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MATHEMATICS: E. J. WILCZYNSKI 
This equation may be written 
Consequently the integral 
will not change its value under the transformations (4). Thus we 
see that the integral p is intrinsically connected with some geometric 
property of that arc of the curve Q which corresponds to the interval 
a -^x -^h. It is also clear that this integral and its geometric signifi- 
cance will remain unaltered by any projective transformation of the 
plane, since it is expressed entirely in terms of the coefficients pi, pi, pz 
of (2) which are invariants of the projective group. 
Therefore the integral p, defined by (5), is a projective integral invariant. 
If / is any absolute differential invariant of the curve Cy, the integral 
Sidp is again an integral invariant, and all integral invariants are 
expressible in terms of those obtained in this way. 
We wish to explain the geometrical significance of the invariant 
integral p. For this purpose we need one further preliminary notion, 
namely that of the eight-pointic nodal cubic of a given point of a given 
curve. 
A cubic curve is in general determined by nine of its points. If eight 
points only are given, there exist infinitely many cubics, forming a 
pencil, which pass through these points. In particular there exists a 
pencil of cubics, such that each cubic of the pencil has eight-pointic or 
seventh-order contact with the given curve Cy at a specified non-singular 
point Py. One and only one of the cubics of this pencil has P^, the 
point of contact, as double point. We call this cubic the eight-pointic 
nodal cubic of the point Py, or the penosculating nodal cubic of Py."^ 
The significance of the integral p is contained in the following theorem 
which we shall state without proof, but all of the terms of which have 
now been explained. 
Consider an arc of an analytic curve corresponding to the interval 
a X b oi the independent variable. Divide this interval into n 
parts by means of the values Xq = a, Xi, x^, . . . Xn-i, Xn = b, such 
that lim dxk = lim (xk+i — Xk) = 0 as # grows beyond bound. Let 
A, Pi, P2, . . . Pn-i, B be the points on the curve which correspond 
to these w + 1 values of x. Let tk be the tangent and Ck the eight- 
pointic nodal cubic of Pk. The three points of inflection of the cubic 
Ck are on a line ik which interesects tk in a point Ik. Denote by tk one 
of the three inflectional tangents of Ck and let Tk be its intersection with 
