Theory of Equal Roots and Multiple Points. 377 



Suppose, now, the curve to have double points, the (r — l)th 

 evectant (and of course all the inferior evectants) of the discri- 

 minant of $ (meaning thereby the result of eliminating x, y, z 



between —/-$■,-$■) will all vanish, and the rth evectant will 

 dx ay as / 



be of the form 



(a 1 x + b l .y + c ] z) n x{a 2 x + b 2 .y + c^z) n ...x{a ) .x + b r .y + c r .z) n } 



where a x : b x : c„ a 2 : J 2 : c 3 , . . . a r : b r : c r are in the ratios of the 

 coordinates at the respective double points. If there be cusps 

 the multiplicity of each such will be 2 ; and calling the total mul- 

 tiplicity m. to every cusp will correspond a factor of the 2nth 

 power in the ?«th evectant ; and so on in general for various 

 degrees of multiplicity at the singular points respectively. The 

 like theorem extends to conical and other singular points of sur- 

 faces; so that there exists a method, when a locus is given 

 having any degree of multiplicity, of at once detecting the amount 

 and distribution of this multiplicity, and the positions of the 

 one or more singular points. In conclusion I may state, that 

 precisely analogous results {mutatis mutandis) obtain, when, in 

 place of a single function having multiplicity, we take the more 

 general supposition of any number of homogeneous functions 

 being subject to the condition of pluri- simultaneity, i. e. being- 

 capable of being made to vanish by each of several different 

 systems of values for the ratios between the variables. Multi- 

 plicity in a single function is, in fact, nothing more nor less 

 than 'pluri-simultaneity existing between the functions derived 

 from it by differentiating with respect to each of the given va- 

 riables successively. But as I purpose to give these theorems 

 and their demonstration, which I have already imparted to my 

 mathematical correspondents in a paper destined for reading 

 before the Royal Society, I need not further enlarge upon them 

 on the present occasion. 

 2fi Lincoln's-Iiin-Fields, 

 March 23, 1852. 



P.S. In the above statement I have spoken only of cusps of 

 curves which are the precise and unambiguous analogues of three 

 coincident points in point-systems, in order to avoid the neces- 

 sity of entering into any disquisition as to the species of singu- 

 larity in curves or other loci corresponding to higher degrees of 

 multiplicity in point-systems, a subject which has not hitherto 

 been completely made" out. I may here also add a remark, which 

 - a still higher interest to the theory, which is (to confine 

 ourselves, for the sake of brevity, to functions of two variables), 

 that if any root of x : y, say a : b, occur 1 +fi times, the total 

 multiplicity of the equation being supposed m, and its degree n, 



