Wave Resistance. 584 
We substitute from (2) and (3) in (1). Carrying out the integrations in 
f. f’; h and h’, we obtain, after some reductions, 
256 gob*l 
Tp® 
+ (cos" fae cos? # £08 ( psec d) — cos ¢ sin (psec ¢)} dd, (4) 
ar /'2, 
R= | (1 emesecnt 2 J feos! $ + cos? ¢ 
0 
where p = 2gl/c?, and « = gd/c”. 
3. In reducing this expression to a form suitable for calculation, we take 
first the terms which are non-oscillating regarded as functions of c. 
A typical integral is 
Pp 
{ cos’ ge FP"? dd. (5) 
0 
Changing the variable, this becomes 
e*| aten be-PP de — IniBe-# W_, _. (8), (6) 
0 es 
where W is a confluent hypergeometric function. We can obtain an expansion 
by using the contour integral for the general hypergeometric function of this 
type. In this case we obtain 
Segaas (26 IN (e) IM —=sa-s) In (eset 
€ s s+i s+} 
Wo) | Bil) Use) 0s ea) aa (7) 
Qret ey Y (3) I (4) 
the contour separating the poles of I’ (s) from those of [ (—s+4)  (—s+4). 
We have, therefore, to evaluate the residue of the integrand at the simple 
poles s = 4, 3, 3, and at the series of double poles s = 7, $, 44,.... The 
lattet residues give logarithmic terms. Carrying out the calculation, we 
obtain the expansion 
Ws, -3(8) = sn ast op! te! —ga! 
: eae n+1/2 ] 9 SEAN p ee )) 
en (n+1) TP 0 (le. lace 2 el 1p z p I)? (8) 
with log y = 0-57722 ... . 
We obtain thus 
“2 WANs gue 113 
B sec? > eS i = poe ea (0 
fy Cas ye ay 15) Ng “Be a ee +e * 3073" 
263 
20480 
Oa + (SOM apg Ph Gggg ht Joe GHB) 
232 
