Mr. J. J. Sylvester on a new Theory of Multiplicity. 461 



tained as a factor in all above it*. When a function of only two 

 variables is in question, there is no difficulty in understanding 

 what property of the function it is which is indicated by the 

 allegation of the existence of multiplicities m 1} m 2 , . . m r ; as 

 already remarked, this simply means that there are r distinct 

 groups of equal roots, such groups containing 1 + m v 1 + m 2 , . . 

 1 -f- m r roots respectively. So for curves and higher loci, the 

 total distributive multiplicity is the sum of the multiplicities at 

 the several multiple points. But the true theory of the higher 

 degrees of multiplicity separately considered at any point remains 

 yet to be elaborated, and will be found to involve the considera- 

 tion of the theoiy of elimination from a point of view under which 

 it has never hitherto been contemplated. 



Confining our attention for the present to curves, we have a 

 clear notion of the multiplicity 1 : this is what exists at an ordi- 

 nary double point. As well known, its analytical character may 

 be expressed by saying that the function of x, y, z, which cha- 

 racterizes the curve, is capable, when proper linear transforma- 

 tions are made, of being expanded under the form of a series de- 

 scending according to the powers of z, such that the constant co- 

 coefficient of the highest power of z, and the linear function of x, y, 

 which is the coefficient of the next descending power of z, may both 

 disappear. Again, when the multiplicity is 2, the 3rd coefficient, 

 which is a quadratic function of x and y, will become a perfect 

 square. This is the case of a cusp, which, as I have said, is the 

 precise analogue to that of three equal roots for a function of two 

 variables. Before proceeding to consider what it is which con- 

 stitutes a multiplicity 3 for a curve, it will be well to pause for 

 a moment to fix the geometrical characters of the ordinary double 

 point and the cusp. 



If we agree to understand by a first polar to a curve the curve 

 of one degree lower which passes through all the points in which 

 the curve is met by tangents drawn from an arbitrary point 

 taken anywhere in its own plane, we readily perceive that at an 

 ordinary double point all the infinite number of first polara 

 which can be drawn to the curve will intersect one another at the 

 double point. Again, at a cusp all these polars will not only all 

 intersect, they will moreover all touch one another at the cusp. 

 Now we may proceed to inquire as to the meaning of a multi- 

 plicity of the third degree, which, strange to say, I believe has 

 never yet been distinctly assigned by geometricians. 



This is not the ca3e of a so-called triple point, i. e. a point 



* The constitution of the quotients obtained by dividing all the other 

 evectants of the discriminant by the first non-evanescent one, presents 

 many remarkable features which remain yet to be fully studied out, and 

 promise a wide extension of the existing theory. 



