529 Prof. T. H. Havelock. Some Cases of 
the expansions were introduced, in (41) and (43), it follows that the contri- 
bution of the second term in (44) is one-third of that of the first term ; 
hence, summing up the result so far as the first term of (49) is concerned, we 
have 
co 7 2 Le 
mR’ = —5ec?atp | c7F dic | cos db dp | dp { sin a Py (cos a) cxasinasine 
0 0 0 0 
X cos (ka sin « cos PB sin d) 
x {Asin (ka cos « cos 6) + B cos (na cosacos)} de. (52) 
Taking the integration with respect to 8, we find it is equal to 
2 ( exasinacesB cos (Ka sin asin gd sin 8)dB = 2rIo(kasinacos¢), (53) 
0 
where Ip («) is the Bessel function Jy (iz), a result which may be obtained by 
direct expansion and integration term by term. For the integration with 
respect to « the term in A in (52) obviously gives zero, and we are left with 
2a (. Ip (Ka cos ¢ sin «) cos (Ka cos ¢ cos «) Ps (cos «) sin « da. (54) 
0 
Here also we may expand in powers of xa and integrate term by term ; 
it can be shown that the integral of the coefficient of («a)" vanishes except 
for the single term x?a?; thus we find that (54) reduces to 
—(20r/5) x?a?cos*¢. 
7. We have now to consider the term mX in the value for p in (49). We 
might omit this term, on general grounds, as giving no contribution to R 
ultimately when mw vanishes; for X is the velocity potential for a sphere at 
rest in a given field X;. However, it may be left in, and we have a similar 
calculation. Taking the second integral in (44), we find it is now only the 
term in Y; which counts; hence the contribution of this part is one-half of 
that of the first integral in (44). Further, it is the term involving A which 
gives a value different from zero when integrating with respect to «, and 
instead of (54) we have 
20 \, Ip («a cos ¢ sin «) sin (Ka cos d cos «) P; (cos «) sin « da, 
0 
which reduces to (47/3) xa cos d. 
8. Collecting the various results, we have now 
Rv—2caep [ory atte [ic Beos d+ pA) cos? ¢ dg, (55) 
0 0 
a form which may be compared with the corresponding expression for the 
cylinder in (29). 
128 
