336 
MATHEMATICS: M. B. PORTER 
fication to any integral function of finite class p whose Weierstrass pri- 
mary factors are of the normal form 
/'(2) = n,(i- 
where 
and where the as are all real, or more generally to the integral functions 
^{z)=f{z) exp(Ti3^+' + 72sO, 
where 71 and 72 are real and 71 is negative if p is odd and 72 positive or 
zero if p is even. The essential thing in Bocher's proof is that the 
vectors which occur in the square brackets all lie inside an angle of 
180° and hence their sum cannot vanish if z is not on the axis of reals. 
Thus we obtain a much more general result than that obtained by 
BoreP or Polya^ who employed methods that seem applicable only to 
functions of class zero and one, the theorem at which we have arrived 
being this: All the zeros oj the derivative of the integral function 
are real, if the as and ys are real and 71 is negative, if p is odd and 72 
positive if p is even.^ 
1 These Proceedings, 2, 247 (1916). 
2 Bocher, Some Propositions Concerning the Geometric Representation of Imaginaries, 
Annals of Mathematics, 1892. 
3 Bocher, A Problem in Statics and its Relation to certain Algebraic invariants, Proc. 
Amer. Acad. Arts Sci., 40, No. 11, 1904. Neither of these papers by Bocher was listed in 
the bibliographies of Fejer and Hyashi, which I cited, and hence were overlooked by me. 
Fonctions Entieres, pp. 32 et seq. 
Bemerkung zur Theorie der Ganzen Funktionen, Jahresber. D. Math-Ver, October- 
December, 1916. 
' The same proof can be further applied to 
4^(z) =f (z) exp. (70 z + 71 2^+' + 72 2O. 
