245 Prof. T. H. Havelock. 
gpk: = 2 (ios (@) . Ko (Ko) COS Kom das = Ko? {d (ko) }?, 
or if we work directly from the integral (7), we have 
i, ix? { («) }? dk (12) 
mgpRi = Ko Lim MG= ere 
ud 
where the real part is to be taken. Under the general conditions specified 
for $(«), or, in particular, for the case given in (9), it can be shown that this 
leads to the same expression (10). The wave resistance in general is the 
sum of two parts, the steady value Ri, as given by (10), and the deviation Rs. 
Using the definition (11) with the second integral in (6), we find 
co —tix(V—c)t in (V-+c)t 
fi it ca |) a2 | Pot C se ee patna os ik | 
2 pat a cm \. CN) =e “Seoenanle 
(13) 
5. Consider first a special case in which the pressure system is such that 
there are no regular waves left in the rear, a type which Kelvin called a 
waveless system. It follows from (10), (12), and (13), that this is the case 
when the system is such that @(«) is of the form («—xo) W(«), where yr («) 
remains finite. We have then 
| “ F@ cos xa da = b(«) = («—Ko) Wr(«). (14) 
If this system is made to travel with the velocity c, for which 27/xp is the 
free wave-length, there will be no regular train of waves in the rear. The 
integrals (12) and (13) now remain finite and determinate with mw zero; we 
can thus simplify the expressions by making yw zero. The integral (12) 
vanishes, as does also the equivalent expression (10). Then, taking the real 
part of (13), we find for the total wave resistance of this system at any 
time ¢ 
—rgpR = nt | (4-00) {1 («) 
0 
x {<4 sin «Vt cos xct—xot cos «Vt sin Ket} dx. (15) 
It is of interest to examine this solution when the integral can be evaluated 
exactly in finite terms. Burnside* suggested some years ago a method of 
building up exact solutions of certain wave problems, and similar forms 
have been analysed in detail by Kelvin, after obtaining the solutions by a 
different method. The cases in which we can carry out the integrations in 
(15) lead to similar functions; we obtain them by taking 
ap («) = Kte-"*, r > 0: (16) 
* W. Burnside, ‘Proc. Lond. Math. Soc.,’ vol. 20, p. 31 (1888). 
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