multiplicity is intermediate between 

 that of the cusp and that of a so-called triple point, when the cha- 

 racteristic of the curve admits of being written under the form 



2 n-2 x Z + ^-3(^3 + J lx <2y + i xy <t} + 2 n-i &<._ . 



or in other words, when over and above the vanishing of tbe con- 

 stant and linear coefficients, and the quadratic coefficient being 

 a perfect square, as in the case of an ordinary cusp, this square 

 has a factor in common with the next (the cubic) coefficient ; or 

 again, in other words, a curve has a point for which the mul- 

 tiplicity is 3 when its characteristic function admits of being 

 expanded according to the powers of one of the variables, in 

 such a manner that the first coefficient and the second (the 

 linear) coefficient vanish, and that the discriminant of the third 

 and the resultant of the third and fourth are both at the same 

 time zero. This being the case, it may be shown that the first 

 polars will all have with each other a contact of the second 

 degree ; and moreover, that all the evectants of the discrimi- 

 nant will have as a common factor a linear function of the 

 variables, raised to a power whose index is 3 times that of 

 the characteristic function. As, then, there is but one kind of 

 ordinary double point, and but one kind of point with multi- 

 plicity 2, so there is one, and only one, kind of point with a 

 multiplicity 3. A cusp is a peculiar double point ; a flex-cusp 

 (as for the moment I call the point last above discussed) is a 

 peculiar cusp. This law of unambiguity, however, appears to 

 stop at the third degree. A so-called triple point (which ought 

 in fact to be called a quintuple point) is a point for which the 

 multiplicity, as shown above, is of the fourth degree ; but it is 

 not the only point of that degree of multiplicity. Without 

 assuming to have exhausted every possible supposition upon which 

 such a degree of multiplicity may be brought into existence, it 

 will be sufficient to take as an example a curve whose character- 

 istic is capable of assuming the form 



z n ~ 2 .z' 2 +Zn- 3 {gx 3 +hx' 2 y)-\-z n - i .(kx' i +lx 3 y+mx' 2 y' 2 +nxy 3 )+z n - !i Scc. 

 It may readily be demonstrated that the first polars of this 

 curve have all with one another at the point x, y a contact of a 

 degree exceeding the 2nd, i. e. of at least the 3rd degree (and, I 

 believe, in general not higher). Now the point x, y is evidently 

 not a triple-branched point, but a cusp with three additional 

 degrees of singularity; so that we have evidence of the existence 

 of a point whose degree of singularity is 5, and whose multipli- 

 city is at least 4, but which is in no sense a modified triple point. 

 It is probably true (but to demonstrate this requires a further 



