Egan — Linear Complex, and a certain class of Twisted Curves. 53 



differential equation of the complex is then xdy - ydic = Kdz, and hence, 



for a curve of the complex, 



.r/3, - ?/a: - K7: = 0. . . . (20) 



Differentiating with respect to the arc, usiug Freuet's formulae aud dividing 



by c, we get 



sjjSo - yui - K72 = 0. (20a) 



Differentiating again, we get 



. dy dx 



X (t7/33 - c/ji) - y ((703 - coi) - K ((T73 - cy ,) = (H -[^ ~ P'-^ =~ Ts- 



Hence xfi, ya^ - kjs = - 73/0-. (206) 



Equations (20) and (20a) give 



* V K^ /-^ ; ; 



-«=- = - = y-t^ +f+ k'. 



Hence 73 = «:/v/a;^ + 2/^ + k^ 



Also, 73/(7 = - iP/Bs + yos + K73 



= -/s^TfT^ifi', + a\ + y\) = ^x" + / + k\ 

 Hence a = kI(x' + y' + k'), 7^3 = <7k. (21) 



The second of the equations (21) is Professor McWeeney's theorem (A). 

 The first is Lie's expression for the torsion. 



Again, the osculating plane at (xyz) is Xy -Yx + k(Z-z) = ; hence, if 

 A is (xiyiZi), and /x is {x^yiZi), we find 



ttV/^Va = {^1 + fi + i-'j/KTi + y\ + K^j (21a) 



38. Coi'ollary. — If X and fi are two ordinary points on any twisted ciu've, 

 the equation (19) is the necessary and sufficient condition for the existence 

 of a linear complex containing three consecutive tangents at each of the two 

 points. 



For, choose as a-axis the axis of the complex determined by three con- 

 secutive tangents at X and two at n, and let xfi^ - ya^ - nji =/(s). Then at 

 A and /x we have /(s) =/'(s) = 0, and at A we also have f"(s) = 0. 



If /"(s) vanishes also at n, then the proof just given establishes equation (19). 

 The condition is therefore necessary. 



Suppose, on the other hand, that we are given equation (19). Equation (21a) 

 follows from the fact that /(s) =/'(s) =0 at A and yu. Since f"(sj = at X, 

 we have <7^ = K/(.r"i + y\ + k^). I'rom this, (19) and (21a), we find that 



<^i. = Kl{x\_ + y\ + K-): 

 and it is easy to verify that this involves the vanishing of /"(*) at yu ; the 

 Complex, therefore, contains a third coustcutive tangent at fx. 



