448 Wisco7isin Academy of Sciences, AHs, and Letters. 



But this series can only be subotitiitcd if it is convergent, which 

 is the case if a function of the variable r of the form kr~ is 

 substituted for x. By this operation the series is changed to 



— — -v — y — . , . , 



6 ISO 



in which the first term increases, and all the others diminish, 

 for increasing values of ^ provided A is assumed to satisfy 



H > '^ >H 



Tor then the exponent of r will be positive in the first term, and 

 negative in all the others. For exanipie, let A=>^ and we 

 obtain the series 



Hence if in the integral written above, we take the limits g = Icy'~^ 

 instead of the limits — cr. and -f go , we shall approach the more 

 nearly to the true value of the integral the greater the value of 

 r. For we have 



sill ex 



and if we put ^ - ^V ^ 



sin o^X/ ?_ 



/<2?'3 1 V 7 



