52 Frederick C. Ferry. 



9 = (aV - b>^) . ü^3_, + V . Vp_, + vl V^,_, + 



+ v^^V =0 



' p— q 



and obtain for the tangent conies to cp = at the point P', 

 regarded as lying on the generator aX -f- bji. = and the 

 generator aX — b[x = respectively, the equations 



aX -f bjx -f >^qV = and aX — b[j. -|- x^^v = 0, 



where x,t== -—, — . 



" 2aX'U;_2 



If V -,= 0, the curve cp = has each branch at P' 



p— 1 ' ' 



lying in the direction of the generator at that point in its 

 own sheet. 



If P' be the pinch point |j,' = v' = 0, the above cp = 

 may be put 



^ A U,_, + V . V^_, + V"-. V„ „, + + v". V^^, = 



where V ^-^ 4= 0' ^^^^ ^^^® would have a case already con- 

 sidered. Here the tangent conic at P' is given by V_j^v = 

 or v=0, and hence the bsanch in either sheet becomes tangent 

 to the double director as P' moves to the pinch point; i.e., 

 the two branches come together and take the same direction 

 there. 



The directions of the curves cp = at points on the pinch 

 point generators, other than the pinch points themselves or 

 the points where the pinch point generators meet the linear 

 director, are not of especial interest; the former particular 

 case has been already considered, the latter will now be 

 treated. 



