Geometry on the Cubic Scroll of the First Kind. 



31 



If S and 2 have multiple lines, let them be, — 

 X , y an f -tuple, x , z a g' -tuple, y ,s an h'-tuple, 



and z , s a h'-tuple on S , and 

 x,y a double, x,z a single, y,s a single, 



and z,s a single on 2, making 

 X , J an f -tuple, x,z a (g' — l)-tuple, y , s an Qi' — l)-tuple, 



and z,s a Qc' — l)-tuple on ^. 

 If further S and 2 have contact g^ times along x , z and 

 h^ times along y,s, then will <b and 1 have x,z occurring 

 g' + So^ — -'- t™6S and y,s occurring h' -j- h^ — 1 times in 

 their intersection; i.e., the lowast terms in x,z in O give 

 g' -|- gQ — 1 and not g' + go — 2, as is evident if in the 

 determinant of O = we put the values of g' and g^^ in 

 place of the corresponding elements in the last row but one 

 and insert the values from 2 in the last row; we will place 

 a dash over the g' or g^ which suffers the reduction in any 

 case and thus we obtain 



+ g^— l,g' + g^, 

 2xs , — 2yz , 



' 2 ff'-ko-' 



Here g' 4~ go — ^ must have its term multiplied by either 

 2xs , — 2yz, or x^, and in any case is raised to g' + gg — 1- 

 The same holds for y , s on <b , which line occurs h' + hjl, — 1 

 times there. 



S and 2 intersect in a locus of order 3m', from which 

 must be rejected the multiple lines as many times as they 

 occur, giving 3m' — (2f ' + g' + go -f h' -\- h^ -j- k') which we 

 put ^ p -|- q as the order of the curve (p , q). 



This curve (p , q) meets = in (p -f- q) (m' -f- 1) points, 

 from which number those lying on multiple lines of <I) ^= 



