determinant whose constituents are limited. 243 
_ that is, Cg GIES (OSG 066 Orrn))p 
whence HOE << [oe 
and if m > 2, Geaaile 
for both are positive. 
Hence if true for m — 1 the theorem holds for m when m > 2. 
But when m= 2, 
C= Cree — Cn Cn, 
= Cy, Cop — lGeil? 
< Cy Cop. 
The theorem therefore holds when m= 2, 3, 4... and universally. 
When n =m, the result gives the theorem of Hadamard used 
by Fredholm in the theory of Integral Equations, that the absolute 
value of a determinant of order m cannot exceed the mth power 
of the absolute value of its greatest constituent multiplied by m2”. 
