partially Lagally's theorem only to calculate the force exerted in some special cases of 

 submerged bodies. Some authors have used the particular forms of Lagally's theorem, 

 for the case of point singularities, as developed by Betz, but sometimes with improper 

 interpretations that exceed its true physical meaning, or ignoring the true power of the 

 theorem, which avoids some usual considerations — [Dickmann: "Krafte auf seine 

 Senke," "Widerstand der Senke," [5] etc., (pp. 468, 649, etc.). Also in references [7] and 

 [8].] It is evident that forces are exerted only when there are bodies which experience 

 pressure, and no more. There are not "forces acting between singularities." These con- 

 siderations are not justified in any strict field theory of hydrodynamic phenomena. 



The new formulas given [3] for the "thrust deduction" phenomenon, as essentially 

 an interaction phenomenon, not only signify the correct application of Lagally's theorem 

 in its original form, but also arise from the direct construction of the total velocity field 

 satisfying Oseen's equations, as extension of Burgers' solution for the open-water pro- 

 peller [9], containing an "interaction field." 



Dr. Lerbs, in his interesting article entitled "Marine Propulsion" [10] has well ob- 

 served that (for infinite bladed propeller) ". . . the tangential component of the in- 

 duced velocity is of minor interest in these considerations." Indeed, in reference to the 

 equivalence of the velocity field to a flow with infinite helical vortices over the slip- 

 stream, there is an interesting result. It has been proven [3] that even with the vortices 

 over a non-cylindrical slipstream, the total velocity field has zero tangential component 

 (in cylindrical coordinates) around bodies with a plane of symmetry. This result is valid 

 in the "first approximation" in Oseen theory. 



In this velocity field, the singularities are those for the flow grad (C/c£ ; *) around 

 the body alone, plus a simple layer over the propeller circle with potential <fi 1 , plus a 

 simple layer over the hull. The latter corresponds to the "interaction field" grad 

 (c<£;*) with which the hydrodynamic boundary conditions remain satisfied over the 

 body, subjected also to the propeller's inflow, grad V This modification of the field 

 around the hull is the cause of a resistance to the advancement of the hull. This deter- 

 mines with precision the meaning of the term "interaction." 



The whole field, which has a region of singularities, satisfied the conditions for 

 which the D'Alembert paradox is removed; and the hull is subjected to a hydrodynamic 

 resistance (or a "thrust deduction," if a reduction of the thrust of the propeller is re- 

 ferred to by this resistance), which is expressed, as derived directly from Lagally's 

 theorem, by [3] 



where <£* — U<$>* + cfa*. The total velocity potential is <£ — Ux 3 + U<j>j* + c§* 

 + </>!• The propeller thrust over the support is: T = — p c(U + c/2)A + AP, where 

 AP is an increment due to the rotation of water in the slipstream, T depending only 

 on the values at infinity, U and U + c, of the velocity field in the exterior and interior 

 to the slipstream respectively. The net thrust T + AT of the propeller, at a certain 

 steady velocity of advancement, —U, is assumed to be equilibrated by the frictional 

 resistance transferred to the hull by the boundary layer. The substantial agreement with 

 observed values of the time average thrust deduction shows the plausibility of the 

 theory (ref. [3] p. 40). 



Without further comment, Dr. Lerbs has mentioned [1] (also ref. [10], p. 283), 

 a type of approximation for the velocity field of the propeller. It consists in an unjusti- 

 fied conjecture replacing the helical free vortex sheets, that corresponds to a non-steady 

 theory, by an infinite number of equidistant vortex disks along the propeller axis. 

 Besides the latter field is computed as steady, we have to recognize that this arbitrary 

 geometrical assumption cannot give a representation of the propeller field with its 

 natural slipstream. Otherwise, in the steady case of the infinite number of blades, the 

 mentioned "representation" fails completely since the actual flow around the body is 



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