Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in an Inclined State
Tsutomu HORI and Manami HORI

viXra : 2301 . 0159 [ Classical Physics ]30 Jan. 2022

p. 1 / 14

Tsutomu H ORI and Manami H ORI( Nagasaki Institute of Applied Science, Japan ) Jan. 2023, pp. 1 ~ 14

Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in an Inclined State
Tsutomu H ORI † and Manami H ORI ††Summary
In this paper, a theoretical treatment on the hydrostatic stability of ships is presented.

As thesimplest hull form, a columnar ship with rectangular cross-section, which is made of homogeneoussquared timber with arbitrary breadth and material, is chosen.
In the Part 2 of this theme, the stable attitude in an inclined state of the ship, which is not stable in
the upright state with horizontal deck, is analyzed by means of ship's hydrostatics.

By doing so, thedependence of the inclined attitude on the breadth and material of the ship will be clarified.
Keywords : Hydrostatic Stability, Columnar Ship, Rectangular Cross-Section,
Arbitrary Breadth and Material, Stable Attitude, Inclined State

1. Introduction
In the previous paper (1) as a typical example problem (2),(3) related to the hydrostatic stability of ships,
we solved the condition under which the ship floats stably in the upright state with horizontal deck, interms of the positional relationship among the center of buoyancy, center of gravity and metacenter.

Atthat time, the target hull form is a columnar ship with a rectangular cross-section, which is made ofhomogeneous squared timber with arbitrary breadth and material.
On the other hand, if the above conditions are not satisfied, under what inclined attitude does the ship
float ? is also of interest from a mechanical point of view.

Igarashi et al. of the National Defense Academyof Japan have elucidated this problem in detail based on geometrical considerations concerning the centerof buoyancy and the center of gravity for the squared timber with square (4) and rectangular (5) crosssections.
In this 2nd paper, as an extension of the previous 1st paper (1), we describe a theoretical treatment forsolving the stable attitude of a columnar ship with a rectangular cross-section in an inclined state.

The

one of the authors gave an solution for the inclined attitude and published it in the journal (6) “ NAVIGATION ”
of Japan Institute of Navigation at 2021.
We summarize the above solutions consistently and introduce them in this paper.
† Professor ,Naval Architecture Course, Department of Engineering, Faculty of Engineering,Nagasaki Institute of Applied Science. 536 Aba Machi, Nagasaki, 851-0193, Japan†† Jewel Manami HORI of Five Stars JP, Daughter of †viXra. org Classical Physics , E-mail : HORI_Tsutomu@NiAS.ac.jp
Submitted : 30 January 2023 , HomePage : http://www.ship.nias.ac.jp/personnel/horiken/ (in Japanese)

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Tsutomu H ORI and Manami H ORI

2. Material  and Breadth  as Setting Variables
In this paper,  and  are defined as the setting variable, as in the previous paper (1).

( hereinafter called the material ) is the ratio of the specific weight  t of the columnar ship ( t in thesubscript is the initial letter of timber ) to that  w of water ( w in the subscript is the initial letter of water )，and  ( hereinafter called the breadth ) is the aspect ratio of the breadth  h to the depth h of the crosssection as follows :

t
w



breadth ････････････････････････････････････････(1)( where ， > 0 ) ( where ，0    1 )

depth



h
h

Here, when  w is fresh water,  represents the specific gravity of the columnar ship.

3. Stable Conditions in the Upright Statefor a Columnar Ship with Rectangular Cross-Section
In the previous paper (1), the condition for stable floating in the upright state with deck horizontal canbe written as the relation between  and  in Eq. (1) as follows : 2  6 (1   )  0････････････････････････････････････････････････････････(2)Hence, it was explained that the above condition can be divided into the following cases (1) :● Stable conditions for breadth  with fixed material 

6  (1   )e.g.  1  
2● Stable conditions for material i ) In the case of  

6
2



 ･･････････････････････････････････････(3)6≒ 1. 225 
2with fixed breadth for wide breadth,the floating body is always stable regardless of material  .
ii ) In the case of  

6for narrow breadth,2

it is then stable in both lighter and heavier materials than wood with  1
as shown below :2

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Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in Inclined State




1

     1 （ Heavy Material ）2 ･････････････････････････････････(4)
3 (3  2  2 )

 


6
where, 

3


e
.
g
.


