46 Proceedings of the Royal Irish Academy. 



Again, if I) and A have the same meaning as m the last section, 



D = -xt, A = - io'\t = - U\t, and D/A = .r,/ U\t = 1. (14) 



From this . identity we can deduce as in the last section the rest of the 

 identities^(13). 



23. Hence an algebraic P-curvc of the nth degree is of the rdli class, and can 

 he represented hy two sets of functions .-r,- and m, rational of order n on a Biemann 

 surface, and, such that (a) iieither set has a common zero, (6) tlie 'poles of %a{Xi 

 are the sam^e as those of 'SbiUi, v:here the a,- atul 6; are ariitrarg consta/iits. 



Tliese functions satisfy all the identities (13) of the last section. 



In proving that D/A is constant, we imposed a more stringent condition 

 on the functions (a;) and (a) than the condition ih) : but the extension is easy, 

 and it is not necessary for our purpose to justify it. 



24. To show the necessity of the restriction (a), that neither set of 

 fimctions must have a common zero, consider for example the rational quartic 

 for which the a-,- are f, f, t, 1, and the a, are 1, - 6;;-, 8^^ - 3^'. It is not a 

 P-curve, as we can see by examining the point ^ = 0, or the point t =x. The 

 same cui've can he represented by the ,r, t^, t', P, t and the a, 1, - 6^^ d-f, - 3f . 

 For the latter forms we find Z)/A constant, (22) = 0, &c. ; but this teUs us 

 nothing about the curve, since B contains a factor t (due to the common zero 

 ^ = of the .r.) which has no geometric meaning. 



Metrical Signifieayux of tJie Function w. 



25. If X, y, z are Cartesian coordinates in par. 19, w vsill have the value 

 hZjt^, where .^is the pei-pendicular fi-om the origin on the osculating plane 

 at t, <T is the toi-sion at t, and h is a constant. This can be inferred fi'om the 

 expression for the torsion of a P- curve given in section vn, par. 46, or it may 

 be directly proved thus : — 



Jlx,y,z are not rectangular, replace them by rectangular axes. This is 

 equivalent to multiplying v:^ = Xt-Xt by a constant. "We then have XtjXt = XsjXs. 

 Also Xg = &(y, where c is the cm'vature. Again, the direction -cosines 03, jSs, -ya 

 of the binormal are equal to Z\, Zfx, Zv. Hence we easUy find Aj = Z'^ (as),, 

 where (03)5 is the determinant foiined by replacing X, ju, v by as, /Ss, 73 in As. 



Evaluating this detenninant with the help of Frenet's formulae (vn, par. 36), 

 we find Z^\s = c'o^, c being the curvature. 



Also Xs = c^<7 ; hence, remembering that le has been multiplied hj a 

 constant, 



Jc*iv^ = XiJKt = Xg/Xs = Z'-i'a', ku- = Z/ai. 



