li 1. >- ryV-- fyxntll UtttotJSiitg piln:ax;g BOUGHT WITH THE INCOME^ FROM THE SAGE ENDOWMENT FUND THE GIFT OF Henrg W. Sage 1891 ..A-itfJAr jj/^^/irp. X ,\A Cornell University Library VM755 .N99 ^ii!»?i?iV,S,?..,?" screw propellers and their olin 3 1924 030 903 466 ■/?; DATE DUE < z o UJ 1- 1 > a a >■ < Cornell University Library The original of tiiis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924030903466 A TREATISE SCREW PROPELLERS AND IHEIE STEAM-ENGINES. A .TREATISE SCKEW PEOP"ELLl]RS AND THEIE STEAM-ENGINES PRACTICAL EULES AND EXAMPLES HOW TO CALCULATE AND CONSTRUCT THE SAME FOR ANY DESCRIPTION OF VESSELS, ACCOMPANIED WITH A TREATISE ON BODIES IN MOTION IN FLUID, EXEMPLrFIED TOE PEOPELLEKS AND VESSELS; ALSO, A FULL DESCRIPTION OF A CALCULATING MACHINE. BY J. W. NYSTROM. PHILADELPHIA: HENRY CAREY BAIRD, No 7. HAKT'S BUILDING, SIXTH STREET ABOVE CHESTNUT. 1852. fir ^■ \i.5\%^ Entered according to Act of Congress in the year 1852, by HENRY CAREY BAIRD, in the Clerk's OflSoe of the District Court of the United States in and for the Eastern District of PennaylTania. PHILADELPHIA: T. K. AND P. 0. COLLIlfS, PKINTEKS. PEEFACE In this treatise on screw-propellers and the engines employed to drive them, the Author offers the results of several years' experience and observation respect- ing their performances. One of the objects aimed at has been to obtain formulae to follow the variations that arise in practice; which formulae are here introduced and exemplified. Theory and philosophy have been followed as far as they correspond with results; but, when they were found to differ, proper co-efl5.cients were intro- duced to make the formulge simple and practical. The difference between theory and practice may have arisen from circumstances not fully understood, and which, if entered into, might complicate the calcula- tions with no remuneration. Algebraical formulae may appear difficult to those not used to their application; but, when they are accompanied by examples, almost any one can insert numerical values and perform the computation. Algebra is becoming every day more popular, and will soon be a stranger to none who are at all inte- rested in calculations. VI PREFACE. The Author would apologize for the manner in which the work is written, which is partly owing to his not being perfect in the English language; but, «ven in his native' tongue (Swedish), possessing no merit as a writer; but he claims originality in the scientific part. The Treatise on Bodies in Motion in Fluids is not what the Author desired to make it; but, without proper experiments expressly for the subject, it could not be materially improved. However, the scientific part of the matter is a fair field for the investigator. In the description of the calculating machine, are introduced matters, with examples, not always acces- sible to persons who frequently feel the want of them. CONTENTS Introduction Suggestions for experiments on screw-propellers and their steam-engines .... The propeller ..... The steam-engine ..... Eotary motion by crank and connecting-rod Tables for the momentum of the rotary motion Length of the connecting-rod Tables for useful effect from different length of connecting-rods Centripetal propeller, description of (Nystrom's) Loss and gain of effect by straight-bladed and centripetal propellers ..... Table of formulsa .... To construct a propeller, and the arrangement of its steam- engine ..... The propeller ..... The steam-engine- ..... Table of formulae. ..... Arrangement of the steam-engine in the vessel A table for the pitch of propellers Propeller-engine patented by Loper and Nystrom . Manoeuvring the engine .... Leper's propeller and the steamer S. S. Lewis English propellers on S. S. Lewis . Centripetal propeller, construction of Carlsund's improvements of steam-engines « " " patent claims Treatise on bodies in motion in fluid Friction and cohesion in fluids Figures with curved lines .... PAGE xi 9 11 22 26 33 40 44 48 58 61 68 63 72 80 82 83 85 91 94 97 98 102 113 114 131 147 VUl CONTENTS. Table for angle of resistance Examples for vessels and propellers Tonnage of vessels (custom-house measurement) Table of results for three different vessels . Description of a calculating machine (Nystrom's) On logarithms Multiplication Division . Proportion Interest . Involution Evolution To compute algebraical formulae by machine Trigonometry .... Plane trigonometry Trigonometrical computation by the machine Spherical trigonometry Navigation Appendix .... PAGE 149 165 174 176 179 182 187 189 191 192 193 194 195 197 200 208 210 218 227 TABLES. Single and opposite cranks Different lengths of connecting-rods Table of equations Depth of bodies in fluid . Proportion of friction and resistant areas Angle of resistance Results from different vessels Natural sin. cos. tan. sec. Table from the mariner's compass . 32—39 44—47 61 80 121 143 149 176 202—207 223 PLATES. Plates I., Plate V. Plates VI. Plate VIII. IX. Plates X. Plate XII. XIII. XIV. XV. XVI. XVII. XVIII. XIX. II., III., and IV. For rotary motion by crank and connecting-rod Centrifugal force and VII. Centripetal propellers Section of a vessel . Pitch table . and XI. Steam-engines Slide valve motion . Air-pump and valves Loper propeller English propeller on S. S. Lewis Centripetal propeller Carlsund's engines XXI. " " XXII. « " XXIII. Bodies in motion in fluid XXIV. « " XXV. " " XXVI. " " XXVII. Friction and cohesion XXVIII. Resistance to figures XXIX. " " XXX. « " XXXI. " " XXXII. Calculating machine 32 50 58 82 84 88 88 88 96 96 100 104 104 104 104 112 113 116 124 125 130 132 148 152 160 176 180 ERRATA. Page 56, line 1, 66, " 7, 76, Mnes 9 aoad 10. 80, line 29, 81, for "Strengthen," rcai/ Straighten. 131, 132, 135, 136, 136, 166, 201, 17, 24, 14, 10, 14, "be L in,'' "18," "18," "inTerse," "a," " a. sjn.f," "a," he in. 0.55. 0.55. immerse. a. a sin.if). a. a. 'c. = 6°.55.6," " = 6° 38.4'. 15, " "bsin.B," " c sin.B. On page 27, the pressure^ means the force in the direction of the conneet- iag-rod ; or the pressure on the steam piston miHtiplied hy sec.w. This makes a slight difference in the momentum and friction. INTEODUCTION. Peopellee. — By the term propeller, we mean an instrument which is used in navigation for the pro- pelling of vessels. It has a centre-line from which a number of blades extend to a circumference, in which the aforesaid line is the centre. The propeller is generally fitted in the stern of the vessel, so that its centre-line is parallel with the length of the vessel, and wholly below the surface of the water. Peopellee-blades. — The blades have the form of a helixoidical surface of the thread of a screw, so that each blade forms a separate thread, and, therefore, bears the name of screw-fropdler. Its mode of action is such that, when placed in the stern of a vessel in water, and revolved in the same, the water within the circuit of the propeller acts as a nut for the ^rew, and the propeller screws the vessel in the direction of its centre-line. Pitch. — The 'pitcli of a screw is the space which the screw moves in its nut while turning it round once (supposing the nut to be stationary). If the water in which the propeller acts was a stationary nut, when the propeller makes one turn the- vessel should move XU INTRODTTCTIOir. a space equal to the pitch of the propeller; but, as the water is a free movable body, it will also move a space, so that the two motions of the water and vessel together will be equal to the motion of the pitch in a unit of time. Those two motions are termed advance and slip. Advance is the motion of the vessel forwards. Slip is the motion of the water backwards. The advance and slip are measured by the pitch as a unit. Suppose the pitch of a propeller was 100 inches, then, when the propeller makes one revolution, the vessel should advance 100 inches if there was no slip; but, if the vessel is found to advance only 75 inches, then the slip is 25 inches. This slip is gene- rally noted in per cent, of the pitch. A TREATISE SCEEW-PEOPELLEES AND THEIE STEAM- ENGINES. SUGGESTIONS FOR EXPERIMENTS ON SCREW-PROPELLERS AND THEIR STEAM-ENGINES. A QUESTION of considerable interest at the present time is the relative merits of screw-propellers and paddle-wheels; which of them is the most available and possesses the greatest superiority for navigation. Were we to decide from the present aspect of public opinion, we would be almost compelled to say screw- propellers are on the wane. But, perhaps, good rea- sons might be assigned for this seeming loss of popularity, without detracting from the merit of the screw-propeller. Compared with paddle-wheels, if we put on more steam or power the vessel will run faster ; but, for screw-propellers, there is a more deli- cate arrangement between the steam and the action in the water; that if you say, "Put on high steam; it must run," you will be disappointed. Then, is pre- eminence justly on the side of the paddle-wheels, with 2 10 A TREATISE ON the same amount of fuel ? As the screw-propeller is the most suitable and valuable instrument in naviga- tion, the question is worthy of attention, and can be answered onlj by a series of patient and impartial experiments, extending through several different ves- sels and propellers. Experiments, we know, have been made, again and again, but the principal feature has been too often overlooked. In these suggestions, we will demonstrate the prin- cipal features in the following questions. SCREW-PROPELLERS. 11 THE PROPELLEK. 1. PiUih. — On what does the pitch depend? If on the slip and diameter of the propeller, and of the ar- rangement of the steam-engine, what will the proper pitch be when those quantities are given ? Or what will be the difference with more or less pitch to the economy of power? English engineers take a very narrow pitch in pro- portion to the diameter — from li to li the diameter. This, we think, results from experiments and observa- tions of the following two facts : — a. That less pitch gives less slip. But, if the slip is no measure of loss of effect, the slip caused by different pitch should not govern the pitch, as the slip caused by the resistant area of the vessel to the acting are^, of the propeller when brought into action. (See further about those areas.) h. That less pitch does better in head winds. But, if any proper arrangement with more pitch could bar lance the former, it will have a greater advantage in calm and fair weather. (See further.) 