mr. e. w. Barnes on integral functions. 
495 
The identity 
n 
S' 
*-4 
t = l 
G, (z) = 0 , 
in which the G’s are integral functions of simple sequence of any finite or zero order 
not greater than some number p, and the functions H t — H,, are polynomials of order 
greater than p or transcendental integral functions, necessarily involves 
G, (: z ) = G .3 (z) = 
= G u (z) = 0 . 
Since the identity holds for all points in the plane of the complex variable 2, we 
may consider it in the neighbourhood of 2 = <x>. 
Suppose first that the G’s are functions of simple sequence of non-integral finite 
order. 
If p L be the order of G t ( 2 ), we shall have near 2 = 00 the identity 
X e sin 
1=1 
,pi 
H (2) 
= 0, 
where we have neglected in each term terms 01 lower exponential order than those 
retained. 
The identity will hold -for all values of arg 2 such that 2 is not within a finite 
distance of the zeros of the G’s. 
The functions H ( 2 ) by hypothesis cannot be equal to one another. As 2 tends to 
infinity, one of them must become infinite to an order which exceeds the order to 
which all the others become infinite by a quantity of order greater than z p . 
The corresponding term (say) G ] ( 2 ) e H ' fz) is then infinite to an order greater than 
n 
the order of any other term of the identity X G t ( 2 ) e H ‘ (2) = 0. 
t =i 
Since e H(:) cannot vanish, we must then have G x ( 2 ) = 0. 
n 
The same argument may now be applied to the identity X G ( ( 2 ) e lI ‘ (s) = 0, and it 
i=2 
may be proved successively that all the functions G vanish. 
And thus the theorem will be proved. 
When any of the quantities 'p are integral, a suitable modification of the formulae 
in accordance with §73 shows that the theorem is still true. When the G’s are 
repeated functions, a corresponding modification again establishes the theorem. 
When the functions G’s reduce to constants c„ so that p = 0, the theorem is still true, 
the functions H being unequal.. 
§ 92. We pass now to the consideration of the resemblance between an integral 
function of simple sequence and its derivative. 
And I would remark that, in the same manner as Rolle’s theorem is proved, it 
may be established that the real zeros of such a function with real coefficients are 
