1893.] On Operators in P/ii/sical Mathematics. 133 



Integrate to x. Then 



*-l+l'-.... , (130) 



when C is some constant introduced by the integration. To find it, 

 note that the series with the exponential factor vanishes when x is 

 infinite ; so (130) gives 



C = aj-^ + i --.... -log a, with^ = co. (131) 



It is not immediately obvious that the function preceding the 

 logarithm in (131) increases infinitely with x. But by (127) we may 

 regard it as the ratio 



(132) 



a?' 2 



and we see that the terms in the numerator become infinitely greater 

 than those to correspond in the denominator. 



A Formula for Enters Constant. 



62. Next examine whether (131) gives a rapid approximation to 

 the value of C. When x = 1 we get 



1-i + T V- uV + - -0 = 0-77, say. 

 When x = 2 we get 1'3203 -0'6903 = 0'6300. 

 When x = 3 we get 1*6888 I'lOOS = (X5790. 

 So with x = 3 the error is about T fa only. The usual formula 



C = i + i+i.+ ---- +- logr, withr = oo, 



is very slow. Ten terms make 0'62. Twenty make about 0*602, 

 which is still far wrong. We see that (131) will give C pretty 

 quickly with -a moderate value of x. 



63. In passing, we may note that the function. 



is represented by 



.., (134) 



