The function K"{x) — V2irY (x), where Y (x) is Weber's Bessel function of the 

 second kind. Indeed, if in the expression for R, one integrates by parts, once with 

 respect to x and once with respect to £, and makes use of X{ — VzL) = X{ x /iL) = 0, 

 one obtains for the strut (cf. Pavlenko [1]) : 



L/2 



R = - / / dxd£X(x)X(l-)Yo(v(x - ?)): (46) 



-L/2 



Birkhoff and Kotik [2] have succeeded in reducing the general Michell integral to a 

 form involving Y (x). Unfortunately, a statement of the precise result requires con- 

 siderable preparation, so that we refer to the original paper. 



Before passing on to other aspects of Michell's integral, we should like to 

 mention an interesting result of Stoker and Peters [1]. They assume a thin ship, but 

 instead of assuming it towed in a fixed position with constant velocity as we essentially 

 have done, they assume a constant thrust T along the longitudinal direction of the ship 

 and a fixed weight. In the linearized theory this leads to a relation between T and the 

 resulting velocity which again is just Michell's integral, i.e., the linearized theory does 

 not distinguish between thin ships held fixed and ones free to trim. In addition, they 

 obtain expressions for the vertical force and the moment about a transverse axis, as 

 integrals similar in nature, but more complicated than Michell's integral. Thus both 

 the sinkage and trim can be computed also (see also Havelock [8] and Lunde [1, app. 

 5]). The result as formulated by Stoker and Peters is actually more general than this 

 in that they consider a thin ship heading into periodic infinitesimal waves and find 

 the transient forces and moments which are to be added to the equilibrium ones 

 mentioned above. 



Limitations of the Theory. Since ships as actually designed do not seem to 

 have the appearance of what one might expect for a "thin" ship and since they move 

 in water and not in perfect fluids, one may reasonably ask: What are the limitations of 

 Michell's integral in use and what useful information can be obtained from it? 



An estimate by purely mathematical means of the possible error in using the 

 linearized theory for a conventional or even a simplified hull shape has not been 

 attempted, as far as the author is aware. One might try to make such an estimate from 

 the second-order theory in the perturbation series, but it would likely be difficult and 

 crude. In any case, this would still be only for a perfect fluid. Comparison with 

 experiment remains, and has frequently been made. This leads to another type of 

 difficulty, how to separate and measure the resistance caused by waves. If L is the 

 length and c the velocity of a towed ship-like form, one may write. 



R = %pc*L*C(Re,F), (47) 



where Re — cL/v, the Reynolds number, and F = c/(Lg)^, the Froude number. 

 Thus, from tests of a hull model, or of a set of geometrically similar models, one can 

 obtain information about C(Re, F) for some range of values of these two variables 

 (actually, for points on several lines in the (Re, F) -plane radiating from the origin), 

 but no direct information concerning the wave resistance. Considerable ingenuity has 

 been used in trying to make this step. All methods seem to rely on first separating the 

 contributions of the tangential and normal components of the force on a surface- 

 element of the hull, which gives after integration over the hull 



C(Re,F) = C t (Re,F) + C n (Re,F), (48) 



and then assuming 



C, = C t (Re), C n = C n (F). (49) 



The coefficient C t is usually estimated from data for the frictional resistance of a flat 



116 



