RESISTANCE OF A SUBMERGED CYLINDER 423 
5. Before proceeding, we may confirm this process by applying the 
method to the case of uniform velocity c, for which the results have 
previously been obtained by direct consideration of steady motion. We 
require the limiting values of the integrals in (19) and (20) for ¢ becoming 
infinite, the quantities P being constant. Putting in the appropriate form 
for L from (6), we have, for instance, the integral 
t (ve) 
/ ae | {eitwe—gbcbyt—1)__gitec taht cBe—2F de, (23) 
0 0 
Integrating with respect to 7 and taking the limiting value of the 
integral in « for ¢ > 00, it is readily found that (23) has the limiting value 
2g? c~*[ me-* + i{a 1 e-*Hi(a)}], a (24) 
where ky = g/c?, « = 2x, f, and Hi is the exponential integral. The similar 
integral with x? in eee of x? converges to 
Ke C[ me * + tf? +t — e- *Hi(a)}]. (25) 
The integral with a factor x? is not required at this stage; being factored 
by £,, it clearly does not enter into the second-order approximation. 
Hence, to this order, (19) and (20), give, for uniform velocity, 
Pea rete 1+ 2iKG ca re-*-+ fa! e-°Hi (a) ], 
i igh at P| me-% 4 i{a-?+ 0-1 e-*Hi(a)}]. (26) 
The resistance R, is zero in this case; and from (17) and (26) we obtain 
Ry = 417? p06 ate-o[ 1— 2? a2fy-2 + 2a-1— 2e-*Hi(a)}]. (27) 
This result agrees, to the second order, with the more general expressions 
obtained previously for the wave resistance at uniform velocity (3). 
6. Returning to the general expressions (19) and (20), we shall examine, 
in particular, motion with uniform acceleration y, starting from rest; thus 
we have c = yt, s = }yt”. The first approximation to P, is ya2¢( (1—a?/4f?), 
and it is sufficient for the next stage to put P, = ya27 in the integrals in 
(19) and (20). Hence, to the required order, we have 
é 
P= vat oP, bigtaty oe 7 dr { Detrte es dics 
0 
P ta" p AAG * $0 —2Kf 2 
— — 5p 1— 397ay | cdr | Lexie dk, (28) 
when L* = etttey—7?)-98 ee rele 7) +98 %(-7)}. (29) 
549 
