EXTENSION OF SERRET-FRENET FORMULAE. VVe may develop the idea of principal lines at any point on a curve of (n -l)-triple curvature geometrically in the following way: Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n -l)-space perpendicular to the tangent and we shall call this space the first normal space at the point;

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... intersection of the first normal space with the osculating plane is a line 1 which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n -2)-space, the second normal space at the point, and the intersection of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n -l)th normal which is perpendicular to the osculating (n -l)-space at the point. We thus see that the rth normal lies in the osculating (r + 1)-space and is perpendicular to r consecutive tangents. These n -1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal. Secondly, let us define the positive directions of these lines. Let the coordinates of any point on the curve be given as functions of a variable parameter: where s denotes the length of an arc of the curve measured from some fixed point on it. We assume, as in the ordinary geometry, that the positive direction of currency along the curve to be that as given by increasing the values of s; we shall assume, moreover, the functions/ 1; / 2 , . . . . , / , , with their derivatives up to the required order to be regular, continuous and finite throughout the range of the par-1 Cayley :-A Memoir on Abstract Geometry : Phil. Trans. Royal Soc, London, 160 (1870) : -"an (n->')-fold linear relation determines anr-omal."