CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 



181 



The value of the first root got by interpolation from Mr. Airy's table is 1.0845. For the 2nd 

 and 3rd roots we get from (45) 



for i m 2, 7T _1 = 1.25 - .00788 + .00020 = 1.24232 ; 

 for i = 3, 7r"'0 = 2.25 - .00438 + .00003 = 2.24565. 

 For higher values of i the last term in (45) may be left out altogether. 



18. The following table contains the first 50 roots of the equation W = 0, and the first 10 

 roots of the derived equation. The first root in each case was obtained by interpolation from 

 Mr. Airy's table ; the series (42) and (45) were sufficiently convergent for the other roots. In 

 calculating the 2nd root of the derived equation, a rough value of the first term left out in (45) 

 was calculated, and its half taken since the next term would be of opposite sign. The result 

 was only - .000025, so that the series (45) may be used even when i is as small as 2. By far 



