TITE CAMM.N. nU I" \i lOR I Al .. I'i ' \ ( -finX . C)1,J 



We may now assume this relation to hold when x is general : 

 and so obtain n.v=I/. 7 — , — x -, — r-rrr as the definition of a 



possible Factorial F'unction. This formula was estabUshed by 

 Gauss, and the function is known as Gauss'. 



Excursus oil ivliat wc propose to call ' simple ' "•- functions. 



It is known that if a^jb^^ ^2/^2, ^3/^3, • • • ai'C fractions whose 

 denoiuinators are all positive, then Zaj'sJ} lies in value among the 

 given fractions. 



It is also true that if nj/^i, ^2/^2, • • • are in ascending order, 

 2„rt/S„6, by which we mean (ai + <^2 + - • • '^«)/(^i + ^2 + - ■ -b,) increases 

 with n: for I,„a/I,„b = (S„_,rt + rt„)/(2„_,6 + /)„), which lies between 

 S„_,<z/i:„_,6 and a„/b„ : but S„_/7/S„_,6 lies between ^1/61 and 

 a„_,/b„_, and is therefore less than a„lb„. 



Moreover., if ^„a,- means the sum of n terms beginning with a„ 

 I,„a,l1,„bj increases with /, when n is constant. 



For, S,a/S„/?, = (^, + 2„_,<?,+.)/(^ + S„-./^>,), which lies between 

 ajbi and S„_,«,+,/2„_,6,+,, whereas 



lies between S„_ia,4.,/2„_j6i+i and «,+„/6j-|-„, therefore the latter is 

 the greater. 



Hence 2„rt,/S„&; increases with / when n is constant. 



Applying these results to the series of fractions 



fxj-fx fx^-fx^ 



' I ' * ' ' ' 



where .v, .Vj, x^ . . . are increasing values of a variable, we find 

 that if these fractions are in ascending order, {fX—fx)/{X — x) in- 

 creases with A' when x is constant, and (J'(x-\-cx)—fx)/cx increases 

 with X when dx is constant. 



These results hold if the intervals between .r, .Vj, X2 . . . are 

 indefinitely diminished, in which case the set of fractions becomes 

 f'x, f'Xi, /'.t'a, . • . Corresponding results hold, of course, if the 

 original fractions are in descending order. Hence, if fx is a con- 

 tinuous function of x such that f'{x) continually increases or 

 diminishes throughout a certain range of .*■ (i.e. if/"(.v) is positive 

 or negative throughout the range). 



• The word ' simple ' is used elsewhere of functions considered simple from 

 other points of view : it is therefore impossible to stereotype the word as used 

 in this paper. 



