Frederick C. FeriT- 



branches of (p=:0 at an 1-tuple point (it being understood 

 that all the branches at a multiple point lie in one sheet of 

 2 unless otherwise specified) P', which is finite. 



If<P = U^ + v.Up^, + vlU^_, + 



+v'-'.ü^_,+, + v".U„_,=0, 



the tangent conic at any ordinary point P' on the double 

 director is given by 



depending only on U _^ and U . 



If U _^^0, the tangent conic consists of a tangent 

 generator and the linear director, and hence the curve has 

 the direction of the generator in the sheet in which the 

 curve lies whenever U _^^0. 



If P' be the pinch point p,' = v' = 0, we may put 



? = !^.Up_, + [x'.Up_2 4-}x-lUj^_g + 



+I^^^U„ + v.V^^^, + vlVp_,+ +v^V^,_^^0 



and thus obtain for the tangent conic 



V- ' Up_i + V . V^_^ = 0, depending only on Up__^ and Y^^^. 



This tangent conic is cut out by the plane s . U'^-^ -f" 

 y.V'_^ = 0, the residue being the pinch point generator 

 p, = 0. To find in what relation this tangent conic stands to 

 the generator at the pinch point, when both are cut out by 

 the plane s. U'^ j^-f-y'^',_i = 0, which plane we will repre- 



sent by y — xs = 0, where x = — ^ — ; — Eliminating s 



