Mr. J. J. Sylvester on a new Theory of Multiplicity. 465 



Without pushing these observations too far for the patience of 

 the general reader, it may be remarked by way of setting foot 

 with our new theory upon the almost unvisited region of the 

 singularities of surfaces, that by the light of analogy we may 

 proceed with a safe and firm step as far as multiplicity of the 

 third degree inclusive. 



The function characteristic of the surface being supposed to be 

 expressed in terms of the four variables x, y, z, t, and expanded 

 according to descending powers of t, then when x, y, z is an ordi- 

 nary double point of the first degree of multiplicity, the constant 

 and the linear coefficient disappear; when the point has a mul- 

 tiplicity 2, the discriminant of the quadratic coefficient will be 

 zero, i. e. this coefficient will be expressible by means of due 

 linear transformations under the form of x~ + y 2 ; and when the 

 multiplicity is to be of the degree 3, the cubic coefficient will, 

 at the same time that the quadratic coefficient is put under the 

 form x^ + y' 2 itself (for the same system of x and y assume the 

 form of a cubic function of x, y, z, in which the highest power of 

 z, i. e. z 3 , will not appear) ; or in other words (restoring to x, y, z 

 their generality), not only will the first derivatives of the qua- 

 dratic function be nullifiable simultaneously with each other, but 

 likewise at the same time with the cubic function itself. These 

 three cases will be for surfaces, the analogues so far, but only so 

 far as regards the degree of the multiplicity, to the double point, 

 cusp, and flex-cusp of curves*. The analogue to the so-called 

 triple point of the curves will be a point whose degree of singu- 

 larity (depending upon the vanishing of the six constants in the 

 3rd coefficient (which is a quadratic function of x, y, z) at the 

 same time as the three constants in the linear factor) would seem 

 to be but 6 more than for a double point, i. e. in all 1 + 6 or 7, 

 but whose multiplicity, as inferred from the nature of the con- 

 tact of its first polars, which will be of the 7th order, would 

 appear to be 8 (a seeming incongruity which I am not at present 

 in a condition to explain) f ; so that there will apparently be 4 



* At an ordinary conical point of a surface for which the multiplicity is 

 1, every section of the surface is a curve with a double point. When the 

 multiplicity is 2, the cone of contact becomes a pair of planes, through the 

 intersection of which any other plane that can be drawn cuts the surface in 

 a section having an ordinary cusp of multiplicity 2, but which themselves 

 cut the surface in sections, having so-called triple points, so that for these 

 two principal sections (which is rather surprising) the multiplicity suddenly 

 jumps up from 2 to 4. All other things remaining unaltered when the 

 multiplicity of the conical point is .'-f, the cusp belonging to any section 

 of the surface drawn through any intersection of the two tangent planes 

 passes from an ordinary cusp to a flex-cusp. 



t So, too, at a so-called quadruple point in a curve, the degree of the 

 contact of the 1st polars is H, and therefore the multiplicity of the curve at 

 such point is '.); but the number of constants which vanish for this case 



Phil. Mag. S. 4. Vol. 3. No. 20. June 1852. 2 H 



