'I'lll--. CAM.MA, OK I'.\( TDRIA!., IUNlTIDN. 



log/(.V + .v)-log/(iV) 



^'.^3 



X 



by II and III above, < !, r/y,y,\ , /-/xf', x 



^ ' \ ; log /(iV + .r) - log f{N + .V - I ) I 



I.e. 



log(iV+i) 

 log (.V+.V) 



<^^=iog ; 



.v+ 



I 



N 



X 



/(A^+-v) 



^v/A' 



and therefore 



Hence, Gauss' Definition II.v = Lt 



N+x\ 

 -^> I as A^ increases. 



A'* . nA^ 



(.v+i)(.r+2) ...{x+N)^'"'^^ 



the only function with a ' simple ' logarithm which satishes the 

 factorial law, and has /'(o) = i. 



That other (not ' simple ') functions satisfy the conditions is 

 evident from the form n.r , jcos 2.V7r + (const.) sin 2.V7r[. 



One aim of this theory of ' simple ' functions is to reduce to 

 logical order the somewhat nebulous ideas implied in the phrase 

 ' drawing a. fair curve through given points,' which one meets with 

 in graphical work. 



PROPOSITION III. 

 .4 complete analytical expression for IT.v. 

 Since log n(A;+i) — log Il^r^log (.v+i), 



•. |logn(.v+i)-(.v+i + c,)log(.v+i)} 



= I log Ux — (.v+ «) log {x + 1) } n arbitrary 



x-\-i 



= \ log n.v -(.V + a) log .V } - (.r + a) log — 



n.v 



••• log -^a - (ditto .v+i) = {x + a)Av\-^^ = i+Av^^^ 



a-i+e 



= I+^2't 



X+I-6 



• It follows at once from the definition of ' the logarithm ' that log x / 

 V- i) being always Ihe upper limit. ^ 



x-i 



I 



I— - 



X 



t ' Avf{Q)' is here and elsewhere used for \f{Q)dQ. The notation is less 



cumbersome, and the special result Avd" =i/(n+i) is constantly useful in expan- 

 sions (as here). 



