'K^■^ 



TIIV: CA.AIMA. OR FACTORIAL, FUNCTION*. 



(I)- 



II. 



III. 



and .*. f'x 



fX-fx 

 X-x 



x-k 



if A' > .V and I < x. 



ex 



increases (or diminishes) as ex increases, 



when .r is constant. 



f{x-\-^x)—fx . 



^x 



increases (or diminishes) as .v increases, 



when ex is constant. 



A function for which f"{x) is always positive or always 

 negative we propose to call a ' simple ' function. 



The more familiar types of continuous functions consist of a 

 succession of ' simple ' functions : in other words, they may be 

 represented by curves with only occasional points of inflexion. 



A useful application of (I) is to the approximate summation 

 of series whose terms are of the form /'{x). 



x=X 



Thus, 



> t X . ex, when ex is constant, < .V, .J ; 

 Z-J \ /A-/(t-ci') 



,v=f 



C.J,. 



{a + ir 



The 'summation theorem' of Integral Calculus is a special 



case. 



By means of II and III we will now prove Proposition II, 

 that Gauss' assumption, in regard to the factorial function, that 

 n(A''+^) — > A'*', n.r is the only assumption which makes log n.r a 

 ' simple ' function (from .v = — i to .v = oo). . 



PROPOSITION II. 



f(x+N) 



We have fx = -, — r-^ / , ,., , and for positive integral 



-^ (-v+i) . . . (a-+A^)' ^ ** 



values of .r, fx is the ordinary factorial, .r ! 



Since 



log fix + 1 ) - log /v = log (.r + 1 ) , 



log/, if ' simple,' must be an increasing function. 



Considering x to vary only from o to i and using N for a 

 positive integer, 