2. Advantage and disadvantage with more or less pitch. — How will fair and head winds affect more or less pitch in propellers ? It is a fact that propellers 12 A TREATISE ON with more pitch and slip employ the effect better for propelling in still water or fair winds ; but, in head winds, propellers with less pitch and slip will do best, owing to the slip of the more-pitched one ; exceeds S and a portion of the power is expended in agitating the water. Then the question comes, is this formula correct? and how will it affect the coefficient 200, if the propeller is centripetal, as described in these pages? For vessels which are exposed to much head wind, a less-pitched propeller is preferable ; but if a well-pro- portioned centripetal propeller with more pitch could balance the less-pitched one, in those circumstances it should be preferred, because, when the vessel has a fair wind, and the sails are set, the less-pitched pro- peller is often of no use for propelling,. and sometimes has what is called a negative slip, while a more-pitched one would still act to propel the vessel with great effect. From this we come to the conclusion, that a propeller in the Pacific Ocean should have more pitch than one in the Atlantic Ocean; because in the latter there is greater liability to storms than in the former. Propellers running between San Francisco and China should be more pitched. Inventions have been patented, both in America and Europe, to change the pitch of propellers, while in the water, by some arrangement of cog-wheels at the hub to twist the blades at pleasure. SCREW-PROPELLERS. 13 3. Slip. — On what does the slip depend? If on the diameter and pitch of the propeller, brought into action with the greatest immerse section area, form, and friction area of the displacement, what will the slip be when those quantities are given ? 4. Loss of effect hy slip. — Is the slip a correct mea- sure of loss of effect or quality of the propeller, as thus far has been a generally-received opinion ? We think experiments will prove that more slip will be more economy of power until a certain limit, depending upon the proper arrangement, and that the slip is merely a measure of the power which propels the vessel. 5. Acting area of the propeller. — On what does the acting area of the propeller depend ? If on the pitch and diameter, what will it be when those quantities are given ? The acting area means a plane at right angle to the centre line of motion; and having the same velocity as the slip, will sustain a resistance equal to that of the propeller, in the direction of motion parallel to its centre line, at the given immersion. 6. Resistant area of the vessel. — On what does the resistant area of the vessel depend ? If on the greatest immerse section area, form, and friction area of the displacement, what will the resistant area be when those quantities are given? The resistant area of a vessel means a plane at right angle to the direction of motion, and, having the same velocity as the vessel, will sustain a resistance equal to that of the vessel. 14 A TREATISE ON 7. Yehoity and resistance to the acting and resistant areas. — Is the resistant to those areas in proportion as the square of their velocity? If so, or whatever the proportion may be, when brought into action it de- termines the slip of the propeller and velocity of the vessel. It is evident that the same propeller must have the same acting area, independent of its velocity, and the same must be the case with the resistant area of ves- sels ; but it is found that, for propellers, this resistant does not increase as the square of its velocity, and it is probable that it will come nearer that proportion as the pitch is less in proportion to diameter. The resistant area of the vessel contains the two quantities, greatest immerse section area multiplied by the sine for its angle of resistance ; and the friction area. The resistance to the former is, as the square of its velo- city/, but to the latter more direct as the velocity. There- fore, the total resistance will be nearer the square of its velocity, as the friction area is less in proportion to the greatest immerse section area. 8. Velocity of the propeller. — On what does the pro- per velocity of a propeller depend ? If on the pitch, slip, and diameter of the same, what will the proper velocity be when those quantities are given ? In regard to the proper velocity, it is known from experience, especially from canal and tow-boats, that, when the velocity of the propeller increases, the speed of the vessel will also increase until a certain limit ; SCREW-PROPELLERS. 15 when exceeding that, the speed of the vessel will not increase, which may be for the following reasons : — a. When the propeller exceeds a certain velocity, the hydrostatic pressure of the water is not sufficient to supply solid water into the circuit of the propeller. This can be partly overcome by giving the propeller an expanding pitch in two directions, so that it ex- pand from ihefore to the after edge; and, in the same proportion, expand from the centre to the periphery. The former expansion is measured by the angles v v', and the latter by the angles w° w' w'. (See Plates V. and VII.) b. The centrifugal force of the water acts to agitate itself, in proportion as the square of the velocity of the propeller. This has led to the suggestion of pro- pellers with curved blades, so proportioned to the centrifugal force that, at any distance from the centre within the propeller, the water will obtain an helix- oidical motion backwards, parallel to the centre line of the propeller. It is not theory that has led to this idea, but actual practice, by observation of the instru- ment's operation in the water, of which there has been a very good opportunity of judging. The re- sults of the observations have been worked out by theory, as wiU be seen in these pages. 9. Centripetal propeller. — Is there any advantage in the blades being curved? And, if so, what does the curve depend upon? If on the pitch, slip, and velocity, what will be the curvature when those quan- tities are given? 16 A TREATISE ON It is evident that the curve should be an arithmetic spiral ; but it is probable that the pitch of this spiral should be independent of the slip or velocity of the propeller for the following reason : — When the generatrix for a screw is an arithmetic spiral drawn on a plane at right angle to its axis, it is the same as if the generatrix for the screw was a straight line with an inclination = U to the axis. This angle ?7will be found by the formulas tang. U— —^^, .... (1) in which the letters denote P = pitch of the propeller, and D = diameter, w° = the angle io° in degrees. (See Plate VI. & VII.) When this inclination U= 45°, the centrifugal force acts with its greatest advantage to propel the vessel, independent of the slip or velocity of the pro- peller. Then tang. U= tang. 45° = 1, and '^ =— ^ (2) A propeller constructed on this principle will be seen hereafter, with its calculations. 10. Mcjpandmg pitch. — Is there any advantage in the expanding pitch, and on what does the expansion depend ? If on the slip, what will it be when the slip is given ? The expanding pitch changes the inclined generatrix to a curved line, drawn on the same plane SCREW-PEOPELLEES. 17 as the axis for the screw, and momentarily changes its form. 11. Length of the propeller. — On what does the length of the propeller depend ? If on the number of blades, and proportions of pitch, diameter, and slip, what will be the proper length when those quantities are given ? The length is of some importance for the strength and room in the stern of the vessel. 12. Nv/mber of hlades. — On what do the number of blades depend? If particularly on the slip, and proportions of pitch and diameter, what will be the proper number of blades when those quantities are given? It is very well known that propellers with two or three blades will do as well, or better than those with four or five blades, but in those comparisons the quantities on which the number of blades depend are often neglected. We do not mean to say that a more- bladed propeller will do better because there are more blades; in that case, there is no hope (except when the slip is very slight) ; but, if we can make it do as well as the less-bladed, we will gain the following advantages : — a. The more-bladed propeller occupies less space in the stern of the vessel, because it is shorter than the less-bladed, if well proportioned ; and the vessel can be made neater and stronger in the stern, especially when made of wood. h. The more-bladed propeller is stronger and safer, 18 A TEEATISE ON" and does not shake the vessel as much as the less- bladed, which increases the comfort of the same, and makes it more durable and safe. c. When the vessel is running in a hard sea, the waves will lift up the stern of the same ; so that often more than one-half of the propeller will be above the water; then, if it is a two-bladed propeller, it can hap- pen that both the blades come over the water, which causes a violent shock in the steam-engine, and in danger of losing the propeller-blades. This has caused Englishmen to apply governors to the engines where the two-bladed propeller is used ; but even that will not fully answer the purpose. If the propeller has three or four blades, there must be at least one blade in the water, but then the whole power from the steam-en- gine is acting upon that blade, which then stands in danger of breaking, and, if it breaks, it is probable that the others will follow. When- the propeller has five blades, there must be at least two blades in the water; but those blades are smaller in proportion to the three-bladed one, which still more increases the safety of the five-bladed one, and, if supported by two narrow bands (as shown in Plate VII.), when only two blades act in the water, the resistance will still act on the other three blades through the bands, and render the five-bladed propeller perfectly safe. In the above statement of number of blades, there must be an exception when the propeller has a very narrow pitch in proportion to the diameter— as the SCREV-PROPELLEBS. 