Waves round a Larye Sphere. 161 



But when /o = 0, 



fi = I dw cos w*= — r-r(A 



Jo 2 V3 v -> ; 



/■'/ = I w <Zm? sin io z = — : — — r( % ), 

 /; = j 1 <Z«, (sin w> 3 + e~«») = i rQ, 



//= | tt7dt0 (cost* 3 + *-«*) = |r(|Y 



so that, by use of a property of the Gamma functions, 



/i/4'-AA'=J»=W-FA' . . , (77) 



in a similar manner, and thus from (74-76) 



^B^-R- 1 -*? (78) 



Otherwise, it is a known property of the asymptotic sub- 

 stitution in H n and (f) n that 



and since p — (m^-z)(6/z)i 



the accent denoting differentiation with respect to /o, and 



3#^s= -(6/*)* (l-ri-s/3*)£/, 



and since the second term of this formula may be neglected 

 if n — s is of lower order than z, even though p have the 

 highest order consistent with this, it follows that 



z- im d<j> u fdx=-~R 



i 



and the value of v' in (78) follows at once. 



The comparison of these two modes of proof supplies an 

 interesting indication of the degree of accuracy possessed by 

 the formulae (70 et seq.). It appears that even if p is of 

 higher order than unity, provided that n — z is of any lower 

 order than s, they may be used in the present case. 



This paper does not propose to tabulate the effect in the 

 transitional region, for in the present state of our knowledge 

 of the subject, the main interest attaches to a determination 

 of the nature of the effect. The formulae are therefore left 

 in a somewhat undeveloped state for the present. 



Phil. Mag. S. 6. Vol. 20. No. 115. July 1910. M 



