Wave Resistance 470 
dividing the integration in 9 into the ranges 0 to « and « to $x, where 
tan « = //k. Reducing the various expressions we find that the part 
corresponding to R” in (22) is now given by 
™a ica) 2 in? ois 
R” = BioeME | Eesi0 a0 Ko Sec 9 ain Ai m cos 2mf 
Si: 0 m? + Kk," sect 0 
x eum (l cos 64-k sin 6) m> dm. (36) 
The remaining terms give contributions to both R, and Rg. It is found 
that the complete results for the two resistances can be put into the form 
Ry = Re = R? = i 
4 16epry'M | * sec® Be-2fe*# cog {xy sec? 6(/ cos 0+-k sin 6)} 6, 
—n/2 
(37) 
Ree RCO RE RY 
1/2 4 
+ 169K )*M? | sec’ Be~2*F ="? cos {ky sec? 6 (J cos 8-+-K sin 6)} d6, 
vi (38) 
where R, is given by (20), R’ by (34), R” by (36), and tan « = //k. 
The previous results for A and B in line, and for A and B abreast, are 
particular cases of these expressions with « = 4x and « = 0 respectively. 
The sum of (37) and (38) could have been obtained from expressions 
given previously for the total resistance of A and B considered as one 
system. Perhaps the most interesting difference between R, and Rg, 
compared with simpler cases, occurs in the last terms in (37) and (38). 
It might appear that both A and B experience effects of wave-interference, 
in the usual meaning of that term, although A is in advance of B. 
However, this is not so, and this can be seen most easily if we suppose 
kov/(P2 + k) to be large and apply the Kelvin method of approximation 
to the integrals in question. According to this, the important parts of 
the integral come from narrow ranges of 0 in the neighbourhood of the 
stationary values of /sec 0+ ksec 0 tan 0, that is, near values of 0 
given by 
tan 0 = — } tana +44/(tan? « — 8). (39) 
Such values only exist if tan? « > 8; moreover, even if they do exist, they 
do not contribute to the value of the integral unless the values of 0 given 
by (39) lie within the range of integration. It is easily seen that they do 
not come within the range for the integral in (37); hence the resistance of 
the leading sphere does not exhibit any characteristic interference effects. 
418 
