MATHEMATICS: M. B. PORTER 
247 
ON A THEOREM OF LUCAS 
By M. B. Porter 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF TEXAS 
Received by the Academy. March 10. 1916 
In a paper on the Geometry of Polynomials^ Lucas has an interest- 
ing generalization of Rolle's theorem, to wit: That the zeros of any 
polynomial F'{z) lie inside any closed convex contour inside of which the 
zeros of F (z) lie. 
Many proofs^ of this theorem have been given, but no one seems to 
have pointed out that the theorem is apphcable to integral transcen- 
dental functions of the type /o(2) = 11]°' (1—z/ai) where 'ST \l/ai\ 
is convergent, i.e., functions of zeroth order (genre zero). 
We shall show that this theorem can be generalized so as to give 
information concerning the distribution of the zeros of the derivative 
of certain rational functions and certain transcendental functions of 
the type Io(z)/Io(z). 
We begin by giving a very elementary proof (perhaps new) of Lucas' 
theorem. 
Proof. Since, in the finite part of the complex plane F'{z)/F{z) = 
ttr)"^, where ai, . . . , an are the zeros oi F (s), can vanish 
only when F\z) vanishes, we have only to show that (z—ai)~'^ can 
vanish only inside the convex contour mentioned. Now since the con- 
tour is convex, all the vectors z—ai drawn from a point z outside the 
contour lie inside the arms of an angle less than 180°; the same thing 
will be true of the vectors {z—ai)~^ (obtained by inverting and reflecting 
in the axis of reals through the point z). But such a set of vectors 
cannot form a closed^ polygon, and hence the theorem is proved. It 
is now at once evident that the theorem is true for functions of the 
t3^e Io{z) and we have, for example, a theorem of Laguerre's that: 
If the zeros of Io{z) are all real, so are those of I'q{z). 
If F {z) = Pn {z) / P m (2), where Pn and Pm are polynomials or func- 
tions of the type /o (z) whose zeros ai and respectively lie inside of 
closed convex contours Cn and C which are external to each other, 
the proof given above shows that, if $ {z) = Pn {z)/P,n (z), then 
can have no zeros in the region swept out by such tangent straight 
lines to C„ as can be moved parallel to themselves into tangency with 
