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1901.] © MARINE REVIEW. 19 METHOD OF CALCULATING CURVE OF STATICAL STABILITY. By GEORGE CROUSE COOK.* Among certain notes on stability from the course in naval archi- tecture at the Glasgow university, Glasgow, Scotland, I find the following splendid mathematical demonstration of a method for calculating the curve of statical stability from a basis of the conventional displacement curves plotted for varying angles of inclination. As to the actual prac- tical value of the method I cannot speak positively, having never per- formed original calculations by this method, but without accurate graphi- cal execution it will be obvious that considerable error may occur and it is therefore to be used with much discretion. : With the required body plan at hand the work should begin with the calculation and plotting of a series of displacement curves, that is curves representing the volume of displacement at corresponding draughts for FIGURE | such convenient angles as may be chosen, say 15°, 30°, 45°, etc., up to 90°; the origin of each curve being, of course, the point of tangency of a line of inclination with the largest section on the plan. On each of these curves the displacement for which the stability condition is to be investigated is marked and the limiting draught line drawn. Then by the proposition that the area of a curve of displacement to any draught divided by the displacement at that draught gives the distance of the center of bouyancy below that water line, the vertical position of the center of bouyancy at each angle of inclination may be found and plotted. With these vertical positions of the centers of buoyancy known, the actual distance the centers have moved vertically, or the dynamical arms, may 'be found by simple comparison and measurement with the position of the center of bouyancy when upright. This stage of the work for the 45° inclination is shown in Fig. 1, where W L has been plotted by the draught (d) taken from the corresponding displacement curve, and the vertical position of the new center of bouyancy (B') has been found at ae ee ws . wy ry Y wy za { t | | | 0 f - os 45° 6> TS* ae BO? oe = DEGREE Go. ee 3. : No Ficuee "2. a distance E B' below W L in the parallel line A A. Projecting B upon E B' gives the dynamical arm b B' (g B'-g b) for this condition. 3 Ess e dynamical stability is then plotted from the arms thus found. ee Fig. 2. A Pied well known proposition of the science of theoretical naval architecture shows that the dynamical stability of a vessel at any given angle is equal to the area of the curve of statical stability to that angle. This is equally true when applied to curves of stability arms only and leads naturally to the following demonstration, which may be readily understood with an elemental knowledge of the calculus: Let the dyna- mical arm at P in Fig. 2 be represented by s; let the corresponding statical arm be h. Then by the proposition: *Associate Member Society Naval Architects, New York; Associate Member In- stitute Naval Architects, London. saa/h. deb. ds=h. d D. ds hs = tangent of slope at P. dD From this, then, it is seen that the ordinate of the statical anm curve at any point is equal to the tangent of the slope of the corresponding dynamical arm curve at the same point or that this ordinate (h) is equal to the tangent of the angle R P Q. Therefore, if P Q is drawn to such a length as to represent unity, that is 57.3° on the scale of the base, Q R will be the measure of the tangent of this slope or the statical arm at the angle of P. So, by drawing a series of these tangents and measuring the Q Rs the ordinates for the curve of statical arms for the required condition may be plotted. The acuracy of the statical curve found by this method depends, of course, upon the skill with which the tangent is drawn to the dynamical curve, as the slightest error in the direction of the tangent at P will cause considerable variation at Q R. I have checked this method in cases where both curves were already determined by the usual calculations and found a sufficiently low percentage of error to justify its judicious use when the usual instruments are not to be had and a fairly accurate idea of the stability condition is desired. The writer would be gratified to have correspondence with anyone interested in the use of this method or with anyone failing to fully under- stand the principles and application of any part trom the description. NOVEL TYPE OF STEAMER. Messrs. John I. Thornycroft & Co., Chiswick, England, have just completed to the order of Dr. Grattan Guinness the stern-wheel steamer Livingstone for the service of the Regions Beyond Missionary Union on the Congo. The vessel, which is the largest of its: kind built for mission- ary work, is lll ft. long over all, 19 ft. 3 in. broad, and will draw 2 ft. 6 in. of water when carrying 20 tons. Her speed when so loaded is to be 10 miles an hour, and this is guaranteed under penalties. A further guar- STERN-WHEEL STEAMER LIVINGSTONE FOR THE REGIONS BEYOND MISSIONARY UNION. i Built by John I. Thornycroft & Co., Chiswick, England. antee is given by the firm that the vessel is to steam 9 miles an hour at the same draught when burning wood fuel, as wood only can at present be obtained on the Congo for fuel. The engines will be compound, having cylinders of 17 and 28 in. diameter and a stroke of 48 in., and these will be supplied with steam from two large locomotive boilers. The vessel is handsomely fitted up. The sleeping cabins, dining saloon, bath rooms and lavatories are on the upper deck, and also a hospital. The windows to cabins are all fitted with mosquito-proof gauze panels, which can be used when the glass panels are open, the object being to reduce the risk of malarial fever which is spread by insects. Special care has been taken by the builders to reduce the amount of strutting and tying, which is fre- quently carried to such excess on stern-wheel steamers as to greatly en- cumber the deck and to make it difficult to get cargo in and out. The vessel will be taken to pieces and shipped to Matadi, from whence it will be transported to Stanley Pool by the new Congo railway. LAUNCH OF SUBMARINE TORPEDO BOAT SHARK. The submarine torpedo boat Shark was successfully launched at Lewis Nixon's ship yard, Elizabethport, N. J., last Saturday. The Shark is one of five torpedo boats building by Lewis Nixon for the United States government. The ones already launched are the Adder, Porpoise and Mocassin. The Plunger is almost ready and will be launched in the course of a few weeks. The Shark is 63 ft. 4 in. in length and 11 ft. 9 in beam. The sections of the boat are in circular formature, and the dis- placement submerged is 120 tons. She has a 160 H.P. gasoline engine for surface running and a 70 H.P. electric motor for driving her under water. The power is obtained from storage batteries carried on board. She is expected to make a speed of 8 knots an hour on the surface and 7 «nots when submerged. She is armed with five Whitehead torpedoes, which are discharged underneath the water either while the vessel is submerged or upon the surface. The only exposed part of the boat when not sub- merged is the conning tower, which is made of harveyized steel 24 in. in diameter and 4 in. thick.