128 PROFESSOR STOKES, ON THE DISCONTINUITY OF ARBITRARY CONSTANTS. 



Hence we have altogether for the inferior term, 



log. mod. = 2-1838 ; amp. = + 183" 45'. 5 



Hence reducing each imaginary result from the form p (cos 9 + v - 1 sin 9) to the form 

 a +v/— 1 6, we have for the final result, obtained from the descending series: 



For amp. a? = gO". For amp. x = 150«. 



From superior term - 14-98520 + 43-81046\/- 1 ; - 45-43S60 - 892767 \/- 1 

 From inferior term - 0-01524 - 0-00100 \/- 1 



- 45*44884 - 8-92867 \/- 1 

 Had the asserted discontinuity in the value of the arbitrary constant not existed, either 

 the inferior term would have been present for amp. x = 90^, or it would have been absent 

 for amp. x = 150", and we see that one or other of the two results would have been wrong in 

 the second place of decimals. 



In considering the relative difficulty of the calculation by the ascending and descending 

 series, it must be remembered that the blanks only occur in consequence of the special values 

 of the amplitude of x chosen for calculation: for general values they would have been all 

 filled up by figures. Hence even for so low a value of the modulus of a? as 2 the descending 

 series have a decided advantage over the ascending. 



