'z- {asX + P^y + y^z), 



Egan — Linear Complex, and a certain class of Twisted Curves. 61 

 mte 1. 



52. Since Wi(ti 0) has a zero of order six for 8 = t, it is to be expected, 

 in virtue of equation (24), that the expression TT/e/o-^a + 7reil<yt- should have 

 a zero of order six in terms of the arc for Q = t at an ordinary point of any 

 twisted curve. This is seen to be true if we take the trihedron at t as the 

 coordinate axes, and suppose B to be {xyz). 



We have, then (writing S for tr^ and tr for at), 



TTet = z, - TT^e = "iX + (isy + y^z ; 



and we have to find the order of 



when the arc diminishes indefinitely. 

 We calculate z by the formulae 



dz dyi dy-i dy-i 



^ = 7., ^ = «7., -^ = .y.-cy., -^=--7^- 



(Lt yi, 72 = 0, Lt 7a = 1) 

 Again, if Z = a^x + jSa y + yaZ, 



-J- =-a {a^cc + (5oy + y^z) = - (tY, 



dY 



-J- = ctZ- c(a,.r+ j3i2/ + yiz) = aZ - cX, 



-^r- = cSooS: + 5ai -y- = cF+ 1. 

 as as 



We shall write (Ti, (T2 \ c^^Ci . . , to denote the values of 

 da d^a dc , , 



• ^' &^ • • • ^ • • • "* ^^^ °^^^^° '■ 



^ = (2/,t)J = 1 + v,s + v2.sV2 ! + z/ssV3 ! + ..., 

 where Vi = (Ti/2(r, vn = a^j'Ia - ai^/Aa', 



Vi = (T3/2(T - 3(7i 0-2/4(7° + 3<Tl78<T^ 



We find that the firs^ five powers of s disappear from the expansion of 



vz - Z. The coefficient of s^S ! is 



n o - o 3 ^ 3Ci(Tj° 9ct7icr2 IScffi' 



t'c = C(73 - (ToCi + ia'Cax - WCi + &a^ + — + „ ■ (25) 



The vanishing of Ce is therefore equivalent to the equation (33) = 0. We 

 can see from the form of Cs that it vanishes if c and c, are both zero ; in other 

 words, if g'/r > 3 (par. 43). This shows that an undulation (p = 5, §' = 4, ?• = 1) 

 on an algebraic P-curve annuls (33) ; an inflexion does not, unless the coefficient 

 of c, vanishes, i.e. unless laa-i = 3(Ti^ - ia*. 



