(40) 



124 PROFESSOR STOKES, ON THE DISCONTINUITY 



Comparing the equations (38), (39) with (37), we get 



C= ^J] [eV-1-^F+ {,r-^r'(i)+log2| x/IT/^, 



z»= g)\£ + {,r-ir'(i)+iog2}r]. 



Eliminating E, F between (36) and (40), we get finally 



C = (27r)-^[\/3T^ + {(^-.n^'(l) +log8) V'::! -7r( B], 

 Z> = (2,r)-i[^ + {tt-^T (^) + log8| 5]. 



Concltision. 



23. It has been shewn in the foregoing paper, 



First, That when functions expressible in convergent series according to ascending powers 

 of the variable are transformed so as to be expressed by exponentials multiplied by series 

 according to descending powers, applicable to the calculation of the functions for large values 

 of the variable, and ultimately divergent, though at first rapidly convergent, the series contain 

 in general discontinuous constants, which change abruptly as the amplitude of the imaginary 

 variable passes through certain values. 



Secondly, That the liability to discontinuity in one of the constants is pointed out by the 

 circumstance, that for a particular value of the amplitude of the variable, all the terms of an 

 associated divergent series become regularly positive. 



Thirdly, That a divergent series with all its terms regularly positive is in many cases a 

 sort of indeterminate form, in passing through which a discontinuity takes place. 



Fourthly, That when the function may be expressed by means of a definite integral, the 

 constants in the ascending and descending series may usually be connected by one uniform 

 process. The comparison of the leading terms of the ascending series with the integral 

 presents no difficulty. The comparison of the leading terms of the descending series with 

 the integral may usually be effected by assigning to the amplitude of the variable such a 

 value, or such values in succession, as shall render the real part of the index of the expo- 

 nential a maximum, and then seeking what the integral becomes when the modulus of the 

 variable increases indefinitely. The leading term obtained from the integral will be found 

 within a range of integration comprising the maximum value of the real part of the index 

 of the exponential under the integral sign, and extending between limits which may be supposed 

 to become indefinitely close after the modulus of the original variable has been made in- 

 definitely great, whereby the integral will be reduced to one of a simpler form. Should a 

 definite integral capable of expressing the function not be discovered, the relations between 

 the constants in the ascending and descending series may still be obtained numerically by 

 calculating from the ascending and descending series separately and equating the results. 



G. G. STOKES. 



