Prof. Stokes on the Discontinuity of Arbitrary Constants. 71 



March 9.- — Mr. Hopkins gave an account of some experiments on 

 the conductivity of various substances, and pointed out the bearing of 

 the results on theories of terrestrial heat. 



April 27.— Mr. Humphry read a paper "On the Proportions of 

 the Human Frame." 



May 1 1 . — A paper was rea'd by Professor Stokes, " On the Dis- 

 continuity of Arbitrary Constants which appear in Divergent Deve- 

 lopments." 



In a paper " On the Numerical Calculation of a class of Definite 

 Integrals and Infinite Series " printed in the ninth volume of the 

 •Cambridge Philosophical Transactions/ the author succeeded in put- 



ting the integral I cos - {w^—mw)dw under a form which admits of 



receiving every numerical calculation when m is large, whether po- 

 sitive or negative. The integral is obtained in the first instance under 

 the form of circular functions for m positive, or an exponential for 

 m negative, multiplied by series according to descending powers of 

 m. These series, which are at first convergent, though ultimately 

 divergent, have arbitrary constants as coefficients, the determination 

 of wliich is all that remains to complete the process. From the 

 nature of the series, which are applicable only when m is large, or 

 when it is an imaginary quantity with a large modulus, the passage 

 from a large positive to a large negative value of m cannot be made 

 through zero, but only by making m imaginary and altering its am- 

 plitude by TT. The author succeeded in determining directly the 

 arbitrary constants for m positive, but not for m negative. It was 

 found that if, in the analytical expression applicable in the case of m 

 positive, —m were written for m, the result would become correct 

 on throwing away the part involving an exponential with a positive 

 index. There was nothing however to show d. priori that this pro- 

 cess was legitimate, nor, if it were, at what value of the amplitude of 

 m a change in the analytical expression ought to be made, although 

 the occurrence of radicals in the descending and ultimately divergent 

 series, which did not occur in ascending convergent series by which 

 the function might always be expressed, showed that some change 

 analogous to the change of sign of a radical ouglit to be made in pass- 

 ing through some values of the am])litude of the variable m. The 

 metliod which the author ai)i)lied to this function is of very general 

 application, but is subject throughout to the same difficulty. 



In the present paper the author lias resumed the subject, and has 

 ])ointed out the character by which the liability to discontinuity in 

 the arbitrary constants may be ascertained, which consists in this, 

 that the terms of an associated divergent series come to be regularly 

 positive. It is thus found that, notwithstanding tiic discontinuity, 

 tlie complete integrals, by means of divergent series, of tlie diftercn- 

 tial equations which the functions treated of satisfy, are expressed 

 in such a manner as to involve only as many unknown constants as 

 correspond to the degree of the e(|uation. 



Divergent series are usually divided into two classes, according as 