19 Englisli propellers ; but such a proportion cannot be adapted to direct-action steam-engines. d. When the vessel is to run by sail, and the pro- peller is coupled off from the steam-engine, and runs by the resistance of the water caused by the speed of the vessel, this resistance is greater to a propeller of less pitch and number of blades, and has caused another arrangement in the stern of the vessel to hoist up the propeller when the vessel is to run by sail. This complicated mechanism increases the ex- pense, and spoils the stern of the vessel, and is, after all, nothing to depend upon. If the coupling can be made strong enough, it will be clumsy in its place ; and, in case of war, the vessel might come in an unsafe, turbid, and shallow water, where some obstacle would fasten in the coupling and prevent the proper manoeu- vring. In a propeller with more pitch and blades, and a narrow length, the resistance will be so trifling that the propeller can, without detriment, remain in its proper place. The disadvantage with the more-bladed propeller is when first starting the vessel, as it does not give so quick starting as the less-bladed, owing to the slip, but, when started to its full speed, there will be no difference in their performance, if properly arranged. 13. Resistance of the prapeller, — What will be the resistance of the propeller, when the vessel is running by sail and no steam, under the following circum- stances ? 20 A TREATISE ON" a. The propeller standing stationary in its place to the vessel. h. Coupled loose from the steam-engine, and can revolve freely. c. What will be the difference in the resistance when the propeller has more or less pitch and num- ber of blades, remains in its place, and hoists up ? d. "Will the resistance be less when the propeller is centripetal? which we think it will be, owing to a disadvantage by such a propeller for backing when worked by steam. 14. Regular screw. — Shall the propeller be a regu- lar screw ? This question will certainly be answered, first by mathematicians, that it shall he a regular screw. It is evident that, if there was no slip, the propeller should evidently be a regular screw, but, as such an instance never can exist in water, the pro- peller shall accordingly never he a regular screw. If the irregularity depends on the slip, what will it be when the slip is given? The question may be asked, if the propeller can be a screw at all if it is not regular ? In the answer to that, we have three different propellers, viz. : — a. Regular screw, is a propeller which has a uniform pitch on all its helixoidical surface. h. Irregular screw, is the propeller which has an expanding pitch, in one or two directions, so that the generatrix for the screw runs through the centre of the propeller. SCREW-PROPELLERS. 21 c. No screw at all, is the propeller where the pro- peller-blades form an angle with the centre line in the centre of the propeller, and the generatrix for each propeller-blade has a centre line not common to the propeller or to themselves. The irregular screw is the one which should be adopted for propelling in fluids. Frenchmen have come to the result, from experiments, that the more the propeller-blades are cut out at the hub, the more effective is the propeller. In those experiments, the propellers have certainly been regular screws. If a propeller has an expanding pitch, in the direction of the radix, proportioned to the slip, the cutting off the blades at the hub would show no difference. 15. What power is required to give propellers a certain number of revolutions per minute, when the diameter, pitch, and slip are given ? In what propor- tion does the power differ with the pitch and slip ? 22 A TREATISE ON THE STEAM-ENGINE. The steam-engine is one of the most valuable agents in navigation; and, as we thus far have no other pro- pelling agent in view, which can substitute the steam, there is perhaps both time and room to improve the arrangement of the steam-engine, particularly for screw-propellers. "We have for years had high ex- pectations of the electro-magnetism as a motive power, but have hardly noticed our disappointment. If it comes slowly, we hope it will be sv/re, and we do not relinquish the hope that yet, in our day, it wiU not always he steam navigation. But we steam-en- gineers will go a-head with the steam-engine, and wish the electro-magnetic engineers a soon and good success. In regard to the propeller-engines, the first point which requires attention is the arrangement of the air-pumps and their valves. To obtain a simple and compact direct-action pro- peller-engine, the air-pump ought to be applied direct to the cross-head or steam-piston; but then it is found that the air-pump works too fast, therefore, the air- pump is geared to run slower, or, perhaps, rather gear the propeller, and a complicated and heavy machinery will be the result. When the engine is direct action, the air-pump is SCREW-PEOPELLEES. 23 often worked by a walking-beam, with a reduced stroke of the steg,m^iston, and thereby obtain a slower motion of the same. If we know the law which governs it, perhaps we can work the air-pump at any reasonable speed. It is evident that the vacuum, or, more correctly, the pressure in the condenser, is what governs the proper speed of the air-pump, and by those two the area of the air-pump valves must be proportioned. To each pound of pressure per square inch answers a column of water about 27 inches; then, when the vacuum in the condenser is 10 pounds, the pressure in the same will be about 4i pounds, which answers to a column of water 4i x 27 = 121 inches, which is the space from w:hich the velocity of the water through the valves is to be ascertained (deducting the friction and slip through the valves, air, &c., about 40 or 60 per cent.), 121 inches is about 10 feet, fallen through by a body will obtain a velocity of 25 feet, deducting 50 per cent., will be about 12i feet per second, the velocity of the water through the valves. Then the velocity of the air-pump piston is to 12J as the area of the valves is to the area of the air-pump. Call the area of the air-pump = 2, and the area of the valve = 1, the proper velocity of the air-pump piston will be 6i feet per second. Capacity of the air-pump, and area of its valves. — What will be the proper capacity of the air-pump, when the capacity of the steam-engine, and density 24 A TREATISE ON of the exhausted steam are given, to obtain a certain vacuum ? And what will be the proper area of its valves, when the velocity of the air-pump piston and vacuum in the condenser are given ? Rules have been laid down, but they are incomplete for direct- action propeller-engines, where the air-pumps are at- tached direct to the cross-head, and, to make such rules simple and applicable for any circumstance, they should be accompanied by tables as follows : — For the valves, a table should be constructed so that the vacuum on the top of the columns in pounds per square inch, and the velocity of the air-pump pis- ton in feet per second in the first left column, and where they meet, should be the coefficient for multi- plying the area of the air-pump piston to obtain the area of the valves. For the capacity of the air-pump, the vacuum should be marked in pounds per square inch on the top of the columns, and, in the first left column, the density of the exhaust steam in pounds per square inch ; in the columns where these meet, should be the coefficient for multiplying the capacity of the steam- cylinder to obtain the capacity of the air-pump (when fixed a mean temperature for the injection-water). Sometimes it is used from 40 to 60 pounds of steam, and cut off the steam first at i or fths the stroke; then the exhaust steam will be from 20 to 30 pounds, excluding the atmosphere. For another arrange- ment, the exhaust steam being only 6 or 8 pounds. SCREW-PROPELLERS. 25 This, of course, makes a considerable difference in the arrangement of the air-pump and its valves, and, when attached direct to the crosshead, it is of great importance that proper attention is paid to it. An ingenious engineer invents some new and sim- ple arrangement of engines, and knows he can apply the air-pump direct to the crosshead, but can obtain only a certain proportion of areas of the valves and air-pump piston, and the stroke and velocity are given : then it is of no use making the air-pump larger than is necessary to work well. By reference to the rules before spoken of, it can easily be regulated to suit the arrangement. These rules could easily be calculated, but, to place confidence in them, it is necessary to lay them down only from experiments, and, in those experiments^ there must be proper arrangements in order to make the rules applicable for any circumstances; which, for propeller-engines, are the most variable. Cut off the steam. — In order to save steam, or, more correctly, to employ the effect of steam to a higher degree, it is common to shut off the admittance of steam to the steam cylinder when the piston has moved a part of the stroke. From that cut-off point, the steam acts expansively with a decreased pressure on the piston, and causes an irregularity in the mo- mentum, by transferring the straight linear motion to the rotary. This irregularity is of no consequence where there are some heavy revolving pieces or fly-* 3 26 A TREATISE ON wheel to regulate the motion; but for screw-propellers it is worthy of attention, and can be regulated by giving a little more or less steam on one side of the piston, particularly where two engines are working at one or opposite cranks, which are the engines we will allude to in this work. Rotary Motion, hy Crank and Gonnecting-Rod. When the straight linear motion is to be transferred to the rotary by crank and connecting-rod, there ex- ists some irregularity in the momentum for the rotary motion. Momentum is a force multiplied by the length of the lever on which it acts : see Fig. 1. The line I represents the connecting-rod from a steam-engine, attached to the crank r at d, which revolves around the centre C. p is the force from the steam-engine, acting on the lever a, which is drawn from the centre O, at right angles to the connecting-rod I. The mo- mentum for that force will be expressed as Momentum = pa. Let ,5' denote the distance which the piston has tra- velled when the crank stands in the angle v, and ex- pressed in a fraction of the stroke. r = 1, the length of the crank. 