40 Frederick C. Ferry. 



B). Determination of the curves (p , q). 

 The general equation 



I p-q-rl ' p-q 



contains 



P + I + P + P — -I-+ 



+p — q + 2 + p — q+ i = pq4-p — iq(q— i) + i 



terras and hence any curve (p , q) is determined hy 

 pq -}- p — ^q(q — 1) 2^oints. Or since the general p-thic 

 in the plane is determined by ^p(p-|-3) points and the 

 equation above has all the terms of the general p-thic save 



q+l 



[Vq-i + ^-üp-q-.+ + V^-^-\Uj = 



^(p — q)(p~q + i) 



terms, the curve (p , q) is determined by 



lp(p + 3) — l(p — q)(p — q 4- 1) = pq + p — iq(q— 1) points. 



Thus in general a generator (1,0) is determined by a 

 single point, any combination of generators (p , 0) by p points, 

 any conic (1,1) by 2 points, etc. Any rational curve, being 

 of the species (p , 1) is determined by 2p points. 



The following table gives the number of points in general 

 Avhich determine a curve of the given species, — 