1



≒
0.
289


6

0 

1
  （ Light Material ）2

4. Stable Attitude for an Inclined Columnar Shipwith Rectangular Cross-Section
In this chapter, we will try to find out what kind of inclined state is stable when the stable conditionin the upright state described in the previous chapter is not satisfied.

For this purpose, let’s analyze theinclined attitude, i.e. the heel angle, of the columnar ship.
As shown in Fig. 1, we shall assume that a columnar ship of length L with a rectangular cross-sectionof depth h and breadth  h , which is made of homogeneous material and of squared timber of specificweight  t , floats stably in a lateral inclined state of heel angle θ to the starboard side from an uprightstate.

h
b

h
2

h

M

h
2

h
2cos 

G

G





B



B

5

h

B

2

o

tan 
Z f  (  cos  


2

d h


Fig. 1

The columnar ship, with rectangular cross-section of length L,
floating stably in a lateral inclined state.

sin  ) h

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Tsutomu H ORI and Manami H ORIFirst, in order to determine the draft, we need to find the cross-sectional area Aw under the water
surface at lateral inclination.
Since its underwater shape is a trapezoid with height  h , the lengths of its upper and lower bases
can be calculated by taking into account the increase or decreaseh2
submerged lengths with respect to the draft d in the upright state.

tan  of the port and starboard
So, the underwater area Aw

is obtained as follows :

1 
h
h

tan  )  ( d tan  )    h
 (d 
2 
2
2


Aw 

  hd･･･････････････････････････････････････････････････････････････(5)Here, the above result is equal to the area of the rectangle, which is the underwater shape in the
upright state.
The weight W and the buoyant force F B of this columnar ship can be obtained as follows, respectively :W   t Vt   t   h  h  L ･･････････････････････････････････････････(6)FB   w V w   w Aw L   w   h d  L 

Here, the weight W of the former is obtained as the product of the specific weight  t and the total
volume V t of the columnar ship.

And the buoyant force F B of the latter is obtained as the product of thespecific weight  w of water and the displacement volume V w of underwater portion, according toArchimedes' principle.

Then V w is obtained by the product of the cross-sectional area Aw in Eq. (5)and the ship’s length L.
The floating body is stable under the following conditions where the weight W and buoyancy F B are in
equilibrium.

W  FB････････････････････････････････････････････････････････････････････(7)Substituting W and F B in Eq. (6) into both sides of the above, we obtain as :

 t hh L   w hd  L･･････････････････････････････････････････････････(8)By solving the above equation, the undetermined draft d in the upright state can be determined as times the depth h of the ship, as follows :d th  hw･････････････････････････････････････････････････････････････(9)In this paper, to simplify the problem, it is assumed that the deck, i.e. upper side of a rectangularcross-section, is in the air and the bottom, i.e. lower side of a rectangle, is in the water over the entire
breadth even when the ship is laterally inclined, as shown in Fig. 1.

That is, we will discuss the case inwhich the cross-sectional shape under the water surface is trapezoidal, as calculated in Eq. (5).

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Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in Inclined State
The above assumptions would impose the following conditions, where the increase or decreasehtan  of submerged length due to the lateral inclination does not exceed the freeboard h  d or the2
1
draft d in the upright state, while divided into two cases around  .
2 h  d  (1   ) hhtan   2
 d  h( for Heavy Material of  

1( for Light Material of  

1

2
2

)

･････････････････(10))

Therefore, the heel angle θ is limited to small inclination within the following range.

1  2 (1   )  tan 





  
 tan 1  2  



  


( for Heavy Material of  

1( for Light Material of  

1

2

2

)

･･･････････････････････(11))

For example, it means the following setting range.

