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



where three branches of the curve intersect. Supposing #=0, 

 y=0, to represent such a point, the characteristic of the curve 

 must be reducible to the form (gx 3 + hx^y + kxrf- + ly 3 )^ 1 ' 3 + See., 

 which, as is well known, involves the existence of four conditions. 

 This, however, would not in itself be at all conclusive against the 

 multiplicity at a triple point being only of the third degree ; for 

 it can readily be shown that there may exist singular points of any 

 degree of singularity (as measured by the number of conditions 

 necessary to be satisfied in order that such singularity may come 

 into existence), but for which the multiplicity may be as low as 

 we please ; as, for instance, if at a double point (which is not a 

 cusp) there be a point of inflexion on one branch or on both, or 

 a point of undulation, or any other singularity whatever, still pro- 

 vided there be no cusps, the multiplicity will stick at the first 

 degree and never exceed it ; for only the discriminant itself will 

 vanish on these suppositions, but no evectant of the discriminant. 

 The reason, on the contrary, why a so-called triple point must 

 be said to have a multiplicity of the degree 4, and not merely of 

 the degree 3, springs from the fact that the 1st, 2nd, and 3rd 

 evectants of the discriminant all vanish at such a point. 



It is clear, then, that there ought to exist a species of multi- 

 plicity for which the 1st and 2nd evectants vanish, but not the 

 3rd. In fact, as at a double point the first polars all merely in- 

 tersect, but at a cusp have all a contact with one another of the 

 first degree, so we ought to expect that there should exist a 

 species of multiple point such that all the first polars should have 

 with each other a contact of the second degree (or if we like so 

 to say, the same curvature) at that point. When the curve has 

 a triple point, all its first polars will have that point upon them 

 as a double point ; and it is not at the first glance, easy a priori 

 to say what is the nature of the contact between two curves which 

 intersect at a point which is a double point to each of them : we 

 know upon settled analytical principles, that when one curve 

 having a double point is crossed there by another curve not having 

 a double point, that the two must be said to have with one another, 

 a contact of the 1st degree ; and we now learn from our theory of 

 evection, that if each have a double point at the meeting-point, the 

 degree of the contact must from principles of analogy be con- 

 sidered to be of the 3rd degree*. Now, then, we come to the 

 question of deciding definitely what is a multiple point for which 

 the degree of multiplicity is 3. It is, adopting either test, whether 



* This may easily be verified by direct analytical means ; as also the more 

 general proposition, that two curves meeting at a point where there are (m) 

 branches of the one and (n) branches of the other, must be considered to 

 have mn coincident points in common, i. e. if we like so to express it, to 

 have a contact of the degree mn— 1. 



