Contacts of Lines and Surfaces of the Second Order. 139 



a factor in common, which would involve the satisfaction of 



r—\ 

 r . — jr 1 conditions only ; but over and above this, that the 



complete determinant, instead of containing this common factor, 



as it needs must, (r— 1) times shall contain it r times : this gives 



?■ — 1 

 one condition more, making up the entii-e number to r — — . 



The total number of different species of singularity for conju- 

 gate fimctious of a given number of letters, can only be expressed 

 by aid of formula containing expressions for the number of va- 

 rious ways in which numbers admit of being broken up into a 

 given number of parts. 



The computation of this number in particular cases, upon the 

 principle of the foregoing method, is attended ^\'ith no difficulty. 



^Ye have seen that this number for two, three and'four letters, 

 is respectively one, foui", twelve. 



I have found that for five letters the number is twenty-fom*, 

 for six letters fifty, for seven letters a hundred, and (subject to 

 further examination) for eight letters one hundred and ninety- 

 three. The series, therefore, as far as I have yet traced it, is 1, 

 4, 12, 24, 56, 100, 193. The last number must not be relied 

 upon at present. 



It will be observed, that the foregoing table for the contacts 

 of surfaces of the second order contains no form coiTesponding 

 to a complete intersection in two non-intersecting lines and an 

 un degenerated conic. In fact, if two such lines form part of the 

 intersection, at least one other right line intersecting them both, 

 must go to make up the remaining part. This is easily verified ; 

 for it is readily seen that the most general representation of two 

 conoids intersecting in two non-meeting lines will be 

 \]=.x])-\-zt 



V = axy + hzt -\- cxt -|- cyz, 

 where the two lines in question are 

 (57=0 0=0) 

 (y=0 ^ = 0). 



Now it will be found that the first mmors of U + A,V formed 

 from the above equation will all contain the common factor 

 (a-f X)(A-|-X) — c^, showing that the contact is quadi'ilinear or 

 linear-indefinite, i. e. bilinear, according as the roots of 

 X^+{a + b)X-{-ab — ce=0 



are distinct or equal ; which explains how it is that only one 

 species of bilinear contact (that is to say, the case corres])onding 

 to the two conoids agreeing in the two right lines in which each 

 is cut by a connnon tangent plane) comes to find a place in the 

 preceding enumeration. 



