32 Frederick C. Ferry. 



must be rejected, since their polar planes qua S and 2, 

 having the same lines multiple, are indefinite; hence if (p,q) 

 has p points on x , y , q on x , z , q on y , s , and p — q on 

 z , s , there will remain, if f ' ^ 1 , g' > 1 , h' > 1 , k' > 1 , 



(p + q)(m' + l) — pf — q(g'4-go— 1) — 



q (h' + h^ — 1) — (p — q) (k' — 1) points. 



Also, if f > 1 , at each one of the p points where (p , q) 

 meets x,y, a sheet of S and a sheet of 2 have contact, 

 since an element of the curve and an element of the line 

 x,y at the point in question, taken together, determine a 

 plane tangent to a sheet of each surface there. This tangent 

 plane meets the arbitrary line, but such points should not 

 be counted for r since the tangent to the curve at the point 

 will not in general meet the arbitrary line, although the 

 polar planes qua S and 2 do through their coincidence 

 cut it. Hence a reduction of p points must be made here. 

 In like manner, if g' > 1 , h' > 1 , and k' > 1 , a similar reduc- 

 tion of q + ^l + P — q points must be made. Therefore, if 

 f ' ^ 1 , g' > 1 , h' > 1 , and k' > 1 , the result is 



■r = (P + q)(m' + l)-pf'-q(g' + g^-l) — 



q(h' + h^-l) — (p-q)(k'-l) — 2p — q. 



If f' = this formula still holds; for the term pf then 

 vanishes, and since each point of (p , q) on x , y is then 

 multiple, a reduction of p points as the formula demands 

 gives the correct result. 



If g' = , the term — q (g' + go — 1) becomes + q which 

 is cancelled by the — q introduced at the end; hence the 

 formula holds for g' = 0; and in like manner it is evident 

 that it is still true if h' = or k' = 0. 



