494 
MR. F. W. BARNES ON INTEGRAL FUNCTIONS. 
For, the same conditions still being supposed to hold good, log F ( z ) has to a first 
approximation been proved to be equal to 
( — z) p log z + (— ) p 1 ' + -— -- z? -f lower terms 
And therefore 
(l 
dz 
log F (z) = (— y pz p 1 log z -fi lower terms. 
Hence, to a first approximation, 
N = j" p (— y e in6 r p [log r -f 16 ] idO 
log 0 (r) 
= (- & f- 
A 77 
e lu9 0 dd — r p — 
log r 
which establishes the theorem in question. 
§ 89. In the two preceding paragraphs we have assumed that we were dealing 
with lion-repeated functions. 
From the analysis of Part IV. it is, however, evident that the theorems hold 
in toto for repeated functions, the order being that which has been assigned to such 
functions. 
We cannot, of course, attempt to prove the theorems for functions of multiple 
sequence until we have investigated the corresponding asymptotic expansions. 
§ 90. We may next write down a number of theorems relating to two or more 
integral functions. 
It is obvious that the sum of two or more integral functions of simple sequence is 
an integral function of order equal to the largest order of the component Integra^ 
functions. We may replace additive signs by those of subtraction if the two 
component functions of largest order are not identically equal. The large zeros of 
the compound expression are to the first order of approximation equal to the large 
corresponding zeros of the component function of largest order. 
The product of two or more integral functions of simple sequence is an integral 
function of order equal to the largest order of the component integral functions. 
The number of zeros of the equation 
/q (z) F, (z) + h, (z) F 3 (z) + • • • + K (z) F. (z) = 0, 
where the F’s are integral functions of simple sequence and the h’s algebraic 
polynomials, within a circle of large radius is ultimately to a first approximation 
equal to the number of zeros within that circle of the function of largest order. 
§ 91. The expansions which have been obtained may be utilised to give a proof of 
Bokel’s extension of a theorem due to Picard A 
* Picard, ‘ Annales de l’Ecole Normale Superieure,’ 2 ger., t. 9 (1880). Borel, ‘Acta Mathematica,’ 
t. 20, pp. 382-388. 
