332 Mr. J. J. Sylvester o7i tlie Enumeration of the Contacts 



which leave the form unaltered) required him to consider the 

 nature of the singular relations capable of existing between two 

 linear substitutions, which is precisely the question, differently 

 stated, of the singular relations connecting two liueo-linear func- 

 tions above adverted to ; accordingly, I am indebted to Mr. Cay- 

 ley for making an observation on the effect of my rule for finding- 

 such singularities, which leads to a most elegant forraulization 

 of the number of singularities in question, and which I proceed 

 to introduce to the notice of my readers. 



If U and V be two quadratic functions, each of n variables, 

 and if we call D the discriminant of Uh-\V = D(\), D(X) will 

 be a function of X of the /ith degree. Now, first, I have observed 

 that if any of these n roots be repeated any number of times, 

 there will be a corresponding degree of singularity about one of the 

 points of intersection of the loci represented by U=0, V = 0; so 

 that if the n roots of D(X.) be made up of ?-j roots «„ r^^ roots a^ 

 rg roots ffg, &c., there will be an inclusive singularity ?•, at one 

 point, rg at another, r^ at a third, and so on — by inclusive sin- 

 gularity meaning a number one unit greater than the index of 

 singularity properly so termed ; the inclusive-singularity at an 

 ordinary intersection being called, 1, at a point of simple singu- 

 larity, 3, of double singularity 3, and in general at a point of the 

 (;•— l)th degree of singularity, r. 



Hence the total-inclusive singularity (which is an unit greater 

 than the total-singularity, properly so called) may be broken 

 up into as many partial heaps of inclusive-singularity as there 

 are modes of decomposing n into integers. We may now con- 

 fine our attention exclusively to the different modes in which 

 a given amount of inclusive-singularity at a single point admits 

 of subdivision into distinct species of singularity, for which I have 

 given in my paper referred to the following rule : — The minor 

 systems of determinants corresponding to the matrix of U -1- XV 

 are to be considered in succession ; and if («) be any root of the 

 complete determinant of the matrix occurring r times, every 

 hypothesis is to be exhausted as regards the number of times in 

 which (A. — «) may be conceived to enter as a factor into each of 

 the system of 1st minors, into each minor of the system of 2nd 

 minors, into each minor of the system of 3rd minors, and so on j 

 the number of such hypotheses being limited by the condition 

 that,if quoad the root a, {\ — a) *', (X — a) '''\ (X — a) *'' be the greatest 

 common factors respectively to three consecutive systems of 

 minor determinants, k-^ must be not less than 2kc^ — k^. Here 

 Bteps in the beautiful observation of Mr. Cayley, that the ques- 

 tion of assigning the different species of singularities respondent 

 to the factor [a) supposed to occur (?•) times, is, by virtue of the 

 above condition, tantamount precisely to that of assigning the 



