54 Frederick C. Ferry. 



At an ordinary point P', 



Now tip HI = ^ V'. (.\l^ + b/) (a,[x -f \l) ■ 



(a^,_,fx + b^,_,X)^ÀV^Up_,. 



Up-j-q-i^X^^^^fjL^^^^.Up-q+i, where Üp_q+i is obtained 



from U by interchanging X and [x. Hence the tangent 



conic is 



>^. ^x^^ +^ ^. hv.X jx Up_,^i-0, 



which becomes 



BX' '' 0|j,' ' X'lx' 



(since P' is in general not a point where either X' or jx' = 0). 



But changing X' , tx' , v' to r-; , — , -^ , this equation becomes 



0Üp_q . oUp_q Up_q-j-i „ 



Ojx' oX' X'fj.' 



and resubstituting r > > - for X , p, , v, we have 



Xu, . ^p-q+1 + Xv . "^p-q + ;i.v.Q^p-'i=o, 



X'jx' 5X' oa' 



the tangent cubic (2,1) at any ordinary point P' on the 

 linear director. 



