62 Proceedings of the Royal Irish Academy. 



If the curve belongs to a linear complex, C's vanishes identically. The 

 differential equation so obtained can be written Aci = Bc^ C'(?, where A, B, C 

 are functions of <r, <7i, a^, cr,. This equation for c is easily integrated, and gives 

 the expression for c in terms of c, <ti, a^ ah-eady given ia equation 22. It was 

 thus, in fact, that that equation was obtained in the first instance. 



Note 2. 



53. Mr. J. H. Grace {Proc. Ca/rnh. PMl Soc, voL SJ., pp. 132 sq.) states 

 that a rational curve of the nth degree with 2n - 6 stationary tangents 

 (inflexions) belongs to a linear complex. He gives special proofs for 

 n = 3, 4, 5. For the general case his argument is as follows : — 



The determinant of the sixth order 



I Xif dXijIdt d'Xij\dt- d?Xijldt? \ = J, 



{Xij = x^j - xfki, j>i) 

 is of degree 6 (m - 5) ia t, where wi = 27i - 2 is the degree of the 

 polynomials Xij. This is best seen if we make the Xij homogeneous by 

 introducing a second variable if, which can subsequently be put equal to unity. 



The equation .7=0 gives, clearly, the points at which six consecutive 

 tangents belong to a Mnear complex. 



Xow, according to the argument, a stationaiy tangent accounts for six of 

 these points, and hence J has 6 (2w. - 6) roots : but its degree is 6 (2% - y) ; 

 hence it must vanish identically ; and it is easy to deduce that the curve 

 must belong to a complex. 



This argument is at fault in one particular. If we consider, for example, 



the quartic 



^, <^f + a, f+3, 



which has a stationary tangent at i = 0, we find that J is rap {t - a) 

 where m is a constant. Hence the inflexion only counts for five and not 

 six roots of J. Accepting the argument that the nimiber of roots of J 

 represented by an inflexion is independent of the degree of the curve, since 

 it depends only on the infinitesimal properties of fhe curve at the inflexion 

 (torn, dt., p. 28), the '2n - 6 inflexions account for lOn - 30 roots. This 

 leaves 2m - 12 ordinary points satisfying ./, if n > 6. -7 vanishes identically 

 only if n < 6. 



There are no undulations and no points where p>o on the curve, by 

 hypothesis : hence the 2n - 12 roots of (33) = are ordinary points : 

 these are the roots of J not accounted for by the inflexions. 



In this case the factors of 7 are (03)' (33), since the zeros of (03) are 

 the stationary points each of order j) — 3, which in this case means unity 



