Mr Dixon, On the Order of Certain Systems of Conditions. 459 



shall pass through the q nodes serve to fix say a m _ 5+2 , a m _ 9+3 . . . a m+1 

 in terms of a lt a 2 ... a m _ g+1 , and thus reduce the equation to the 

 form 



a x v x + a 2 v 2 + . . . + a m _ q+1 v m _ g+1 = 0. 



In this let the parametric expressions for x, y be substituted and 

 the factors independent of a 1} a 2 ... divided out : what is left is an 

 equation of degree 2m — 2q in 6, and the roots of this can be made 

 to coincide in pairs in 2 m_ ? ways by our former result. 



Out of p nodes q can be chosen in C lhq ways, and we thus find 

 for the whole number of solutions 



X C pq 2 m -Q.2<2 or 2 m +P, 



q = 



it being assumed that m <£ p. 



For instance the number of curves of degree n — 1 passing 

 through all the ^(n — l)(n — 2) — p nodes, touching the curve 

 f(x, y) — in p + n — 1 points, of which r(<n) are fixed, and 

 passing through n—r—1 given external points, is 2' 2p+n ~ r ~ 1 . 



When m <p, q cannot be greater than m, and some terms 

 of the series 2 C Pt q 2 m ~i . 2? must be left out. For instance, of the 

 curves of degree n — 3 that pass through all the nodes, the number 

 that touch f{x, y) = in p — 1 places is 



V % C pq 2P- l -v.2i or 2*- 1 (2*-l), 



g=0 



as given by the theory of Abelian functions. 



It may be well to discuss the grounds of the convention that a 

 curve passing through q nodes is to be counted 2? times among 

 the solutions of the contact problem. 



Let S = 0, 8' = be two neighbouring curves, of which the 

 former has nodes at A lf A 2 ...A q . Then we may write, to the 

 first order of small quantities, 



8' = S + e x T x + e 2 T 2 +...+ e r T r , (r * q), 



where of the curves T x = 0, T 2 = . . . all but T x = pass through 

 A 1} all but T 2 —0 through A 2 , and so on; e 1; e 2 ... are small 

 constants. 



Let Q = 0, Q' = be tw r o neighbouring curves, of which the 

 former passes through A lt A 2 ... A q , so that we may write 



Q' = Q + Vl R 1 + v . 2 R 2 +..., 



where again of the curves jR x = 0, R 2 = ... all but R l = Q pass 

 through A x , all but R 2 = through A 2 , and so on : small quantities 

 of higher order than t] 1 , rj 2 ... are .igain neglected. Consider the 



