68 Proceedings of the Royal Irish Academy. 



Torsion of Algebraic P-curves. 

 46. We shall write a,.(0) = jii, Xi{d) = yi, a^t) = m, .>\{t) = x^, so that 

 (/3)' iy)' ^2 ^""s functions of H, and [a], {x), s, functions of t. 



Since 2/3 .^i is of the form ■wteF^{si)Fi{Si) (lemma 2), it follows that 



7]- 7\~ 



r-^ log t^fi = — ^- log Km = S«(Te/7r'M. 



If we denote differentiations with respect to t and d by dots and accents 

 respectively, we have 



log 2/3..r, = —^ _- log 2/3,r; - -t^ 1^ J ^ 



The numerator of this fraction is equal to 



/3, i3. /33 i3, 

 (i\ i3\ i3'3 /3', 



= ^BijXij, (j>i), 



lA/i i/jo "^3 '^'i 



Xi «2 «3 rri 

 where Bij, for instance, irf /3.,:/3V - /3//3';. 



Now the fractions BijjY^i are all equal, and in the case of a P-curve 

 their common value /u is a constant (section IV., equations (13), and note). 



We have therefore B/eneh'/e = juSX;;-r}c;/sis'2(2/3;a;i)^ or, substituting 

 the value of ^jBiXi/ine from lemma 2, 



S,e-hs'2Z'{t),p{e)ae = ix%XijYu. 

 Interchanging t and ti we get 



(We suppose / > i, and / - ^ positive or negative according as k + I 

 is odd or even.) 



Hence ^^ - ^^-^ <-23^ 



= const., since < and are two arbitrary points on the curve. 



Hence the torsion at a point t on an algebraic P-curve is equal to 

 AZ''(t)/(p(t), where L and are the rational functions desc7-ibed in lemrrm 2 

 and A is a constant. 



V and have the same poles and with the same order of infinity 

 (lemma 2) : hence the torsion is zero at points on the plane at infinity, infinite 

 at the points of contact of isotropic osculating planes, and finite at all other 

 •points, real or imaginary. 



47. Every curve belonging to a linear complex, whether algebraic or not, 

 has this property. For, since as : jSa : 73 = 3/ : - a; : k (par. 37), the equation 

 a^ + jS's + 7% = 0, which gives the isotropic osculating planes, is equivalent to 

 a? + 2/' + K^ = k/o' = ; and k\<s is infinite only at the points of the curve in the 

 plane at infinity. I have not been able to ascertain whether non-algebraic 

 P-curves have the property. 