 1 ,  12

 tan   1   


4



 ･･･････････････････････････････････････････････････(12)The position of the center of buoyancy B ( B ,  B ) in the inclined state by heel angle θ is obtained by
Hori (7) ~ (11) as the center of hydrostatic pressure.

As shown in Fig. 1, in the inclined coordinate system,

which is fixed to the ship and has its origin at the center of the ship's bottom, the position ( B ,  B ) is
calculated as follows (7),(10),(11) when the draft and half-breadth of the ship in upright state are f and b,
respectively.

 ･･････････････････････････････････････････････････････(13)
2
f
b
2
B 

tan  

2
6f

B 

b2
tan 
3f

This result is consistent with the result of Eq. (A-5) described in the Appendix A-1 of the above Hori’s
papers (10),(11).

In the Appendix, the centroid of the trapezoid, which is the cross-sectional shape under

the water surface at lateral inclination, is calculated geometrically from the areal moment.

Therefore,

it can be seen that Eq. (13) coincides with the well-known center of buoyancy certainly.
Here, in order to conform to the notation of this paper, f and b in Eq. (13) are replaced as follows
respectively.

f  d  h
1
b 
h
2



 ･････････････････････････････････････････････････････････････(14)

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Tsutomu H ORI and Manami H ORI

Thereby,  B and  B can be written as follows : 2 tan 
h
12 




 ･･･････････････････････････････････････････････････(15)22h tan 
B 

h 

2
24 

B 

Next, the center of gravity of the ship is located at the centroid of the rectangular cross-section ( i.e.,at the center of the figure ), even after inclining, since homogeneous materials are assumed. Therefore,husing the fact that the sum of  B and  G is equal to,  G in Fig. 1 can be obtained as follows :
2

h
B
2
1
 2 tan 2 

h
h
2
24 

G 



12  (1   )   2 tan 2 h24 ･･････････････････････････････････････････････(16)In order for the ship to float while maintaining the inclined state shown in Fig. 1, the center ofbuoyancy B and the center of gravity G must first be located on the same vertical line.

Therefore, thefollowing relationship is required between  B and  G .

B tan G ･･････････････････････････････････････････････････････････(17)  B   G tan  

Here, by using Eqs. (15) and (16) for  B and  G , the following relationship is obtained.

 2 tan 2   2  6 (1   )   2  ･･････････････････････････････････････････････(18)The tangent of the inclined attitude θ for a given material  and breadth  is then obtained by thefollowing equation.

tan  2  6  (1   )   2 



･･････････････････････････････････････････････(19)When the interior of the radical symbol of the right-hand side of the above equation is positive, there
exists a solution for the heel angle θ.

This result coincides with Eqs. (1-h) and (4-f) of Igarashi et al. (3).

This requires that the interior of the braces in the numerator of the above equation take positive values,
as follows :6 (1   )   2  0････････････････････････････････････････････････････････(20)The inequality above is the inverse condition in which the inequality sign is opposite to the stablecondition in the upright state in Eq. (2) of Chapter 3, and the validity of the analysis in this chapter canbe confirmed.

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Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in Inclined State
Finally, it is necessary to examine whether the above-mentioned inclined attitude is stable or not.
For this purpose, let's consider determining the location of the metacenter M , meaning the center ofinclination.
The metacentric radius BM can be calculated by using the basic formula (12),(13),(14) of naval
architecture as follows :

I CL

BM 

Vw

･･･････････････････････････････････････････････････････････････(21)Here, I CL is the quadratic moment about the center line of water plane, and V w is the underwater
volume of a ship.

I CL in the numerator of the above formula can be calculated as follows, since the water plane athinclination is a rectangle of length L and breadth
.
cos 
3
1  h 
I CL 
 L ･･････････････････････････････････････････････････････(22)12  cos  
And the denominator V w can be obtained by using d in Eq. (9) for A w in Eq. (5) and as follows :V w  Aw L   h d  L    h 2 L･････････････････････････････････････････････(23)

By using the obtained results I CL and V w into Eq. (21), BM can be determined independently of thelength L of the columnar ship as follows :3

1  h  L
12  cos  
2
BM 

h
 h2 L12  cos 3 ･････････････････････････････････････(24)

BG in the inclined state is then obtained below by using the trigonometric ratio with  B in Eq. (15),as shown in Fig. 1.

