For the section enclosed by a spiral curve and a circular arc, we apply the following torsion stress function:-
Ψ=A(rⁿ-a²ⁿ/rⁿ)cosnθ+B(r²ⁿ-a⁴ⁿ/r²ⁿ)cos2nθ-r²+a² ……(1)
where a=radius of a circle,
n=fraction or integer from 1/2 to 2,
A and B=constants concerning the shape of a s ction.
Boundary lines can be determined by solving the following equation with respect to r_b or cos nθ.
Ar_bⁿ(r_b²ⁿ-a²ⁿ)cosnθ+B(r_b⁴ⁿ-a⁴ⁿ)cos2nθ-(r_b²-a²)r_b⁴ⁿ=0.
The stresses are
Zr/τG=-1/2nsinnθ[Arⁿ+4B(r²ⁿ+a²ⁿ)cosnθ](r²ⁿ-a²ⁿ)/r²ⁿ+1,
Zθ/τG=[r²-1/2nA(rⁿ+a²ⁿ/rⁿ)cosnθ-nB(rⁿ+a⁴ⁿ/r²ⁿ)cos2nθ]/r.
It is necessary to trace the values of stresses on the boundary in our case. As to thecrescent profile, the maximum value of Zθ occurs on the intersecting point of a circulararc and a polar axis, and that of _??_Zr_??_ does on the spiral or outer boundary betweenθ=0 and θ=θa, where θa is the angle, at which two arcs of circle and spiral intersecteach other. The shorter the radius of a circular arc, the greater the value of thismaximum Zθ and consequently the profile becomes wider.
When the radius of a circle is reduced to zero, we get a reentrant profile, which iseasily inferrable from (1). The stress function for this shape of profile is
Ψ=Ar^mcosmθ+Brⁿcosnθ-r²……(2)
where m_??_n, 2_??_m and n_??_1/2.
The values of m and n depend upon the shape of profile. The variations of stressesalong on the periphery are easily seen on Table 1. Severe stress is induced at thereentt ant point as in the profiles (A), (B) and (C). This accumulation of intensivestress can easily be relieved by producing a small circular arc in place or by roundingoff the reentrant part to a small degree, which will be at once understood from thefirst description in this paper.

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By adopting the torsion stress function in a form of
Ψ=rⁿcos nθ-r²,
we can get two kinds of a rod section, one being a kidney profile when 1>n_??_1/2 and the other a leaf profile when 1<n<2.
Then we have two stresses in polar coordinates:-
Zθ=τG[r-1/2nr^n-1cosnθ],
Zr=-τG[1/2nr^n-1sinnθ].
It will suffice to deal with the peripheral stresses in our case. For the leaf profile, themaximum stress lies between θ=0 and±π/2n along the periphery according to the valuesof n. At the reëntrant corner of kidney profile, however, the stress Zr or Zθ accumulatesgreatly, amounting to an infinity, which can be easily relieved by providing asmall circular groove.

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By the analysis of the past towing experimental results with the smooth flat platesand the actual ships, the formula (1) is obtained as the equation for the calculation ofthe frictional resistance of the ship and the formula (2) for the model; and it is foundthat the total resistance of the ships, estimated by the Froude's method, using theformula (1), (2) and the model tests, are in excellent agreement with the thrustmeasurement trial results and the towing experimental results of the actual ships.
(1) Cf=0.00235+0.165/R^0.3125
(2) Cf=0.00235+0.165/R^0.3125-1600/R
where Cf: Mean frictional resistance coefficient: F/1/2ρAV²
R: Reynolds' number: VL/v.
F: Frictional resistance in pounds.
L: Length of ship in feet.
V: Speed of ship in feet per second.
A: Area of wetted surface in square feet.
ρ: Density of water in pounds×second²×feet^-4.
v: Coefficient of kinematic viscosily in feet² per second.

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In this paper it is intended to obtain practical formulae, which are to be used readilyin ship designing office, without resorting to much complicated mathematical treatmentas has been usually the case in these kinds of problems.

