48 Proceedings of the Royal Irish Academy. 



29. Let yi and {z be any points in space. Let 



%y^ta.if.i ^ ^i^a-i/yj = B{yz). 



The equation R{yy) = represents in general a quadrie. The 

 coefficients (ff,-.-, etc.) being functions of t, there "wlH be one quadrie R 

 attached to each point on the curre. 



It is clear that if R{yz) vanishes, the points (y) and (z) are conjugate 

 with respect to ^. Also if we write B{d:x) = R {12), R{tx) = R^}1), etc., 

 equations (15) show that for m, n < 4, 



B{mnj == {mnj -i- (nm) = 0, {mf/i) R{mm) = 'ICmm) = for m = 0, 1, 2. 



Hence the tetrahedron (x) 'x)(sd)(z) is self-conjugate, and the first three 

 vertices lie on R. Also if the point (x) is an ordinary point on the curve, 

 this tetrahedron is of finite volume, since the determinant of the coordinates 

 of its vertices, which is |a;ii.*,-^.-| = B, does not vanish. Hence by the 

 lemma, R reduces to the square of the plane {x)(£) (x), ie. of the osculating 

 plane a. 



Hence ^(]/y) = A(2a^,y, and therefore 



(33) = R{33) = A(2a,i..)- = X(03f. 



Therefore R [yy) = (33) i^a,yOV(03f. 



Hence R 'yy) vanishes identically if, and only if, (33) = 0. (16) 



30. If (33) vanishes identically, so does R [yy), and hence 



a„ = a,-.. + an = 0. (17) 



Supposing this, one of the equations ^ 1-5; becomes a, = ai2^2 + flis^s + «u-?"4. 

 Differentiating this, and taking account of the value of di given by 

 another of the equations (15), we get di^e^ -t- di^ + a^ai = 0. 

 Treating di, ui in the same way, we get 



d^ -h di^ + diiXi = 0, Ori^ T- Oia^ -i- djiXi = 0. 



if ow the determinant (x^b^^ is not identically zero ; hence 



«IS = ai3 = «14 = 0, 



and «!,, ffli3, an are constants. Similarly the other «(, are constants. The 

 equation (01; = can then be written 2«y(iP.i,- - ayi/J = 2«yX,- = 0, and 

 this is the equation of the complex to which the tangents belong. 



Conversely, if the curve belongs to a linear complex we must have 



Ol _ 02 



hiiT-i -r 6,^^ + hiiXi 621*1 + izifz + iiiXi 

 {hi, const., hij + Iji = 0). 