BG 

B
2

h  BM cos 2  ･･･････････････････････････････････････(25)sin 
12  cos 

Thereby, the metacentric height GM can be determined by subtracting Eq. (25) from Eq. (24), as
follows :GM  BM  BG

 2 ( 1  cos 2  ) 2 sin 2 hh12  cos 3 
12  cos 3  BM sin 2   0 ･････････････････････････････････････････････････････(26)From this result, the metacenter M is always located above the center of gravity G, since GM takes
a positive value regardless of the heel angle θ, material  and breadth  .

Therefore, it can be seen thatthe inclined attitude θ determined by Eq. (19) is constantly a stable state.

However, it is necessary tocheck that the calculated θ is within the assumed small heel angle in Eq. (11).

p. 8 / 14

Tsutomu H ORI and Manami H ORI

For example, in the states  and  below, the heel angle θ, BG and GM are calculated as follows
by Eqs. (19), (25), and (26).

 1 ,   ･････････････････････････････････････････････････(27)
2

h 
6


1

 
2
4

 GM  BG This state corresponds to the case where the diagonal line of the square cross-section is aligned withthe water line, and the heel angle θ is also within the setting range of Eq. (12).

And BG and GM also

coincide with the results described in examples of many textbooks (15), (16), (17).
In addition, Fig. 1 shows the following states, and the inclined attitude θ and the positions of B, G and

M are also drawn accurately.

  0.4 ,   1.1    31.7

 GM  0.113 h , BG  0.296 h  ････････････････････････････････････････････(28)Z f  0.629 hHere, Z f in the above Eq. (28) is the water depth at the starboard side of the ship's bottom, and is
calculated by the following equation.

htan  ) cos 2 (  cos  
sin  ) h2

Z f  (d 

5.

･････････････････････････････････････････････････(29)Calculation Results for the Stable Inclined Attitude θ
In this chapter, the dependence of the stable attitude θ at lateral inclined state on the breadth  andmaterial  of the columnar ship is grasped.
Fig. 2 shows the dependence of the above on breadth  when  is a fixed, and Fig. 3 shows that onmaterial  when  is a fixed.

The results in both figures are obtained by calculating the heel angle θ in

Eq. (19) using an Excel spreadsheet.
Since   0 means that the ship floats with its deck horizontal and is the limit point at which theinequality sign in Eq. (2) becomes an equality sign,  and  satisfy the following relationship at thatpoint.

 2  6 (1   )  0･･････････････････････････････････････････････････････(30)Thereby the intersection with  - axis in Fig. 2 is obtained by Eq. (3), and that with  - axis in Fig. 3 isobtained by Eq. (4), replacing the inequality sign in both equations by an equality sign.

p. 9 / 14

Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in Inclined State
In both Figs. 2 and 3 above, the heel angles θ of materials  and 1   are obtained equally, as can
The angle θ becomes smaller as breadth be seen from the factors in the radical symbol of Eq. (19).
becomes wider.

And θ is largest for materials with   0.5 such as wood, and is smaller as  becomes

heavier or lighter than that.
The reason why the point is not plotted in the case of   1 for   0.5,   1.06 for   0.4, 0.6 and  1.04 for   0.3, 0.7 in Fig. 2 is because the heel angle θ exceeds the range of the small inclination in
Eq. (11).
For the same reason, we have placed points for the three states of   1.10, 1.15, 1.20 in Fig. 3, because
in the case of breadth   1.06 , the heel angle θ is within the range of the small inclined state in Eq. (11)over the entire range of material  .
Igarashi et al. (5),(6) provide a detailed analysis of all inclined state, including cases of large heel angles(where part of the deck is immersed in the water or part of the ship's bottom rises into the air), whichcannot be calculated in this paper.