6 = distance from the centre of the rotary motion to the centre of the cross-head. Z = 4 r, the connecting-rod twice the stroke we have and SCEEW-PEOPELLEES. 27 S='-±Jp^, . . . . (1) 6 = i4!^, (2) sin.v r sin.v /o\ sm.w = , . . . . (o) I sin.x sin.w /,, a = -. ... (4) sm.v a == h sin.w (5) x + w + v = lSO°. ... (6) Suppose we divide the circle of motion into 24 equal parts, and draw the radius. From the above formulae calculate the values of a, b, and ;i^. In each of those 24 parts, set off the corresponding value of a from the centre C, and join them with a curved line m n, as represented in Fig. 2. The values of a, h, and S, corresponding to their angles v, w, and x, are set up in the Table I. In any position of the crank, the dis- tance from the centre to where the crank crosses the curved line m n, represents the momentum for the rotary motion, when the force p=\. When the crank stands in the position 3, the mo- mentum for the rotary motion is a, but in the position 9, the momentum is G e. If the connecting-rod was infinite, the curved lines m, n would be two circles, with a diameter equal to the radius of the crank, but the shorter the connecting-rod is, the more it varies from a circle ; when the pressure p is constant through- 28 A TREATISE ON out the whole stroke, but when the steam is shut off at a part of the stroke, the curved line will be a more irregular one; and by calculating the decreased pres- sure on the piston in each 24 positions of the crank, multiplied by the corresponding value of a, is the mo- mentum for the rotary motion. If the steam is shut off" at J of the stroke, it will be a curved line, G, n, o, d, Fig. 2, which is a clear view of the irregu- larity. The two inner circles represent the iTiean mo- mentum for full steam, and cut off" J of the stroke, which is as 620 : 434, see column a, and op. Table II. The column n shows the number of divisions in which the crank stands, and v, the angle of the crank to the centre line of the steam-engine. See Fig, 1 for the angles w and x. When two engines are working at right angles on one common crank, the momentum for the rotary mo- tion will be equal to the sum of the momentum from both the engines, and so divided, that when the mo- mentum of one engine is 0, it is near its maximum of the other engine, and there exists no dead point. Table and Fig. 3. — Set two columns of a together, so that the 6th division in one corresponds with in the other, as seen in the table : the 6th division is 0.970, which, in the column a', corresponds with in the column a. Add the two columns a and a' together for each division, place the sum in the column a + a', which then will represent the momentum from both the engines. SCREW-PROPELLERS, 29 Fig. 3. Draw the two lines G b and C h' at right angles with each other, which then will represent the centre lines of the two steam-engines; proceed, as in Fig. 2, by dividing the circle of motion into 24 equal parts, and start from o at b, in the direction of the arrow i. On each division, set off the correspond- ing value a + a', join them as described in Fig. 2 ; it will be a curved line m, n, m, n. If the connecting- rods were infinite, the curved line would have been four half circles, with a radius = 0.707 times the radius of the crank. In 3 and 15, the curves are nearly half circles, but in 9 and 21, it differs more, owing to the connecting-rod being shorter than infinite, or in this figure only twice the stroke. In the half circle, 15, 21, 3, the momentum is greater than in the other half circle, 3, 9, 15. By observing such a steam-engine in motion, it will be found that, when the crank passes the point o, it runs fastest, and when passing 12, it runs slowest. If there is any additional work applied to the steam-engine, as, for instance, a single acting air-pump, it should be so attached to the engine, that when the crank stands in 21, the air-pump piston should stand on half the stroke when going up. When the steam is cut off at 4 the stroke, the mo- mentum line will be represented by the curve o, o, o, o, a more irregular one. The momentum is greatest and more regular in the half circle, 9, 15, 21; less and more irregular in the half circle, 21, 3, 9, which shows that a little more steam should be admitted on the top 30 A TREATISE ON of the pistons, as to cut oflf the steam a f of the stroke. The drawn circles represent the mean momentums, and the dotted one, the centre circle of the crank. On page 75 is described an opposite crank. Fig. 9. If we, in this same position of the steam-engine, apply the opposite crank, the difference from Table III. will be, that the 18th division in the column a will corre- spond with 0. (See Table IV. and Fig. 4.) The mo- mentum for the rotary motion will still be a + a, but the friction in the bearings will be j+ a "+ a, repre- sented in the column. The momentum line, mn,mn, will be precisely the same as in Fig. 3, but it has another position to the steam-engine, and the maxi- mum momentum is at 3 ; that is, the crank which is attached to the engine h, stands in the third division from 0, but the crank which is attached at h' stands in 15. In the four points where the momentum is greatest, the friction is o. In the bottom . of the columns a + a and 4^ a 4^ a, we have the mean mo- mentum for rotation, which is to the mean momentum for friction as 1.24 : 0.527. When. two engines are working at right angles on a single crank, the mean momentum for friction will be 2 sin. J 90° = 1.414; then we have the friction in a single crank is to the friction in a double crank as 1.414 : 0.527. When the steam is cut off at 4 of the stroke, the momentum line for rotation will be o, o, o, o. For the friction, the momentum line will be /, /, /, /. The arrows in the friction line show the direction in which the shaft SCREW-PROPELLERS. 31 presses in the bearing. The small circle shows the mean momentum for the friction. Fig. 5 shows the momentum lines from two steam- engines working at 120° on one single crank. When working with full steam, the mean momentum for friction will be 2 sin. i 120° = 1.732. Fig. 6 shows the momentum lines from two engines working at 120° on opposite cranks. In the bottom of the column +^a^ a,vfe have the mean momentum for the friction = 0.464. Then, when two engines are working at 120°, the friction by the single crank is to the friction by the opposite crank as 1.732 : 0.464. Fig. 7 shows the momentum line from two engines working at 150° on one crank. The momentum for the friction will be 2 sin. i 150° = 1.9318. Fig. 8. Two engines working at 150° on opposite cranks. The friction in the column 4^ a +" a = 0.285, then, when the angle is 150°, the friction in the single crank is to the friction in the opposite crank as 1.932 : 0.285. The loss of effect by friction with shafts of wrought- iron, in bearings of bronze, will be, in horse-power, about TT D n p* ~ 233000 in which D = diameter of the shaft in inches, n = number of revolutions per minute, j> = total pressure in the bearings. * Morin's Experiments. 32 A TREATISE ON In the steam-engine on Plate VIII., is calculated the values of _p = 61,575 pounds pressure on the piston, D = 12 inches, diameter of the shaft, re = 48.5 revolutions per minute, / = friction coeflGcient, which, for this purpose, will be found in the friction columns in the accompanying tables. Let us, from these values of p, n, D, and /, calculate the difference in effect caused by different arrange- ments of cranks and angles in Figs. 3, 4, 5, 6, 7, and 8, D^j)/_ 12 X 48.5 X 61575 x/ ,., . ^ 233000 ~ 233000 ~ *'■ Then we have the loss of effect, by multiplying the coefficient for friction by 154, which will be for Fig. 3, 154x1.414 == 218 horses. The results of all these calculations are collected and compared in the accom- panying table. Nature of crank. Angle of Loss of Useful Friction Speed of engine. effect. effect. /• the vessel. Single . . 90° 218 570 1.414 9.90 Opposite . 90 81 709 0.527 11.25 Single . . 120 267 523 1.732 9.37 Opposite . 120 71.5 718.5 0.464 11.30 Single . . 150 296 494 1.932 9.6 Opposite . 150 43.9 746 0.285 11.55 This shows a difference of 2 miles per hour gained by the opposite crank. Plate I. PlaJbK n. Plate m. PlateW. f SCREW-PROPELLERS. 