VI. On the Discontinuity of Arbitrary Constants which appear in Divergent 

 Developments. By G. G. Stokes, M.A., D.CX., Sec. R.S., Fellow of Pembroke 

 College, and Lucasian Professor of Mathematics in the University of Cam- 

 bridge. 



[Read May 11, 1857.] 



In a paper " On the Numerical Calculation of a class of Definite Integrals and Infinite 

 Series," printed in the ninth volume of the Transactions of this Society, I succeeded in 



developing the integral / cos — {vf - mw) dw in a form which admits of extremely easy 



•'o 2 



numerical calculation when m is large, whether positive or negative, or even moderately large. 

 The method there followed is of very general application to a class of functions which 

 frequently occur in physical problems. Some other examples of its use are given in the 

 same paper ; and I was enabled by the application of it to solve the problem of the motion 

 of the fluid surrounding a pendulum of the form of a long cylinder, when the internal friction 

 of the fluid is taken into account *. 



These functions admit of expansion, according to ascending powers of the variables, in 

 series which are always convergent, and which may be regarded as defining the functions for 

 all values of the variable real or imaginary, though the actual numerical calculation would 

 involve a labour increasing indefinitely with the magnitude of the variable. They satisfy 

 certain linear differential equations, which indeed frequently are what present themselves in 

 the first instance, the series, multiplied by arbitrary constants, being merely their integrals. 

 In my former paper, to which the present may be regarded as a supplement, I have employed 

 these equations to obtain integrals in the form of descending series multiplied by exponentials. 

 These integrals, when once the arbitrary constants are determined, are exceedingly convenient 

 for numerical calculation when the variable is large, notwithstanding that the series involved 

 in them, though at first rapidly convergent, became ultimately rapidly divergent. 



The determination of the arbitrary constants may be effected in two ways, numerically or 

 analytically. In the former, it will be sufficient to calculate the function for one or more 

 values of the variable from the ascending and descending series separately, and equate the 

 results. This method has the advantage of being generally applicable, but is wholly devoid 

 of elegance. It is better, when possible, to determine analytically the relations between the 



• Comb. Phil. Trans. Vol. IX. Part U. 



Vol. X. Part I. 14 



