42 Proceedings of the Royal Irish Academy. 



It follows that D and A each contain (03) as a factor, since the exponents 

 of {t - ts) in D and A, namely ^ 4 $' + r - 6 and 2,p - q- r - Q, are each at 

 least equal to ^ - 3. 



The degree of (03) is seen to be N+n-& by writing d = t+ {d-t) in 

 %ai{t)xi (d) and expanding in powers of (d-t). We find that {6-tf divides 

 Soi [t) Xi [0), and that 



2ai (0 Xi (6»)/(0 - ty = (03)/3 l + (e-t)G, 



where G is finite or zero for 6 = t. The left-hand side is of degrees iV - 3, 

 n - o in t and 9 respectively. Hence, putting 6 = t, we find that (03) is of 

 degree iV" + m - 6. 



Hence 2 (2^ - 3) = i\^ + n - 6. 



If we write w = jj - 3, k = q - 2, p = r - 1, we have 



^{k+p-zi) = 2N-2n; - (^^ 



hence 2 (^ + p) = oN - n - &. 

 For a P-curve these results coincide with those already found* 



The n - ^, 71-3 Relation on a Curve of a Linear Complex. 



16. If the oscillating plane at a point ^ on a curve of a linear complex 

 passes through another point B on the curve, that at B will pass through A. 

 AB will be a line of the complex. If the curve is a rational n - ic, the para- 

 meters of A and B are connected by a symmetric n - S, n - o relation, say, 



V(t, 6) = 0. 

 V{t, t) will vanish at the points where f > 3. It is of degree 2n - 6, and is 

 easily seen to be equal to (03), apart from a numerical factor. In fact, 



V{t,e) = ^oi {t)xi {6)1(6 -ty. 

 Hence ■ V(t, t) = Lt V{t, 9) = (03)/3 ! 



It follows that V(t, t) has a zero of order ^ - 3 at a point t^ where ^ > 3. 



Again, it is easily seen that V[t, t„) has a zero of the same order at t^. 



For if we write t = t„ + (t- 1^) and expand Xi(t) in powers of (t - 1^, we find 



^(<.g = (03)o/3! + (i-g(04)„/4! + . . . 

 Now at the point <„ we have 



(00) = (01) = (02) = . . . (Cm) = 0, (ffi^p- 1), (0^) + 0, 

 which proves the proposition. 



* A number of questions connected with non-linear cycles on twisted curves are treated by 

 W. A. Tersluys in three papers : — Proceedings of the Royal Academy of Sciences, Amsterdam, 

 vol. viii. (1905), p. 498 ; and vol. ix. (1906), p. 364 (English series). Archives du Mus^e de Teyler, 

 Haarlem, ser. 2, vol. X. (1907), pp. 263-365. (On curves of the type x = al", y = bt"*<-, z = ct"*<-*"'.) 



