Frederick C. Ferry. 



The pinch points are given by X = , v = and 

 jjL = , V = ; the corresponding generators have for their 

 equations X = and ji- = respectively, the two generators 

 from either pinch point coinciding. 



IV. Curves on 2. 



Any irreducible curve on 2 is given by a homogeneous 

 equation in X, [x, v. Let such an equation be represented by 

 cp = and call its degree p. cp = may be written according 

 to powers of v, if its degree in v be q, thus: — 



'.^U^^4-v.U^,,4-v^U^,_,+ 



_j_v'i-\U ,-fv'\U =0. 

 ' p— ci+1 p— q 



q<p 



Ü =0 is a homogeneous equation of degree p in X and [x 

 and gives the points of intersection of the curve cp = with 

 tlje double director; consequently cp = has p points on 

 v = or the curve cp = meets the double director in p imints. 

 A point of the doubhs director is here to be regarded as 

 lying on the generator for which X : jx has the same value 

 as for it, and as lying in the sheet in which that generator 

 lies, but not as lying on the other generator through this 

 point, for which generator X : [i has the opposite value, nor 

 as lying in the sheet of this second generator. That a point 

 aX -|- b|x = , v = be regarded as lying in both sheets and 

 on both generators at that point of the double director, it 

 is necessary that U ^ (aX" -j- bjx") . V „ ; and that every 

 point of the curve cp = on the double director be regarded 

 as lying in both sheets there, it is necessary that U ^ f (X"^, [x^). 



