14 Frederick C. Ferry. 



intersection and will so occur whenever in every term of 

 (o . CD the substitution of z for X" or s for [j,"' be involved as 

 a part of the entire substitution ; this will be the case when 

 the highest power of v in œ . cp be less than half the 

 degree of co . cp, — i.e., when q + n^ <; ^(p-j-ii')j hence the 

 linear director will in general occur ^(p — 2q -f- n' — 2n') = 

 1 (p + n') — (q + n^) times as a part of the residual inter- 

 section. Any one of these lines can occur as a part of the 

 intersection only if it is a line of multiplicity > 1 on S, 

 and it may occur more times in the intersection than the 

 product of its multiplicities on 2 and S only when the two 

 surfaces have contact all along the line in question. Hence 

 let the multiplicities of the lines under consideration on S 

 be denoted thus, — 



f ^ of the double director, v=0, or x==0, y=0, on the surface S, 

 g'^^ „ „ pinch point generator X=0, or x=0, z=0, „ „ „ „ 



"^ = » » » >} }> ^''^^^^"> )j y^^^j s=u, „ „ ,f „ 



V V 



k' ^ „ „ linear director r-=- = cxD or z=0, s=0, „ „ „ „ 



Å II 



1' ^ „ all the other generators together which occur in the 

 intersection; and in case it be necessary to distinguish between 



them let 1' , 1" , l'" , denote the multiplicities of the first, 



second, third, etc. Similarly let f^ , g^ , h^ , k^ , 1^ , 1" , 1^", 



respectively denote the number of times the lines occur in 

 the intersection from the existence of contact of a sheet or 

 sheets of the two surfaces all along the line. The tangent 

 planes to 2 all along the double director have as their 

 equations x^s^ — y^z^ ^ 0, and that fg > the lowest terms 

 in x,y in the equation of S must contain x^s — y^z as a 

 factor; such a factor could be found in S only from the 

 occurrence in to . cp of the expression (hif. }x" — (|jlv) . X" ^e 

 X^[jlV — X^ix^v^ ^ , and we suppose to . cp to contain no such 