And they have perfectly elucidated the dependence on  and  byorganizing all cases in maps and tables and verifying them experimentally, so we encourage to read theirpaper for anyone interested.

θ (deg)50
α = 0.5
α = 0.4 , 0.6
α = 0.3 , 0.740θ (deg)β = 1.1040

β = 1.15
β = 1.20

30

30

20

20

10

10

β

0
1.00

Fig. 2

1.05

1.10

1.15

1.20

1.25

The dependence of the stable
inclined attitude θ on breadth  .

α

0

0.2

Fig. 3

0.3

0.4

0.5

0.6

0.7

0.8

The dependence of the stable
inclined attitude θ on material  .

Fig. 4 illustrates the attitudes of the four states when the material is fixed at   0.5 and the breadth  1.0, 1.1, 1.2 and 1.3, including the positions of B, G and M. It can be seen how the heel angle decreases as the breadth  increases.
Fig. 5 shows the five attitudes for material   0.25, 0.3, 0.5, 0.7 and 0.75, with the breadth fixed at  1.06. It can be found that the heel angle  decreases symmetrically around   0.5 even if thedraft increases or decreases as the material  becomes heavier or lighter than that.

p. 10 / 14

Tsutomu H ORI and Manami H ORI 1.0 1.2 1.1

M

M

M

G


G

B

Fig. 4BM  2.9  0.5





0.5.

  0.7

  0.75

M.

B  0.0

1.0, 1.1, 1.2, 1.3 with the material fixed at

M
G

G

B

  16.1

The four attitudes for breadth  0.3

MG



G  34.7

  0.25

Fig. 5B  45.0 1.3

GBB  26.2

The five attitudes for material  39.3

M



G
B

  26.2

M



G
B

  2.9

0.25, 0.3, 0.5, 0.7, 0.75 with the breadth fixed at

1.06.

6. Verificational ExperimentFig. 6 compares the model experiment ( left ) and the calculation results ( right ) for the case of material  0.458 and breadth   1.15 .
The model of the columnar ship is length L = 30 cm, depth h = 10.0 cm, breadth  h = 11.5 cm, and weight

W = 18.09 N. Two pieces of chemical wood were pasted together in the center at the top and bottom, andthe model was manufactured by Space Model Co., Ltd. in Nagasaki, Japan.

The verificational

experiment was conducted by floating its model in a small water tank.
The inclined attitude was   27. 5 in the experiment and the calculated results are as follows by
Eqs. (19), (25), (26) and (29).

  26.7 (   0.458,   1.15 )GM  0.068 h , BG  0.269 h  ････････････････････････････････････････････(31)Z f  0.668 hp. 11 / 14

Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in Inclined State
We consider that the reason why there is a difference of about 1 ° between the two is that the heel
angle θ in the experiment was obtained by measuring tan  from photographs and that the center ofgravity position G may be slightly off-center due to the manufacturing process of the model.

Therefore

we are able to verify that the theory in this paper can correctly calculate the actual inclined state.

  26.7

 52.65 
1tan 1   tan (0.5146)  27.23
 102.32    27.0   26.7

  27. 5


M G
5

B

DSC_6895

Fig. 6The Comparison of experimental ( left ) and calculated ( right ) results
for material   0.458, breadth   1.15.

7. Afterword
In this paper, as an applied example which is an extension of the previous paper (1), a theoreticaltreatment for solving the stable attitude of a columnar ship with a rectangular cross-section in a lateralinclined state is explained in an easy-to-understand manner, in which the inclined states are limited toa small heel angle ( where the deck is not submerged and the ship’s bottom is not floated ), in order tounderstand the stability theory of ships.
The authors would be very happy if this paper could be of assistance to teachers and students who willteach and learn this field in the future, going one step forward from the basic examples in the previous
paper (1).

Acknowledgments
In closing this paper, we would like to pay tribute to two valuable papers (4),(5) written by Tamotsu
IGARASHI, Professor Emeritus of the National Defense Academy of Japan.

