38 Proceedings of the Royal Irish Academy. 



rV. — Eational P-Curtes. 



8. We shall use the term P-curve to denote a curve at every point of 

 which p - q - r vanishes : in other words, a curve of which every cycle has 

 its degree equal to its class. Every curve belonging to a linear complex is a 

 P-curve ; but the converse is only true, as we shall see, when the degree of 

 the curve is less than six. 



9. A rational curve of the nth degree is specified by four polynomials in t, 

 Xi (t) (i = 1, 2, 3, 4), which are proportional to the homogeneous coordinates 

 of a point on the curve. The polynomials «; have no common factor. The 

 coordinates of the osculating plane will be proportional to four polynomials 

 ai(t), whose degree iVis equal to the class of the curve. 



We shall need the determinants 



^ = I a; a; o; ttj | . 



In the neighbourhood of any point U we can write, by changing the 

 tetrahedron of reference and putting t for t - to, 



yi = afi + a'tP*^ + . . . \ 



3/4 = d + d't + . . . (i) > q > r > 0, abed = / 



where the yi are linear functions of the «,-. It is easily seen that the deter- 

 minant D (y) has a zero of order p + q + r - Q at the point. Again, D {i/)/D 

 = const. ; hence Z> has a zero of order p + q + r - d at every point where this 

 number is greater than zero (its value at an ordinary point of the curve is 

 zero, and it is never negative). 



Again, at the point to, let /3; be the transformed plane coordinates corre- 

 sponding to yi, the j3i being equal to linear functions of the a,-. From equations (6) 

 we see that /3i//3i, jSaZ/Bi, /Sa/jSi are of orders p, p - r, p-q respectively in t. 

 Again, j3i cannot have a zero at the poiut, otherwise the /3; would all vanish 

 at the point, and the new tetrahedron of reference woidd vanish also. Hence 

 we can write 



/3i = «o + o.\t + . . . 



/3. = h^P't + . . . 



iSa = C.P-' + . . . 



/34 = djf + . . ., a^hoC^do^ 



and the determinant A (j3), and therefore A, will have a zero of oider 



p + {l3-r) + {p-q)-Q = ip-q- r -d 

 at the point. 



(8) 