33 Tables I. & II. One Engine working on one Granh. (Figs. 1 & 2.) Position of crank and connecting-rod. 3 team full-stroke. Cut off at |. n V w X b 8 a P qp 0° 00' 180° 00' 6.000 0.000 0.000 1.000 0.000 1 15 3 43 161 17 4.974 0.015 0.321 1.000 0.321 2 30 7 11 142 49 4.830 0.085 0.604 1.000 0.604 3 45 10 11 124 49 4.680 0.160 0.827 1.000 0.827 4 60 12 30 107 30 4.400 0.300 0.955 1.000 0.955 5 75 13 58 91 2 4.135 0.432 0.998 0.771 0.769 6 90 14 29' 75 31 3.875 0.562 0.970 0.587 0.575 7 105 13 58 61 2 3.610 0.695 0.875 0.483 0.420 8 120 12 30 47 30 3.400 0;800 0.738 0.416 0.307 9 135 10 11 34 49 3.220 0.890 0.570 0.375 0.214 10 150 7 11 22 49 3.100 0.950 0.388 0.355 0.136 11 165 3 43 11 17 3.025 0.987 0.196 0.333 0.066 12 180 00 00 00 3.000 0.000 0.000 1.000 0.000 13 195 3 43 11 17 3.025 0.013 0.196 1.000 0.196 14 210 7 11 22 49 3.100 0.050 0.388 1.000 0.388 15 225 10 11 34 49 3.220 0.110 0.570 1.000 0.570 16 240 12 30 47 30 3.400 0.200 0.738 1.000 0.738 17 255 13 58 61 2 3.610 0.305 0.875 1.000 0.875 18 270 14 29 75 31 3.875 0.433 0.970 0.770 0.749 19 285 13 58 91 2 4.135 0.562 0.998 0.587 0.586 20 300 12 30 107 30 4.400 0.700 0.955 0.476 0.455 21 315 10 11 124 49 4.680 0.840 0.827 0.398 0.328 22 330 7 11 142 49 4.830 0.915 0.604- 0.364 0.219 23 345 3 43 161 17 4.970 0.985 0.321 0.338 0.109 24 360 00 180 00 5.000 1.000 0.000 0.333 0.000 Meatt momentum = *0.620 0.669 0.434 34 A TREATISE ON Table III. Two Engines worMng at 90° on one Granh. (Fig. 3.) n V Steam full-stroke. Cut off at \. a a> a+a'. Friction. pa pa' p{a+a') Friction. 0.000 0.970 0.970 0.000 0.576 0.575 1 15 0.321 0,875 1.196 0.321 0.420 0.741 2 30 0.604 0.738 1.842 0.604 0.307 0.911 3 45 0.827 0.570 1.397 0.827 0.214 1.041 4 60 0.955 0.388 1.343 0.955 0.136 -1.091 •^ 5 75 0.998 0.196 1.194 0.769 0.066 0.835 ir- es 6 90 0.970 0.000 0.970 0.575 0.000 0.575 O 7 105 0.875 0.196 1.071 ^ 0.420 0.196 0.616 II 8 120 0.738 0.388 1.126 II 0.307 0.388 0.695 "2 9 135 0.570 0.570 1.140 0.214 0.570 0.784 CD 10 150 0.388 0.738 1.126 o O 0.136 0.738 0.874 X 11 165 0.196 0.875 1.071 Hc4 0.066 0.875 0.941 o 12 180 0.000 0.970 0.970 0.000 0.749 0.749 o 13 195 0.196 0.998 1.194 03 0.196 0.526 0.722 -*i 14 210 0.388 0.955 1.343 CM 0.388 0.455 0.843 .| 15 225 0.570 0.827 1.397 II 0.570 0.328 0.898 to 16 240 0.738 0.604 1.342 g 0.738 0.219 0.957 1! 17 255 0.875 0.321 0.196 '43 0.875 0.109 0.984 18 270 0.970 0.000 0.970 o '^ 0.749 0.000 0.749 g 19 285 0.998 0.321 1.319 fa 0.586 0.321 0.904 .2 20 300 0.955 0.604 1.559 0.455 0.604 1.059 fa 21 315 0.827 0.827 1.654 0.328 0.827 1.155 22 130 0.604 0.955 1.559 0.219 0.955 1.174 23 345 0.321 0.998 1.319 0.109 0.769 0.878 24 360 0.000 0.970 0.970 1.414 0.000 0.575 0.575 Mean momentum = 1.240 0.868 0.974 SCREW-PROPELLERS . 35 Table IV. Two Engines worTcing at 90° on opposite Cranks. (Fig. 4.) n V Steam full-stroke. Friction. Cut off at |. Frictibn. a a' a-\-a' +a+ffl' pa pa' p{a+a') p{±a+af) 0.000 0.970 0.970 0.970 0.000 0.749 0.749 0.749 1 15 0.321 0.998 1.319 0.677 0.321 0.586 0.907 0.265 2 30 0.604 0.955 1.559 0.351 0.604 0.455 1.059 0.149 3 45 0.827 0.827 1.654 0.000 0.827 0.328 1.155 0.499 4 60 0.955 0.604 1.559 0.351 0.955 0.219 1.174 0.736 5 75 0.998 0.321 1.319 0.677 0.769 0.109 0.878 0.660 6 90 0.970 0.000 0.970 0.970 0.575 0.000 0.575 0.575 H 1 105 0.875 0.321 1.196 0.554 0.420 0.321 0.741 0.099 8 120 0.738 0.604 1.342 0.134 0.307 0.604 0.911 0.297 9 135 0.570 0.827 1.397 0.257 0.214 0.827 1.041 0.613 10 150 0.388 0.955 1.343 0.567 0.136 0.955 1.091 0.819 11 165 0.196 0.998 1.194 0.802 0.066 0.769 0.835 0.709 12 180 0.000 0.970 0.970 0.970 0.000 0.575 0.575 0.575 13 195 0.196 0.875 1.071 0.679 0.196 0.420 0.616 0.224 14 210 0.388 0.738 1.126 0.350 0.388 0.307 0.695 0.081 15 225 0.570 0.570 1.140 0.000 0.570 0.214 0.784 0.356 16 240 0.738 0.388 1.126 0.350 0.738 0.136 0.874 0.602 17 255 0.875 0.196 1.071 0.679 0.875 0.066 0.741 0.809 18 270 0.970 0.000 1.970 0.970 0.749 0.000 0.749 . 0.749 19 285 0.998 0.196 1.194 0.802 0.586 0.196 0.782 0.390 20 300 0.955 0.388 1.343 0.567 0.455 0.388 0.843 0.067 21 315 0.827 0.570 1.397 0.257 0.328 0.578 0.898 0.242 22 330 0.604 0.738 1.342 0.134- 0.219 0.738 0.957 0.519 23 .345 0.321 0.870 1.191 0.554 0.109 J.875 0.984 0.766 24 360 0.000 0.970 0.970 0.970 0.000 0.749 0.749 0.749 Mean momentum = 1.240 0.527 0.868 0.454 36 A TREATISE ON Table V. Two Engines working at 120° on one Crank. (Fig. 5.) n 1 2 8 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 V 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 Steam fnU-stroke. Cut off at \. a a' a+a' FrictioiL pa pa^ p{a-\-a') Friction. 0.000 0.321 0.604 0.827 0.955 0.998 0.970 0.875 0.738 0.570 0.388 0.196 0.000 0.196 0.388 0.570 0.738 0.875 0.970 0.998 0.955 0.827 0.604 0.321 0.000 0.738 0.576 0.888 0.196 0.000 0.196 0.888 0.570 0.738 0.875 0.970 0.998 0.955 0.827 0.604 0.321 0.000 0.321 0.604 0.827 0.955 0.998 0.970 0.875 0.738 0.738 0.897 0.992 1.023 0.955 1.194 1.358 1.445 1.476 1.445 1.358 1.194 0.955 1.023 0.992 0.891 0.788 1.196 1.574 1.825 1.910 1.825 1.574 1.196 0.738 CO II o O (M I-H >*1 "1 II '■+3 o 0.000 0.321 0.604 0.827 0.955 1 0.769 |0.575 0.420 0.307 0.214 0.136 0.066 0.000 0.196 0.888 0.570 0.788 0.875 0.749 0.586 0.455 0.828 0.979 0.109 0.000 0.807 0.214 0.136 0.066 0.000 0:196 0.388 0.570 0.738 0.875 0.749 0.586 0.455 0.328 0.219 0.109 0.000 0.321 0.604 0.827 0.955 0.769 0.575 0.420 0.307 0.307 0.535 0.740 0.893 0.955 0.965 0.963 0.990 1.045 1.089 0.885 0.652 0.455 0.524 0.607 0.679 0.788 1.196 1.858 1.413 1.410 1.097 0.794 0.529 0.307 o t-i II § o X o O !M r-l •i 11 g o IV [ean momentum = 1.240 1.732 0.868 1.195 SCREW-PROPELLERS. 37 Table VI. Two Enginei worldng at 120° on opposite Cranks. (Fig. 6.) n V Steam fuU-stroke. Friction. Cut off at J. Friction." a a' a+ffl' ± 14 210 0.388 0.000 0.388 CO 0.388 0.000 0.388 s 15 225 0.570 0.321 0.891 II 0.570 0.321 0.891 S 16 240 0.738 0.604 1.842 0.738 0.604 1.342 (M 17 255 0.875 0.827 1.702 O "43 0.875 0.827 1.702 D 18 270 0.970 0.955 1.-925 2 0.749 0.955 1.704 1 19 285 0.998 0.998 1.996 S 0.586 0.769 1.355 "S 20300 0.955 0.970 1.925 0.455 0.575 1.080 ■| 21315 0.827 0.875 1.702 0.328 0.420 0.748 rH 22'330 0.604 0.738 1.342 0.219 0.307 0.526 23 345 0.321 0.570 0.891 0.109 0.214 0.323 24 360 0.000 0.388 0.388 0.000 0.136 0.186 Mean momenttim = 1.240 1.9318 0.868 1.333 SCREW-PROPELLERS. 39 Table VIII. Two Engines working at 150° on opposite Cranks. (Fig. 8.) V Steam full-stroke. Friction. Cut off at J. Friction. u a' a+a^ +aipa'' pa pa' p(a+a') P (±«+a^) 0.000 0.604 0.604 0.604 0.000 0.219 0.219 0.219 1 15 0.821 0.321 0.642 0.000 0.321 0.109 0.430 0.212 2 30 0.604 0.000 0.604 0.604 0.604 0.000 0.604 0.604 3 45 0.827 0.321 1.148 0.506 0.827 0.321 1.148 0.506 4 60 0.955 0.604 1.559 0.351 0.955 0.604 1.559 0.351 5 75 0.998 0.827 1.825 0.171 0.769 0.827 1.596 0.058 6 90 0.970 0.955 1.925 0.025 0.575 0.955 1.580 0.380 7 105 0.875 0.998 1.873 .0.123 0.420 0.769 1.189 0.849 8 12P 0.738 0.970 1.708 0.232 0.807 0.575 0.882 0.268 9 135 0.570 0.875 1.445 0.305 0.214 0.420 0.634 0.206 10 150 0.388 0.738 1.126 0.350 0.136 0.307 0.443 0.171 11 165 0.196 0.570 10.766 0.374 0.066 0.214 0.280 0.148 12 180 0.000 0.388 0.388 0.388 0.000 0.136 0.136 0.136 13 195 0.196 0.196 '0.392 0.000 0.196 0.066 0.262 0.180 14 210 0.388 0.000 ;o.388 0.388 0.388 0.000 0.388 0.388 15 225 0.570 0.196 ,0.766 0.374 0.570 0.196 0.766 0.374 16 240 0.738 0.388 1.126 0.350 0.738 0.388 1.126 0.350 17 255 0.875 0.570 1.445 0.305 0.875 0.570 1.445 0.305 18 270 0.970 0.738 1.708 0.232 0.749 0.738 1.487 0.011 19 285 0.998 0.875 1.873 0.128 0.586 0.875 1.461 0.289 20 300 0.955 0.970 1.925 0.015 0.455 0.749 1.204 0.294 21 315 0.827 0.998 1.825 0.171 0.328 0.586 0.914 0.258 22 330 0.604 0.955 1.549 0.351 0.219 0.455 0.674 0.236 23 345 0.321 0.827 1.148 0.506 0.109 0.328 0.487 0.219 24 360 0.000 0.604 0.604 0.604 0.000 0.219 0.219 0.219 Mean momentum = 1.240 0.285 0.868 0.265 40 A TREATISE ON LENGTH OF THE CONNECTING-ROD. The length of the connecting-rod in a propeller-en- gine deserves some attention. It is evident that the longer it is, the better; but the question is, how will it affect the quality of the steam-engine, and economy of power? In reference to the arrangement of the steam-engine, a short connecting-rod has the advantage of occupying less room, and thereby allowing a nar- row space for the steam-engine ; the disadvantage is, it gives a more irregular motion to the machinery, which two points are of some consequence when the engine is direct action, and works fast. To the economy of power, we make a calculation for four different lengths, 1, 2, and 3 times the stroke, and one infinite. It is evident that the loss of effect consists, first, in the mean momentum for rotation ; second, the additional friction in the guides. The former will be calculated, as before described, by the formulge. r sin.v stn.w = a = I sin.w sin.x s = I -{- r I sin.x 2 2 sin.v' In the accompanying Tables, IX., X., and XI., are the values calculated for three different connecting- rods, with their frictions in the guides. When the con- necting-rod is infinite, the momentum a = sin.v and the SCRE-W-PROPELLEES. 41 friction in the guides = o, because the angle w = o. When the connecting-rod is short, in every position of the crosshead in the guide, will be a friction = tang.w' calculated and collected, as seen in the tables. s = the space which the piston moves between each 24th division. g- = J) tang.w pressure in the guides. p = total pressure on the piston. q s == friction in each division. The sum of those frictions, multiplied by 0.06 = coeflScient for frictions in guides, will be the true fric- tion in a fraction, of the total effect. In the column where the connecting-rod I = oo infinite, the mean momentum for rotation is 0.6325 (proper number = 0,63694). When the connecting- rod is equal to the stroke, the mean momentum = 0.577. z = momentum from the short connecting-rod com- pared with the infinite one. 0.6325:0.577 = l:h connecting-rods of equal the stroke ^z = ^'-g^r = 0.911 twice three times " 'z = infinite " °°z = 0.620 0.6325 0.626 0.6325 0.6325 0.6325 = 0.981 = 0.990 'useful effect omitting friction. 