The reason is that the authors

were deeply impressed by both of their papers.
We then would like to express my heartfelt gratitude to Dr. Yoshihiro KOBAYASHI, former professor of
Sojo University and current president of Como-Techno Co., Ltd. in Nagasaki, Japan.

He always gave

warm encouragement to the author's research and recommended that this study should be published in
English.

We are greatly inspired by the vigorous academic spirit with which he writes about theresults of his research in books (18),(19).

p. 12 / 14

Tsutomu H ORI and Manami H ORI

Finally from the 1st author (20), let me express the following thanks.

I would like to communicate mydeepest gratitude to my late teacher, Pr. Masato KURIHARA (21),(22),(23), who cordially taught me the theoryof “ Hydrostatics of Ships ” with detailed figures and formulas on the blackboard when I was a 1st year
undergraduate student and learned my 1st specialized subject of naval architecture in the College of Naval
Architecture of Nagasaki in Japan.

Therefore, I am following the appearance of my teacher at that timefrom more than 40 years ago as an exemplary example, when I currently lecture Hydrostatics of
Floating Bodies (24),(25) and Theory of Ship Stability (26) ~ (30) to 2 nd year students at my university (31),(32).

References ‡(1) Hori, T. , Hori, M. : “ Theoretical Treatment on the Hydrostatic Stability of Ships ( Part 1 : ) Stable
Conditions for the Upright State ”, 2022 (March), viXra.org ( Pre-print Repository ), viXra : 2203.0180[ Ver.3 ], Classical Physics, pp. 1 ~ 16.

(2) Hori, T. : “ A Typical Example on Ship’s Stability Theorem ” (in Japanese), NAVIGATION (Journal of Japan
Institute of Navigation), 2021 (July), No.217, pp. 39 ~ 46.

(3) Hori, T. : “ Theoretical Procedure on the Hydrostatic Stability of Ships ( Part 1 : ) Stable Conditions for
the Upright State ”, The Bulletin of Nagasaki Institute of Applied Science, 2023 (January), Vol.62,

No.2, Research Notes in Mathematical and Physical Science, pp.151 ~166.
(4) Igarashi, T. : “ An Investigation on the Orientation of a Long Square Bar in Still Water ” (in Japanese),
NAGARE (Journal of Japan Society of Fluid Mechanics), 2000 (August), Vol.19, No.4, pp. 253 ~ 267.
(5) Igarashi, T. , Nakamura, T. : “ An Investigation on the Orientation of a Long Rectangular Bar in Still
Water ” (in Japanese), NAGARE (Journal of Japan Society of Fluid Mechanics), 2007 (December), Vol.26,
No.6, pp. 393 ~ 400.
(6) Hori, T. : “ An Advanced Example on Ship’s Stability Theorem ― Solution for Stable Attitude of an
Inclined Ship ― ” (in Japanese), NAVIGATION (Journal of Japan Institute of Navigation), 2021 (October),

No.218, pp. 58 ~ 65.
(7) Hori, T. : “ A Positioning on Ship’s Centre of Buoyancy Derived by Surface Integral of Hydrostatic
Pressure ― Proof that Centre of Buoyancy is Equal to Centre of Pressure ― ” (in Japanese), NAVIGATION(Journal of Japan Institute of Navigation), 2018 (January), No.203, pp. 88 ~ 92.

(8) Hori, T. : “ A Positioning on Ship’s Centre of Buoyancy Derived by Pressure Integral of Hydrostatic
Pressure ― Part 2 : In the Case of Arbitrary Sectional Form ― ” (in Japanese), NAVIGATION (Journal ofJapan Institute of Navigation), 2018 (July), No.205, pp. 28 ~ 34.

