Wave Motion due to a Submerged Obstacle. 530 
A and B are given by (14) when we replace ¢ by ccos¢; putting these 
values in (55), we see that we may change the order of integration. Further, 
as we make pw vanish ultimately, we may use simplified forms of A and B 
corresponding to (16). These give 
a | 2 5) KEW dk 
— 2098 2 ed 
R = 4n)2ca po |" sec*h dp {, Ge naseC 5) EWI ECCA 
To obtain the limiting value for ~ zero we may treat this like the similar 
expressions in (30); or, alternatively, we may put (y/c)sec d = 1/n, and use 
the general result 
Lim 
n—> 2 
(eS =F Ve-9 4/240}. 
al+n?(c—a)? 
The apparent difficulty with regard to values of @ near 7/2 is overcome by 
noticing that with the particular functions involved in R no extra contribu- 
tion arises from such terms near the upper limits of the variables. Carrying 
out the integration in « in this way, and changing the remaining variable by 
putting tan @ = ¢, we obtain 
R = 4rrg*pa8c- Se— 2a \,« 14 22)3/2¢-20fle dt. (56) 
0 
The remaining integral can be expressed in terms of known functions. 
Possibly the simplest method is to use the confluent hypergeometric 
function* defined, for real positive values of « and for real values of & and m 
for which k—m—4 <0, by 
Omelet ene m —$+Mp—u 
Wa, m (a) =a. lime mea el +u] a) s+Me—U Clyy, (57) 
We have now the wave resistance of the sphere given by 
R = 42? pa®f Fail e~ ol? Wy, (a) ; a = 29f/c?. (58) 
8. For purposes of calculation, we require expansions of W,,:(«). This 
function belongs to the logarithmic type of confluent hypergeometric function, 
and general expansions are not available in this case; however, they can be 
obtained without difficulty for Wi,1. In the first place, the differential 
equation satisfied by Wi,; is 
Cy ff We MS ) ae 
Perera) a 9) 
We use the ordinary methods for solving by means of power series. The 
roots of the indicial equation are 3 and —4; hence one of the fundamental 
* E. T. Whittaker, ‘Bull. Amer. Math. Soc.,’ vol. 10, p. 125; also Whittaker and 
Watson, ‘ Modern Analysis,’ Chap. XVI. 
129 