1.00 42 A TREATISE ON Now we will collect the results from the Tables IX., X., and XI., to Table XII., in which we will find that the useful effect by an infinite connecting-rod, is to the useful effect of a short one, nearly as 1 is to cosin. for the greatest angle of the connecting-rod to the centre line of the engine. Useful effect = We see here that the difference between useful effect by connecting-rods 2 and 3 times the stroke, is only 0.0179 ; consequently, when the connecting-rod can be twice the stroke, there will be no remuneration in the effort to make it longer if the space is narrow to place it in. To thoroughly test the above treatment on screw- propellers and their steam-engines, it would require three different vessels ; one very sharp, and one with common proportion, and the third one very full. If circumstances would allow the vessels to have about the same draught of water, so that the same diameter of propellers could be used on them, it would save num- ber of propellers which would require at least six dif- ferent kinds. Experiments on smaller boats could be easily tried. From such experiments should be ob- tained results accompanied with rules, so that for any disruption of vessels, or arrangement of engines, the rules would give a corresponding propeller. After the experiments, the propellers need not be SCREW-PEOPELLERS. 43 throwii away as useless ; they would all be good for their corresponding arrangement. Money or time should not be spent in making the propeller-blades of any pecuHar shape on their extremities or edges, like a tail of a fish, &c. The steam-engines would suffice with the three, one in each vessel. The two with direct-action steam- engines, and the other with gearing, and, if convenient, to change the gearing one to different gears. 44 A TREATISE ON Table IX. Connecting-rod equal to the Stroke. Position of cranV and connecting-rod. Friction in the guide. n V w a s s tang.v>=g qs 00 0° 00' 0.000 0.000 0.000 0.000 0.00000 1 15 7 25 0.380 0.026 0.026 0.130 0.00338 2 30 14 30 0.702 0.095 0.069 0.259 0.01790 3 45 20 40 0.925 0.188 0.093 0.377 0.03512 4 60 25 40 0.998 0.345 0.157 0.480 0.07550 5 75 28 50 0.974 0.489 0.144 0.550 0.07940 6 90 30 00 0.868 0.632 0.133 0.577 0.07700 7 105 28 50 0.718 0.753 0.121 0.550 0.06670 8 120 25 40 0.562 0.850 0.097 '0.480 0.04660 9 135 20 40 0.413 0.913 0.063 0.377 0.02380 10 150 14 30 0.267 0.966 0.053 0.259 0.01375 11 165 7 25 0.149 0.995 0.029 0.130 0.00378 12 180 00 0.000 0.000 0.005 0.000 0.00000 13 195 7 25 0.149 0.005 0.005 0.130 0.00065 14 210 14 30 0.267 0.034 0.029 0.259 0.00752 15 225 20 40 0.413 0.087 0.053 0.377 0.02000 16 240 25 40 0.562 0.150 0.063 0.480 0.03030 17 255 28 50 0.718 0.247 0.097 0.550 0.05345 18 270 30 00 0.868 0.368 0.121 0.577 0.07000 19 285 28 50 0.974 0.511 0.143 0.550 0.07885 20 300 25 40 0.998 0.655 0.144 0.488 0.07040 21 315 20 40 0.925 0.818 0.157 0.377 0.05930 22 330 14 30 0.702 0.905 0.093 0.259 0.02411 23 345 ' 7 25 0.380 0.974 0.069 0.130 0.00898 24 360 00 0.000 0.000 0.026 0.000 0.00000 Mean momentum = 0.577 0.86649 SCKEW-PEOPELLEES. 45 Table X. Connecting-rod twice the stroke. Position of crank & connecting-rod. Steam full-stroke. Friction in the guide. 1= » V w a 8 s tang.w=q qs siri'Vzz^a 0° 00' 0.000 0.000 0.000 0.0000 0.0000 0.000 1 15 3 43 0.321 0.015 0.015 0.0649 0.000976 0.259 2 30 7 11 0.604 0.085 0.070 0.1260 0.0084 0.500 3 45 10 11 0.827 0.160 0.075 0.1796 0.0135 0.707 4 60 12 30 0.955 0.300 0.140 0.2216 0.0311 0.866 5 75 13 58 0.998 0.432 0.132 0.2487 0.0329 0.966 6 90 14 29 0.970 0.562 0.130 0.2583 0.0350 1.000 7 105 13 58 0.875 0.695 0.128 0.2487 0.03185 0.966 8 120 12 30 0.738 0.800 0.105 0.2216 0.0235 0.866 9 135 10 11 0.570 0.890 0.090 0.1796 0.0162 0.707 10 150 7 11 0.888 0.950 0.060 0.1260 0.00757 0.500 11 165 3 43 0.196 0.987 0.037 0.0649 0.00241 0.259 12 180 00 0.000 0.000 0.013 0.0000 0.0000 0.000 13 195 3 43 0.196 0.013 0.013 0.0649 0.000845 0.259 14 210 7 11 0.388 0.050 0.037 0.1260 0.00467 0.500 15 225 10 11 0.570 0.110 0.060 0.1796 0.00666 0.707 16 240 12 30 0.738 0.200 0.090 0.2216 0.0200 0.866 17 25513 58 0.875 0.305 0.105 0.2487 0.02617 0.966 18 27014 29 0.970 0.433 0.128 0.2583 0.0331 1.000 19 28513 58 0.998 0.562 0.130 0.2487 0.0336 0.966 20 30012 30 0.955 0.700 0.132 0.2216 0.0293 0.866 21 31510 11 0.827 0.840 0.140 0.1796 0.0251 0.707 22 330, 7 11 0.604 0.915 0.075 0.1260 0.00947 0.500 23 345' 3 43 0.321 0.985 0.070 0.0649 0.00456 0.259 24 360: 00 0.000 1.000 0.015 0.0000 0.0000 0.000 Mean momentum =0.620 0.397681 0.6325 46 A TREATISE ON Table XI. Connecting-rod 3 times the Stroke. Position of the crank. Friction in the guide. n V w a s s tang.W7=q qs 00 0° 00' 0.000 0.000 0.000 0.000 0.00000 1 15 2 28 0.300 0.020 0.020 0.043 0.00186 2 30 4 47 0.571 0.070 0.050 0.083 0.00416 3 45 6 45 0.784 0.160 0.090 0.118 0.01065 4 60 8 17 0.927 0.270 0.110 0.145 0.01600 5 75 9 15 0.996 0.400 0.130. 0.163 0.02122 6 90 9 35 0.985 0.543 0.143 0.169 0.02422 1 105 9 15 0.911 0.666 0.123 0.163 0.02012 8 120 8 17 0.784 0.777 0.111 0.145 0.01610 9 135 6 45 0.618 0.866 0.089 0.118 0.01050 10 150 4 47 0.426 0.934 0.068 0.083 0.00564 11 165 2 28 0.217 0.974 0.040 0.048 0.00172 12 180 00 0.000 1.000 0.026 0.000 0.00000 13 195 2 28 0.217 0.026 0.026 0.043 0.00112 14 210 4 47 0.426 0.066 0.040 0.083 0.00332 15 225 6 45 0.168 0.134 0.068 0.118 0.00803 16 245 8 17 0.784 0.223 0.089 0.145 0.01293 17 255 9 15 0.911 0.334 0.111 0.163 0.01814 18 270 9 35 0.985 0,457 0.123 0.169 0.02083 19 285 9 15 0.996 0.600 0.143 0.163 0.02334 20 300 8 17 0.927 0.730 0.130 0.145 0.01887 21 315 6 45 0.784 0.860 0.110 0.118 0.01300 22 330 4 47 0.571 0.930 0.090 0.083 0.00746 23 345 2 28 0.300 0.980 0.050 0.043 0.00215 24 360 00 0.000 1.000 0.020 0.000 0.00000 Mean mo men ;um = 0.626 0.24887 SCEEW-PROPELLERS. 47 Table XII. Connecting-rod times the stroke. 1 2 3 00 Mean momentum a Friction qs . . . . Friction 0.06 qs . . . Momentum z . Useful effect 2—0.06 qs . When V — 90°, cos.w z= 0.5770 0.8665 0.0520 0.9110 0.8590 0.8660 0.6200 0.3977 0.0238 0.9810 0.9572 0.9682 0.6260 0.2488 0.0149 0.9900 0.9751 0.9860 0.6325 0.0000 0.0000 1.0000 1.0000 1.0000 48 A TREATISE ON DESCRIPTION OP A CENTRIPETAL SCREW-PEOPELLER. PATENTED BY J. "W". NYSTROM. Philadelphia, Maech 19, 1850. Whereas in screw-propellers the water between the blades is acted upon at the same moment by two forces, the one being the propulsive force, resulting from the oblique action of the revolving blades, and the other being the centrifugal force generated by their rotation ; the first force tending to force the water backwards, in direction parallel to the axis of the propeller, and the second force, tending to force it outwards, in direc- tion at right angle to the axis or centre line. Now, my invention consists in counteracting the centrifugal force by a particular curve imparted to the blades of the propeller in such a manner that the water, instead of being deflected outwards, is delivered in direction parallel to the axis of the propeller. The formulae by which the curvature of the blades is calculated, are deduced from those by which the value of the centrifugal force is obtained, in the fol- lowing manner: — SCREW-PROPELLERS. 49 Let a body B (Fig. 1, Plate V.) revolve (in the di- rection of the arrow) round a point A, at a distance of r feet from that central point, with a velocity of h feet per second ; its centrifugal force will then be given by the equation. jB A* Centrifugal force G = ... (1) gr In which equation B denotes the weight of the body, and g = 32.2, the force of gravity. If the number of revolutions per minute, which may be denoted by the letter n, and the radius r, be known, the velocity per second, or the quantity h in the equation 1, is found by the formula ^_2^r^ (2) 60 ■ ' K ) By substituting this value of h in the equation 1, we obtain — 60'' rg m g the value of the centrifugal force in terms of the quantities, B, r, 7t, n, and g. Now, referring again to Fig. 1, let the line a h re- present the direction and m.omentum of the body B, under the action of the propulsive force, and a c the magnitude and direction of a force equal and opposite to the centrifugal force acting upon it; then, by a well-known principle of mechanics, the diagonal a d of the parallelogram erected on the two forces as sides, will represent the magnitude and direction of their 50 A TREATISE ON resultant force, which will force the body to describe a circle around the centre A, with a radius of r feet, and with a velocity of h feet per second, if then the blades of the propeller at this distance of r feet from the centre line be at right angles to this diagonal line, it will counteract the tendency of the water, which is the body in this instance, to pass outwards from the axis under the action of the centrifugal force. The angle v, which the blades at this distance from the axis make with the radius, and which is equal to the angle d ah, which the diagonal of the parallelo- gram makes with the line a h, is determined by the trigonometrical equation , sin.v tang.v = cos.v In the parallelogram a 6 c cZ, we have a c = sin.v = C, and ah = cos.v =z B, hence tang.v = _ , JS and C= B tang.v. Substituting in this equation the value of C, given by equation 3, we have of which tang.v = 4: r n^ n^ TlattY. SCREW-PROPELLERS. , 51 By replacing the symbols ti and g by their known value, viz. g = 32.2, n = 3.1416, we have . 4: r n^ Dn^ ,.. ^""^■^ = 11740=5870' • • • • W which equation represents a rule for finding the angle V, which may be thus expressed in words : Multiply the number of revolutions of the propeller per minute by itself, and the product by the diameter D of the circle described, in feet; divide the product by 5870; the quotient will be the tangent of the angle which the blades of the propeller should make with the radius at the circumference of the circle described. If the inclination of the blades at various distances from the axis be determined by this rule, and if the inclined lines be drawn and united, a spiral curve, A D (Figs. 1 and 2, Plate V.), will be formed, progress- ively increasing in its inclination to the radii of the propeller as it proceeds from the centre to the circum- ference, and this curve will be contained in the sector of the circle bounded by two radii, A D and A E, and the included arc D E or w°. In order to save the time and figuring required to calculate the inclination of the propeller-blades at all the intermediate points between the axis A and the circumference B, we will have recourse to some formulae of a common arithmetical spiral. (See Fig. 2.) The tangent for the angle v bears the following analogy to the radius y and the circle-arc x that tang.v : 1 = x : y, 52 A TREATISE ON and tang.v = — , (5) By reference to the formulae for the circle-arc x we have 7trW° in.s " = T8r' ^'^ in which r =y, the radii of which the angle v is calculated ; w° = the angle in degrees in which the circle-arc x and the spiral A D are contained. By insertion of this value of x in the formulae, we have tang.v = 180 y 180 As the tang.v is given by the formula 4, we have ''*'^^"= 5870 = 180'- • • • (^) from which we obtain the angle 180 Dn^ Dn^ ,q, ^=^-0^=1021'- • • • (S) This is the formula by which the curvature of the spiral is ascertained, but, before applying it to the propeller, it must undergo a little modification in re- ference to the quantity n, depending on the slip of the propeller. It is evident that, tf the propeller has no slip, it does not act to propel the vessel, as the water is a free movable body, and can give no more resistance than that of its own inertia, and the hydrostatic pres- SCEEW-PKOPELLEES. 