(9) Hori, T. : “ A Positioning on Ship’s Center of Buoyancy Derived by Surface Integral of Hydrostatic
Pressure ― Part 6 : The Proof for Submerged and Floating Body of Arbitrary Form ― ” (in Japanese),
NAVIGATION (Journal of Japan Institute of Navigation), 2021 (January), No.215, pp. 69 ~ 77.
(10) Hori, T. : “ Proof that the Center of Buoyancy is Equal to the Center of Pressure by means of the
Surface Integral of Hydrostatic Pressure Acting on the Inclined Ship ”, 2021 (September),viXra.org ( Pre-print Repository ), viXra : 2109.0008 [ Ver.4 ], Classical Physics, pp. 1 ~ 20.

______________________________________________‡ Bold text in the list means that there is a HyperLink

p. 13 / 14

Theoretical Treatment on the Hydrostatic Stability of Ships( Part 2 : ) Stable Attitude in Inclined State(11) Hori, T. : “ Proof that the Center of Buoyancy is Equal to the Center of Pressure by means of the
Surface Integral of Hydrostatic Pressure Acting on the Inclined Ship ”, The Bulletin of Nagasaki
Institute of Applied Science, 2022 (January), Vol.61, No.2, Research Notes in Mathematical andPhysical Science, pp. 135 ~ 154.
(12) Hori, T. : “ A Consideration on Derivation of Ship’s Transverse Metacentric Radius BM ” (inJapanese), NAVIGATION (Journal of Japan Institute of Navigation), 2017 (April), No.200 (First 200th
Anniversary Issue), pp. 75 ~ 79.

(13) Hori, T. , Hori, M. : “ A New Theory on the Derivation of Metacentric Radius Governing the Stabilityof Ships ”, 2021 (November), viXra.org ( Pre-print Repository ), viXra : 2111.0023 [ Ver.3 ], ClassicalPhysics, pp. 1 ~ 16.
(14) Hori, T. : “A New Theory on the Derivation of Metacentric Radius Governing the Hydrostatic Stabilityof Ships ”, The Bulletin of Nagasaki Institute of Applied Science , 2022 (June), Vol.62, No.1, Research
Notes in Mathematical and Physical Science, pp. 51 ~ 67.
(15) Ohgushi, M. : “ Theoretical Naval Architecture (Upper Volume) ― New Revision ― ” (in Japanese),
Chapter 4 – Equilibrium of Ships, Transverse and Longitudinal Metacenter, Change of Trim,
Section 4.4 – Transverse Metacenter and BM , Questions 2 and 3, Kaibun-dou Publishing, 1971
(June) 1st. Printing, p. 83.

(16) Sugihara, K. : “ Nautical Theory (Ship Mechanics Division) ” (in Japanese), Chapter 3 – Transverse
Stability, Section 3. 3 – Calculation of BM and the approximate value, Example 2, Kaibun-dou
Publishing, 1964 (July) 1st. Printing, p. 58 ~ 59.
(17) Akedo, N. : “ Basic Nautical Mechanics ” (in Japanese), Chapter 3 – Stability of the Ships, Section 3.1.3– Metacentric Radius, Examples 1 and 2, Kaibun-dou Publishing, 1983 (June) 1st. Printing, p. 125 ~132.
(18) Kobayashi, Y. : “ Tank System of LNG - LH2 ― Physical Model and Thermal Flow Analysis by Using CFD― ” (in Japanese), Seizan-dou Publishing, 2016 (December), p. 1 ~ 375.

(19) Kobayashi, Y. : “ Utilization System of LNG - LH2 at Ultra-Low Temperature and Cold Heat ― No
Waste Energy System ― ” (in Japanese), Shouwa-dou Publishing, 2019 (April), p. 1 ~ 226.

(20) Hori, T. : “ Web Page on HORI’s Laboratory of Ship Waves and Hydrostatic Stability ” (in Japanese),
Naval Architectural Engineering Course in Nagasaki Institute of Applied Science,
http://www2.cncm.ne.jp/~milky-jun_0267.h/HORI-Lab/ .
(21) Kurihara, M. : “ On the Rolling Motion of a Buoy in Regular Waves ” (in Japanese), The Bulletin of theCollege of Naval Architecture of Nagasaki, 1974 (June), Vol.15, No.1, pp. 1 ~ 4.
(22) Kurihara, M. : “ On the Motions of a Ringed Buoy in Regular Waves ” (in Japanese), The Bulletin ofNagasaki Institute of Applied Science, 1978 (October), Vol.19 (Commemorative issue of the namechange from former the College of Naval Architecture of Nagasaki), pp. 11 ~ 15.