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The rudder of a singe screw vessel is working amidst of the race behind the screw propeller, while she is propelled by her own engine.
This race is pushed out from the propeller and their stream lines whirl just as thoseof the strands of a wire rope, so that they cross the rudder plane or the longitudinalcentre line plane of the vessel with some acute angle, from left to right on the upperpart of a right hand propeller shaft line and vice versa on the lower part of it.
Therefore, the upper part of the iudder tends the vessel to the port and the lowerpart to the starboard, because they have some helm angle with respect to the streamline of the race in spite of the helm at centre.
These two athwartship components of the upper and lower portions of the ruddernearly balance each other, and the fore-and-aft components of them incre se the ship'sresistance with plus or minus signs according to the angle of incidence of the rudderelements with respect to the stream line of the propeller race.
The ange θ_r between the stream line and the longitudinal centre line plane and thespeed V_r in m/sec of the stream line can be expressed by the following formulae:-
θ_r=tan^-1[{V_p'-V_s(1-W_R)}cot(tan^-1P_r/2πr)/V_p'×nbr/2πr]
and V_r=V_p'secθ_r
where V_p'=Corrected speed of the propeller race in m/sec.
V_s=Speed of the vessel in m/sec.
w_R=Wake fraction for rudder.
P_r=Pitch of the propeller in m.
n=Number of the propeller blades.
b_r=Length of the propeller elements in m.
And the best angle of incidence φ can be got by aid of a section paper plotting thefollowing C_t values on the base of the angle of incidence for a given θ_r,
C_t=C_asinθ_r-C_wcosθ_r
where C_t=Thrust constant.
C_a=Lift constant.
C_w=Drag constant.
Ty the suitable, distribution of the best angle of incidence φ according to the abovementioned θ_r, the best rudder as a propeller i.e. a contra-propeller, as well as the bestone as a rudder can be designed.

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The rolling of a ship up to large angles of inclination in still water may nearly be expressed by
d²θ/dt²+2αsdθ/dt+λs²k²θ=0,
where λs²=Wh/I; W=the displacement; h=the me acentric height; I_s the apparent momentof inertia at any amplitude; k=the metacentric factor; ∫^θ2_-θ12αI_sdθ/dtdθ=the actualenergy lost per swing. Let k_i²=2E_i/Whθ_i², E_i being the dynamical stability at an inclinationθ_i, then we obtain
k²=k₁²θ₁²-k₂²θ₂²/θ₁²-θ₂² and 1-e-2α_sπ/√λ_s2k2-α_s²=θ₁²-θ₂²/θ₁²
If we know the successive amplitudes and the corresponding periods by experiments, we shall have the variation of the apparent moment of inertia in terms of amplitudeby the formula
I_s/I₀=T_s²/T₀²{k²-(α_s/λ_s)²},
where I₀ and T₀ are the apparent moment of inertia and the period of a single rollrespectively when the ship rolls to very small angles. It was found from the experimentalresults that the apparent moment of inertia increases with the amplitude even thoughthe period of roll decreases, and that no theoretical formulae neglecting its variationgive a correct value of period for any specified amplitude, particularly for ships fittedwith deep bilge keels.

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A ship is inclined inwardly toward the centre of a turning circle soon after the rudder is put over, and when she becomes to turn steadily, she inclines in the opposite side. In this paper, the inclinations of those two kinds are considered.
The first inward heel is important for the safety of a ship when helmed over abruptly, and consequently for the determination of the time required to put the helm hard over, which is necessary to determine the capacity of the steering gear. Treating dynamically the motion of a ship helmed over, the expression for the maximum inward inclination is obtained as follows:
θmax=θ₀{1+T_s/πt₀|sinπ/T_st₀|}
where θ₀: static l heel angle due to the heeling moment caused by the transverse force on the rudder plate put hard over.
T_s: natural period of the ship.
T₀: the time required to put the helm hard over.
A similar motion of a ship occu s in an inclining experiment, and the conditions in order to avoid the free rolling, usually accompanied, are discussed. Additionally, the motion of a pendulum on board used in that experiment is investigated.
Lastly, for the outward heeling angle in the stendy turning, the breadth of the ship must be considered, though neglected usually, and the new expression for it is got as follows:
tanθ_??_tanθ₁/1-b/dtanθ₁
where θ: the required angle of heel.
θ₁: the angle of heel which is usually taken, neglecting the effect of breadth.
b and d: half breadth and draft of a ship respectively.
Studying the effect of a bilge keel on the heeling quality experimentally, it is found that the bilge keel is very injurious to that quality in a turning condition.