53 sure of water and air. If there is no slip, the resist- ance by the inertia will be o, and the hydrostatic pressure of the water and air will be equal on both sides of the propeller ; therefore, slip is a necessity for propulsion in water, and, accordingly, a measure of propulsion. If a propeller makes n revolutions per minute, and has a slip = S, the value of propulsion is measured by the product of n S; that if a propeller makes n = 50 revolutions per minute, and has a slip S = i,- the number of revolutions which act to propel will be only n S= 50 x i = 26 revolutions; there- fore, the value n in the formula 7 must be multiplied by the slip = S, when n denotes the total number of revolutions per minute, and -T02X' ^^^ This is the practical rule for calculating the angle w°, expressed in words : — EuLE. — Multiply the diameter of the propeller by the square of the number of revolutions per minute, and by the square of the slip in a fraction ; divide the last product by 102.4 ; the quotient will be the angle w° expressed in degrees of the circle. "When this angle w^ is obtained, set it at the cir- cumference of the propeller, then, as the curve is a regular arithmetical spiral, it is described by dividing the angle w° into a number of equal parts, 6 for ex- ample, as seen in Fig. 2, and number them in regular succession from one extremity D of the arc to the 54 A TREATISE ON" other E; then divide the radii A E oi the propeller into the same number of equal parts (6), and number them in regular succession from the circumference E to the centre A ; through the divisions 1, 2, 3, &c., of the angle w°, draw the radii y', y", y'", &c., and through the divisions of the radii E A, with the axis A of the propeller as a centre, draw the circular arcs x', x", x'", &c.; unite the points of insertion of the arcs x', x", x'", &c., with their respective radii y', y", y'", &c., by a curved line, which will be the spiral curve required in the propeller-blades, moving in the direction indicated by the arrow, with n revolutions per minute, to counterbalance the tendency of the water to move outwards from the axis of the propeller under the influence of the centrifugal force, generated by the rotation. This curve is not influenced by the obliquity of the propeller-blades to its axis, or by what is gene- rally termed its pitch, which may be adjusted at plea- sure to suit the engine and vessel. As the water is discharged by the propeller in a direction parallel with its axis, the slip of the propeller will be less than with those constructed by the ordinary, or with straight blades. The centripetal propeller is a regular screw. Its peculiarity from others consists only in that the gene- ratrix for the screw is a curved line (spiral) drawn on a plane at right angle to the axis of the scre-s^. SCREW-PROPELLERS. 55 while a common screw-propeller has a straight gene- ratrix drawn on the plane at right angle to the axis. For canal propeller-boats, where it is of great im- portance to prevent, as far as possible, the propeller from agitating the water, that is wholly overcome by a propeller constructed upon this principle, because it only touches the water within the cylinder described by the propeller, and delivers the water as solid as it receives it; which is not the case with the straight- bladed propeller, where the water is thrown out by the centrifugal force, and thereby turbids the same, and spoils the bottom and sides of the canal, and causes a partial vacuum behind the propeller, and the rudder comes in a porous mass of water; which cannot be the case when the propeller is centripetal, and then the centrifugal force acts to propel the ves- sel, and the rudder comes in a solid mass of water, while an easier steering of the vessel is obtained. The accompanying drawing, on Plates VII., Figs. 4 and 5, represents a propeller made of wrought-iron. The blades are screwed together at the hub, so that any of the blades can be taken ofi" when required. The hub can be made of cast or wrought-iron, as a round cylinder, to which the blades are screwed. The blades are supported by two bands at the circum- ference, so that, if the propeller meets with any obstacle in the water, the bands serve to keep the blades in their proper position, and, as the blades are curved, the resistance and centrifugal force of the 56 A TREATISE ON water act to strengthen the blades, then, if there were no bands around them, the propeller would obtain a larger diameter which might not be allowed in its determined space. Therefore, a pair of narrow bands around the propeller, particularly when made of wrought-iron, is necessary. If made of cast-iron, the blades would sooner break than bend. We will here give a few examples how to calculate the angles w° and V. Example 1. — Suppose a canal-boat having a pro- peller 6 feet in diameter, and making 65 revolutions per minute, with a slip of 55 per cent., we have from the formula 9 ^0^ 6x65^x0.55 ^ 102.4 from the formula 7 we have 3.14 X 75°. , oi coo tang.v = -_ = 1.31. v = 53°. loO Example 2. — A merchant steam-ship having a pro- peller 12 feet in diameter, making 54 revolutions per minute, and so proportioned to the vessel that the slip will be 30 : What will be the angles vP and v ? """'.'nl^ "■"''- 80-66 = 30° 40-, 102.4 ' y.^ F^.ff 'T^r ':x :zD" _ JV y^'.>5trgTTv j SCEEW-PEOPELLERS. 59 If the blades are straight, the section will be a straight line at right angle to the axis of the propeller or in direction of the radii. The accompanying diagram represents the section of a propeller with an expanding pitch. Let the line a c represent the magnitude and direction of the cen- trifugal force in the point a, draw through the point a the tangent t b, draw from the point c a line c d, at right angle to a b, from d draw the line de, at right angle to a c, then the line d e represents the magni- tude and direction of a force, which acts to propel the vessel in the direction parallel to its motion, and caused by the centrifugal force. From this diagram we obtain be = ab cos.^ ac = ab sin.^ cd = be sin.^ de := ed sin.^ = be sin?^ , de ac 8in.^

^ ^^ ^.^^ ^^^ tang.^ sin.^ but, therefore, That is to say, the force de is greatest when half the sm. n , A de=ae^2q, (10) A 60 A TREATISE ON sine for double the inclination is greatest. The great- est sine is equal to 1, which answers to the angle 90°, therefore de will be greatest when 2^) =90°, or That is to say, the centrifugal force acts to propel the vessel with its greatest tendency when the inclination of the section is 45° independent of slip or velocity. The angle w° will then be simply «' = — p- (11) The mechanical effect developed by de will be, when ac = G. E=CPnS^2^ . . . . (12) I do not mean to say, that this mechanical effect E is created from nothing by the propeller and the cen- trifugal force itself; but, as the steam delivers a certain amount of effect through the engine and propeller. It is my intention to construct an instrument that will employ all this effect in the propelling of the vessel. Before closing this book, we will construct a pro- peller upon this principle, and with an expanding pitch in two directions. SCREW-PROPELLERS. 61 A Table of Formulce collected for convenience and reference. Tlz: G-. tang.v- gr' ' ' 2 H r n 60 ■ iang.v = 5870 It r w° '"180"' J (11) i;=CPn 8 — 2^.(12) 7t r w 360 P . . . (13) P = I 360 r ■ (14) • (15) • (16) G = centrifugal force in pounds. B = weight of the revolving body in pounds. h = velocity of B in feet. n = number of revolutions per minute. r = radii of the circle in which B revolves. V = inclination of the spiral to the radii r. w = angle in which the spiral is contained. D = diameter of the propeller. P = pitch of the propeller. S = slip of the propeller. 62 A TREATISE ON ^ = angle of inclination of the section of the pro- peller-blades to the axis. de = the force which acts to propel the vessel, and caused by the centrifugal force ac. E = the mechanical effect of de. We will here add a few formulae for calculating the pitch and length of an arithmetical spiral. The letters will iienote — I = length of the spiral within the angle w° and radii r. p = pitch of the spiral, which is equal to the radii r when w = 360. SCEEW-PKOPELLEES, 63 SUGGESTIONS RELATION TO THE MANNER OF CONSTRUCTING A PROPELLER, ARRANGEMENT OF ITS STEAM-ENGINE, WHEN THE DIMENSIONS OF THE VESSEL ARE GIVEN. The Propeller. When a propeller-vessel is to be built, we first have the dimensions of tonnage, length, and heam. From them the ship-builder lays out the lines of the vessel, and ascertains the tonnage, greatest immerse section area of the displacement, draught of water, &c. From these the propeller is to be constructed, and suppose the following to be the dimensions, viz. : — Length on deck 230 feet. Beam 35 " Draught of water from bottom of the keel 15.5 " Greatest immerse section area . . 