228 Dr. J. W. Nicholson on the Asymptotic 



Stokes' law to liquid spheres of diameters varying from 30 

 to 50 times the mean free path of air molecules. 



3. The results obtained by this method taken in connexion 

 with Rutherford's experiments seem to constitute experi- 

 mental verification of Stokes' law for these drops. 



4. Positively charged drops of water and alcohol are found 

 by direct measurement to carry charges which are multiples 

 of 4*65 x 10" 10 , and all of the multiples from 2 to 6 inclusive 

 have been obtained. 



5. The mean of the five most reliable determinations of e 

 is 4*69 x 10~ 10 . The corresponding value of n (the number 

 of molecules in 1 cubic cm. of gas at 0° C, 76 cm. pressure) 

 is 2-76 X 10 19 : that of N (the number of molecules in a gram- 

 molecule) is 6'18xl0 2y : that of e (=3/2-^-, the kinetic 



energy of agitation in ergs of a molecule at 0° C, 7G cm. 

 pressure) is 2*01 x 10~ 16 ; that of m (the mass in grams of an 

 atom of hydrogen) is 1*62 X 10~ 24 . 



Byerson Laboratory, 



University of Chicago, 



October 9, 1909. 



XXIII. The Asymptotic Expansions ofBessel Functions. By 

 J. W. Nicholson, M.A., D.Sc, Isaac Neivton Student in 

 the University of Cambridge* '. 



MANY physical problems depend, for their final solution r 

 upon a knowledge of the approximate values of 

 Legendre and Bessel functions for a large range of their 

 argument and order. In the case of the Bessel functions, 

 investigators f have almost entirely confined their attention 

 to those special types in which the order n is small, though 

 the argument z may be large or small. 



A treatment of the more general problem presented when 

 n is also large has been given by Lorenz J, but only when n is 

 half an odd integer. The immediate object of Lorenz was 

 to obtain some expansions necessary for his investigation of 

 the scattering of light by a glass sphere, in which, as in most 

 problems of this type, only Bessel functions expressible in 

 finite form are required. His results were first approximations 



* Communicated bv the Author. Head before the British Association, 

 Dublin, 1908. 



f Poissnn, Journal de VEcole, 1823 ; Stokes, Cainb. Phil. Trans. 1856 ; 

 Hankel, Math. Ann. i. 1869 ; Lipschitz, Crelle. 1859 ; and others. 



X CEuvres Scientifiqnes, vol. i. p, 435 et seq. 