(23) Kurihara, M. : “ On the Motions of a Floating Vertical Cylinder in Regular Waves ” (in Japanese), The
Bulletin of Nagasaki Institute of Applied Science (former the College of Naval Architecture ofNagasaki), 1979 (June), Vol.20, No.1, pp. 1 ~ 5.

p. 14 / 14

Tsutomu H ORI and Manami H ORI(24) Hori, T. : “ Lecture Video Proving that Center of Buoyancy is Equal to Center of Pressure for the
Rectangular Cross-Section (80 minutes in the 1st half ) ” (in Japanese), 2021 (7 January), Regular

Lecture No.13 of “ Theory of Ship Stability ” (Specialized Subject of Nagasaki Institute of Applied
Science), https://youtu.be/Wd7jKMXSqhc .

(25) Hori, T. : “ Lecture Video Proving that Center of Buoyancy is Equal to Center of Pressure for the
Rectangular Cross-Section (90 minutes in the 2nd half ) ” (in Japanese), 2021 (14 January), Regular

Lecture No.14 of “ Theory of Ship Stability ” (Specialized Subject of Nagasaki Institute of Applied
Science), https://youtu.be/bniJ6-9vJPI .

(26) Hori, T. : “ A New Derivation of Metacentric Radius (i) Positioning the metacenter (63 minutes in the
1st half ) ” (in Japanese), 2021 (14 October), Regular Lecture No.3 of “ Theory of Ship Stability ”(Specialized Subject of Nagasaki Institute of Applied Science), https://youtu.be/lUWbQ92zJQQ .

(27) Hori, T. : “ A New Derivation of Metacentric Radius (ii) Calculation formula for metacentric radius (82minutes in the 2nd half ) ” (in Japanese), 2021 (21 October), Regular Lecture No.4 of “ Theory of Ship

Stability ” (Specialized Subject of Nagasaki Institute of Applied Science),https://youtu.be/qAIzLKXSY4U .
(28) Hori, T. : “ Example on the Stability Theory of Ships (i) Stable condition of a columnar ship withrectangular cross-section of different widths (85 minutes) ” (in Japanese), 2020 (19 November), Regular

Lecture No.8 of “ Theory of Shup Stability ” (Specialized Subject of Nagasaki Institute of Applied
Science), https://youtu.be/PNVuRuZWYBM .

(29) Hori, T. : “ Example on the Stability Theory of Ships (ii) Stable condition of a columnar ship with squarecross-section of different materials (92 minutes) ” (in Japanese), 2020 (26 November), Regular Lecture

No.9 of “ Theory of Ship Stability ” (Specialized Subject of Nagasaki Institute of Applied Science),https://youtu.be/eeVg9ThjPd0 .
(30) Hori, T. : “ Seminar on the Stability Theory of Ships (Outline explanation of the theory and modelexperiment in a small water tank) (15 minutes) ” (in Japanese), 2020 (7 August), Inquiry learning forhigh school students (Nagasaki Prefectural Seiryo High School) conducted online,
https://youtu.be/4T6znj1iKPI .
(31) Hori, T. : “ Naval Architecture Course, Department of Engineering , Faculty of Engineering, Nagasaki
Institute of Applied Science ” (in Japanese), Introduction of Educational and Research Institutes,
NAVIGATION (Journal of Japan Institute of Navigation), 2021 (January), No.215, pp. 38 ~ 45.
(32) “ Naval Architecture Course’s Web Site ” (in Japanese), Faculty of Engineering in Nagasaki Instituteof Applied Science, administrated by Hori, T. , http://www.ship.nias.ac.jp/ .

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