247 Prof. T. H. Havelock. 
writing 6 = 47—tan~1(2r/ct), and expanding the various terms, we get, up 
to and including terms in (27/ct)*”, 
—m go R = 73yg'e7 Mt Me 8 cos (gt /4c+47) 
+ Arpgict- 32 (374+ B(18B—75) te Fcos(gt/4c—tm), (22) 
where X» = 2c?/g and 8B =mr/X. If the pressure system has moved 
through n wave-lengths, we have ct = mo, and the ratio of the amplitudes 
of the two terms in (22) is 
1 mr (180r 
—— + 37+ — —75 2 
dan Vo! * ma xa )} ) 
an expression which gives some estimate of the approximation obtained by 
using only the first term of (22). It depends not only upon the distance 
travelled, but also upon the ratio of the effective breadth of the system to 
the free wave-length for the assigned velocity. 
Compare this approximation with that obtained by applying Kelvin’s 
group method directly to the integral expression for the wave resistance. 
Under certain conditions,* an approximate value of an integral of the form 
b 
f F(x) ef de 
is given by oy 
rE («) 
= et f(a)+m/5} (34 
AO) 
the upper or lower sign being taken in the exponential according as 7” () is 
positive or negative, and a being a root of f’(«) = 0. It is assumed that the 
circular function in the integral goes through a large number of periods 
within the range of integration, while F («) changes comparatively slowly ; in 
addition, the quotient f”’ («)/{/’’ (a)}*? must be small. 
Apply this to the form for F given in (13). The second term within the 
square brackets contributes nothing to the approximation; from the first 
term we have, with ct = Xo, 
f(x) = —K(V—c)t = —gbted + ete. 
Hence 
a=gl4e; f(a) = 2/9; f(a) S’ (a) 3 = 3/./ (an). 
From (13) and (24) the group value of R is 
ll2 3/2 Py 
Rees - 1 Lim ennet a/ (2) Ue Lp (4) et (gt/4e—m/4) (25) 
TY Baga 56 ct / gall? — coe — ic 
Taking the real part of this expression and putting » zero, we obtain 
Bs = PP Lb (9/407) P aos (gt/4c+ 47). (26) 
47? pci? AP 
* Lamb, ‘Hydrodynamics,’ 1932 edn. p. 395. 
112 
