But let us take heart. A function devoid of singularities is also devoid of 

 interest: it is a constant. 



A different representation of steady irrotational flow is due to H. Lewy [23]. 

 By proper choice of axes and units the surface condition can be expressed in the form 



dz / dz dy \ 1 



— ( 2t — I H = 0, yp = 0, w = <p + vp. (42) 



dw \dw dw / 2y 



Let us regard this as an equation in the complex domain involving a complex func- 

 tion y = — t](w), real when if/ = o. 



The equation has the solution 



z = — ir)(w) + 



dr} \ 2 



_ 2t] \dw J _ 



dw, (43) 



and provided the integral is real on some segment of if/ = 0, we have 



y = — y(w) on \p = 0, (44) 



and the point z describes a free streamline. 



Lewy has also proved that the flow is analytic on the free surface in the steady 

 case and has thus established that the formula gives the most general steady irrotational 

 motion with a free surface. 



M. J. Vitousek [24] has studied in detail flows obtained by attributing certain 

 forms to 77 (w) and in particular waves of trochoidal and cycloidal profile. 



You will no doubt expect to hear something concerning the motion of a body 

 through a fluid, typically the problem of the submarine well below the surface. The 

 classical case governed by Kirchhoff's equations, when the vessel moves with velocity 

 u = u(t) and angular velocity co in liquid otherwise at rest gives [25] for the force 

 R and the moment M 



dK 



Ro= o> A K, (45) 



dt 



dL 



Mo = co A L - u A K, (46) 



dt 



where 



K = pj<pdS, L — — pj<pr A dS. (47) 



s s 



The integrals are taken over the surface 5 of the body. M. D. Haskind [26] has this 



year extended these formulae to include the case where the liquid has velocity v =r v{t) 



dv 

 at infinity and acceleration / = — . If $ is the velocity potential, he defines a, = $ 



dt 



— v • r and the corresponding force and moment are 



R = Ro + M'f (48) 



M = Mo + v A K + M'r cA f, (49) 



where R ft and M are calculated by the formulae above, M' is the mass of liquid dis- 

 placed and r c is the position vector of the centroid of the volume displaced. 



The motion of a submarine when gravity is taken into account has been dis- 

 cussed this year by P. V. Harlamov [27]. The problem he studies is that of a body 

 totally immersed with buoyancy equal to the weight of the body so that in general 

 the weight and buoyancy form a couple. A thorough going discussion of directions of 



13 