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A simple cutting machine for model propeller was designed and made for the use in the tank belonging to the Imperial Fisheries Experimental Station. It resembles, in general, to the templating apparatus for propeller mould, the moving arm being provided with tracer and cutter. For the tracer, it is necessary to prepare special drawings to show the sections of blade at various radius. Cutting work is nearly the same as in the case of ship model cutting machine. It may be a help for speedy preparation of series models of the same diameter, differing slightly in the pitch, blade width and blade thickness.

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As the drag horsepower of radiator is nearly proportional to the cubic power of air speed, the speed-up of modern aeroplane makes it desirable to save the resistance. Recently Assistant Prof. Fukatsu, Member of the Aeronautical Research Institute, investigated the nature of flow around radiators by photographing the aluminium-powder scattered on the water. From the experimental results he concluded that the aerofoil-cowlings would be very effective for decreasing the resistance due to vortices produced along the side and that the interference of side-walls or body-walls upon airflow at the inlet of radiators would be considerably reduced by taking some clearance between radiators and side-walls as illustrated in Figs. 1, 2and17, 18.
According to the measurements of drag of actual radiators with the wind-tunnel, the aerofoil-type cowlings are very much effective upon the reduction of the drag as shown in Fig. 5 and the experiments for cooling power reveal that the use of cowlings as no serious influence on the heat dissipating capacity of radiators (Fig 11).
From Figs. 19 and 22 it will be seen that only a small amount of drag is diminished by the side clearance between walls and radiato s and that there is hardly any effect upon cooling power.

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Pitot tubes and Bruhn tubes (double Venturi-type) are widely used for aeronautic speed meters at the present time. However, the former are unsuitable for other than for high speed uses because of their low sensitivities, and the latter are likewise unsuitable except for low speed uses because of their too high throat velocities, which attain sound velocity when outside wind velocity reaches to about 270km/hr. The object of the author's reseach is to obtain a new speed meter that is fit for medium speed uses.
From numerous experimental reseaches carried by the author, he has obtained the following two excellent tubes of the single Venturi-type. Both measure 18mm in outer diameter and 105mm in length, and have such merits that their actions are almost unaffected by manufactural errors. The one type of tube has constant value of velocity coefficient K=7.3 above the air speed of V=80km/hr. (In the above K=Δp/(γV²/2g), where Δp means the manometric pressure of the air speed meter.) The other type of tube has constant value of K=5.8 above V=15km/hr, and its critical speed (when the throat air velocity reaches that of the sound) is more than 400km/hr.
These air speed meters might also be used for naval purposes in the water. In this case, in making use of the second type of tube, the pressure difference Δp corresponding to height of 80mm of water column, is probably obtainable at the speed of one knot, and be of sufficient accuracy even under that speed.

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Results of model ship experiments which were conducted at the naval experimental tank in Tokyo have been compiled during last twenty years for about 130 actual high-speed vessels as well as many model ships of methodical experiments which cover very wide range of ship-form beyond those of actual ships.
Ship-form coefficients and “elements of form” for optimum resistance qualities are examined by tracing minimum (c) envelopes for successive values of (K) of standardised “(K)-diagrams”.
In designing a high-speed vessel it is assumed that the first given items are the displacement and the speed of the ship.
A method of estimating principal dimensions and ship-form, having desired amount of GM, as well as horse-power to propel said displacement at intended speed is treated as per table 3 in text and figs. 7, 8.
In analysing steam trial results of actual ships by the aid of their model experiments, two empirical coefficients are introduced, one is to cover the scale effect, weather conditions etc., and the other is to provide for the cavitation effect of actual propellers.
The author is in opinion to arrange those empirical coefficients on proper criterions for the reference of the most successful design of future ships.

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