( 326 ) 
_ 8 dpd 2ermord 
(ps — pa)? de? de! (p — pa)’ (pap) 
X [(p2+p1—2 p)? + (p—pa) (P2—P)] 
so that the locus is everywhere 
convex in the heterogeneous region 
with an inflection-point at the 
beginning, and we get acurve as 
drawn in fig. 3, which clearly 
indicates the different possibilities 
for the empiric isothermal. The 
point where the locus cuts the 
curve p=f (#2) 1s of course de- 
termined by the formula for jz. 
Mathematics. — “Factorisation of large numbers”, by Mr. F. J. 
Vars, Mechanical Ingeneer at Rotterdam. (Communicated 
by Prof. P. H. ScHouts). 
Introduction. 
The history of the research about the divisibility of large numbers 
is very simple. 
ERATOSTHENES (275—194 b. C.) is said to have invented the 
method of the sieve (determination of the prime numbers under a 
given limit by removing from the series of odd numbers those divisible 
by 3, 5, 7, etc.). 
In 1643 Fermar decomposed a number proposed to him by 
MeRSENNE. In a letter dated “Toulouse le 7 Avril 1643” we find: 
“Vous me demandez si le nombre 100895598169 est premier ou 
“non, et une méthode pour découvrir, dans l’espace d'un jour, s'il 
“est premier ou composé. A cette question, je réponds que le 
“nombre est composé et se fait du produit de ces deux: 898423 et 
“112303, qui sont premiers.” 
The method of FermMat has never been published *). 
') In 1640 Fermat believed 22"---1 gives prime numbers for all values of w 
Afterwards Eurer found that 22° +1 (a number of ten figures) is the product of 641 
and §700417. The author is inclined to ask: If in 1643 FerMar really could factorise 
