248 
MATHEMATICS: E. J. WILCZYNSKI 
Cn without cutting either Cm or Cn or passing through the position of 
the Hne infinity. This is a region which can be easily marked out 
on the complex plane and will have inside of it neither of the contours 
Cm and Cn- 
Thus our theorem asserts that if the zeros of P„ are inside of one 
branch of an hyperbola and the zeros of P„ are inside the other branch, 
all the zeros of are inside of the hyperbola, or again, if all the zeros 
of Pn are real and lie in the interval (1) x > a, while all the zeros of 
Pm are real and Ue in the interval {2) x < b ^ a, then (z) has no 
complex zeros and all of its zeros He in the intervals (1) and (2). 
1 /. Ec. Polytech., Paris, 28. 
2 All these proofs save one by Hayashi {Annals of Mathematics, March, 1914) are 
based on dynamical considerations. Fej^r, Ueber die Wurzel vom kleinstein absoluten 
Betrage, etc., Leipzig, Math. Ann., 65, 417, attributes the theorem to Gauss and gives 
a bibliography for it, 
3 If this is not at once intuitionally evident it can be shown by resolving the vectors 
in question into components parallel to the arms of the angle above mentioned. 
INTERPRETATION OF THE SIMPLEST INTEGRAL INVARIANT 
OF PROJECTIVE GEOMETRY 
By E. J. Wilczynski 
li y = f {x) is the cartesian equation of a plane curve, the integral 
which represents the length of the arc of this curve between the points 
Pq{xq, Jq) and Pi(xi, ;vi), obviously remains unchanged when the curve 
is subjected to a plane motion. Therefore we may speak of 5 as an 
integral invariant of the group of motions, or as a metric integral invariant. 
In the present paper we shall show how to find integrals connected 
with a given plane curve, whose values are not changed when the points 
of the plane are subjected to an arbitrary projective transformation. 
We shall speak of these integrals as projective integral invariants. 
Let ji, ji, jz, be the homogeneous coordinates of a point Py, and let 
yi,yi. be given as linearly independent analytic functions of a parame- 
ter X. As X changes Py will describe a non-rectilinear analytic curve 
Cy. There exists a uniquely determined linear homogeneous differential 
equation of the third order 
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO 
Received by the Academy, March 13, 1916 
(1) 
y'" + Spiy" + ipiy'_+ pzy = 0 
(2) 