486 sq. ft. Displacement 1646 tons. Diameter of the propeller. — The diameter depends on the draught of water, and, in this instance, can be made. 14 feet wheji the draft of water is 15.5 feet. Pitch. — The pitch of a propeller depends on the p 64 A TREATISE ON arrangement of the steam-engine, as if it is with, gear- ing or direct action. Let the number of revolutions of the propeller be n, when the number of revolutions of the steam-engine is n', a proper pitch will be = 2.5i)j!i'; (1) of which D = diameter, and P== pitch of the pro- peller in feet. In this we will construct the propeller for a direct-action steam-engine, and take the pitch P = 2i Z> = 2J X 14 = 35 feet. The angle V of the propeller-blades at the peri- phery will be found by the formula ""■'^=.^ (2) cot.V= -^^'-^ = 0.795 = cot.bl° 30' 3.14 X 14 When the angle Fis given, the pitch will be P=cotYnD (3) The pitch of a propeller can be found, when the length of the circle arc e is given, and will be ^=^ (4) in which e = length of the circle arc in the angle v at the periphery, in feet. When the extreme breadth of the propeller-blades is given at the periphery, the pitch will be obtained by P=-^i. (5) SCREW-PKPPELLERS. 65 in whicli h = extreme breadth of the propeller-blades in feet over the edge. Length of the Propeller. — The length must be pro- portioned to the diameter and number of blades. In the propeller we are constructing, we will take five blades, denoted by the letter m; then the proper length L will be, L = dn. V. COS. V (6) m L = sm.51° 30' X cos.61° 30' x ^'^^ ^ ^^ =42.5 ft. 5 Number of Blades. — The number of blades in a pro- peller depends on the slip, and angle V, Fig. 8. The more slip, and larger angle. V, the less the number of blades should be. As to the number of blades in a propeller, it is known that a less number do better, or just as well, as a greater. It may be so in some circumstances, but not always. In canal and tow- boats, where the slip is considerable, a two or three- bladed propeller will, no doubt, do the best execution ; but for vessels which are to run fast, and the propeller has a slight slip, a five or six-bladed propeller should give the greatest effect; but in such cases the propeller must be particularly constructed, and the length L, Figs. 5 and 8, be only one-quarter of the diameter. If this is disproportioned, and the blades are wrong, of course there ought to be a less number of them. When the length of the propeller is given, we have 66 A TREATISE ON the proper number of blades by solving the formula (3) to m = sin. V cos. V — — - (7) m = sm.51.° 30' x cos.61.° 30' ^lH^ = 5 blades. 4.25 Acting area of the propeller. — The acting area of a propeller depends on the proportion of pitch and diameter, so that the more the pitch the less the act- ing area will be L in the same diameter ; which will be seen in the formulae A = ^-^ ^ , (8) in which A = acting area. 2.5 X 14" = 122.5. ^35'' + 3.14'' X 14^ Slip. — The slip depends on the greatest immerse section area, and form and friction area of the dis- placement. When brought into action with the act- ing area of the propeller, it will be found very near by the formulae o 486 + 2V1646* r.... .r, /5> = 16 X 122 5 ^ ' °^ ^*^ ^^^ ■ (S* = 486 square feet, greatest immerse section area. Q = 1646 tons displacement. A = 122.5 square feet, acting area {Jess than 0.785 IF). This is the formula for calculating the slip when SCREW-PROPELLERS. 67 the vessel is new-built and not coppered. When the vessel is coppered, and has run for some time, so that the friction area is very smooth, the coefficient 2 will vary to 1, or perhaps less. Within the common or reasonable proportion of vessels and propellers, this formula follows the variations of slip as near as I, until this, have been able to observe, which has been on a great number of vessels. A formula based upon the principle that the acting area of the propeller, and resistant area of the vessel brought into action, and that the resistance to these areas is in proportion as the square of their velocity, will not follow the varia- tion of slip, and is a complicated calculation.* The slip given by this formula (9) means the slip when the vessel is running in still water, or when there is no other force acting with or against the vessel but that of the propeller. The less the acting area is in pro- portion to the vessel, the greater the slip will be; and, therefore, the slip can be no measure of loss of effect. The slip is nearly constant with different velocities until a certain limit ;" when it exceeds that, the slip will increase by the excess of velocity. Velocity of the propeller. — The proper velocity of the propeller depends on the pitch, diameter, and slip, and can by them be proportioned to suit the steam- engine. The proper velocity, or number of revolu- tions per minute = n; in order to give the vessel the highest speed with the greatest economy of power, is * See bodies in motion in fluid. 68 A TEEATISE ON 200 ,j^ n = 200 (10) ^14 = 51.3 revolutions 0.415x35' per minute. From this number of revolutions, pitch, slip, and diameter of the propeller, the power of the steam- engine is to be ascertained ; but before entering into that, we will finish the propeller. Although it may not be necessary to reach the 51.3 revolutions, but will determine it, for the calculation, to be 50. The pro- peller to be centripetal, and have an expanding pitch in two directions, as mentioned in the former pages, and first calculate the angle w°from formula 9, page 53. FIG. 2. FIG. 1. FIG. 3. Fig. 1. - 2 02 W = Dn^S 102.4 14 X 5 0^ ^ X 0.40'^ = 54.6°. 102.4 (11) Compute two more angles ^w and ^w from the fol- lowing formulas : — ^to = ^ {2-S) (12) Fig. 2. - ^^.=:^ (2-0.40) =^-1:^^1:^=43.8°, n 2 SCBEW-PROPELLEES. 69 in which angle the spiral is to be constructed, for the fore-edge of the propeller, and Fig. 3 ^w is the one foi* the after-edge. Fig. 3. ^w = :!^(2 + S) (13) 'Wz 54.6 (2 + 0.40) 54.6 X 2.4 = 65.5°. 2 ^ ' ^ 2 The fractions or minutes of those results will not be taken in the construction of the curves. In each of these angles °w ^w ^w, construct a curved line, as in Figs. 1, 2, and 3. h FIC. 5, r "^^M L— J fv Place those curved Hues on a common axis, as shown in Fig. 4, so that the planes on which the spirals are drawn be at right angle to the axis. Set the curves ^w and ^w in a distance equal to the length i of the propeller; then set the spiral °w in the middle, between the planes on which the spirals ^w and ^w are drawn ; draw the straight lines from the centre to the ex- tremity of the spirals. Set the spirals ^w and ^w, so that the angle v will be as follows. 360 X Fig. 4. ' V =■ P 360 X 4.25 35 _ 0.3 i^w—'-w) .... (14) —0.3 (65—44) = 37.38°. 70 A TREATISE OK Place the spiral "w between the spirals ^w and ^w, so that the angles v will be "^^. = 1(2 + ^) (15) Kg. 4. 37.38 (2 + 0.40) = 22°.5. « — — ^. ^ '■■■f r f ^=L y f <» t* ly wirz When the spirals are laid down in these positions, make a drawing of the propeller, as Figs. 7 and 8, which represents a propeller made of wrought-iron. If made of cast-iron, the spirals °w ^w and ^w will be laid out in loom. Build to the spirals a box, as re- presented in Figs. 4 and 5; form it in the hub as in the drawing, so that x is equal to *a;. In this box, build a pattern of boards, about 2i by 8 inches; the boards must run from the spiral ^w to the spiral *to, so that SCREW-PROPELLERS. 71 the edges or seams run parallel with each other, or so that the same seam runs to the same figures on the spirals, and forms a regular curve, for instance, a circle arc of a circle, touching the spiral "««. When the slip exceeds 50 per cent., it will be ne- cessary to pay more attention to this curve or circle arc, and lay out 5 w spirals. From this pattern cast a block of cast-iron, represented in Fig. 6. Bore holes in this block, corresponding to the holes for the screws, by which the propeller-blades are screwed together. Fit a pasteboard on the face side of the block, and make it the same shape as the propeller-blades are to be; when fitted precisely, mark the holes on the pasteboard; from this, cut out one propeller-blade from the sheet-iron, and bore the holes accurately from the pasteboard. From this blade cut out the rest of the propeller-blades, and one more than the number of blades in the propeller. Afterwards bend these blades over the block (Fig. 6) so that the holes correspond with each other, then the blades are ready to screw together. If the holes are not bored before the blades are bent, they must be marked on the block, or else they will not come in their proper position, and it will cause much trouble in putting them together. If the pitch of the propeller was not expanding in the direction of the radii, the before-mentioned angle V would be from the formulae 360 L i-ir>\ V = ....... (16) 72 A TREATISE ON therefore, if we will calculate the mean pitch and angle V at the circumference of the propeller, we have p__ 360 L _ ^ ^ n7) V p^ 360 X 4.25 ^ 38 g 37.38 TTie Steam^ngmes. The effect of a steam-engine sufficient to drive the propeller 50 revolutions per minute will be found by this formula in horse-power = H: — 2500 ^'^ ^ ^ 50* X 14* 5"= 2500 ^^^ ^ ^'^^ "" ''^^^ horses. To be two direct-action condensing-engines, let us determine the effectual pressure per square inch to be 24 pounds; say pressure in the boiler to be 20 pounds; cut off the steam at H the stroke, the mean pressure will be 14 pounds, vacuum 10 pounds, effectual pres- sure 10 4- 14 = 24 pounds. By the quantities we now have given, we will be able to ascertain the size of the steam-cylinders, which may be this : — «-7^^Tr-=7r' • ■ * ■ ^ ^ in which a = area of the steam-cylinder in square inches, s= stroke of piston in inches, /= friction and working the pumps per cent., which, in this in- tended steam-engine, will be/= 0.82, or 32 per cent. SCREW-PROPELLEES. 73 Let the proportion of stroke and diameter of the steam-cylinder piston be as 2 : 3, we have the diame- ter in inches. ^_^\ 1^nl^7m .... (20) ^ ^'(1-/) J M76 X 50 X 14? >/35 x 0.40 ., ^. . , Say 56 inches diameter and 36 inches stroke, the eflfect of the steam-engine, in horse-power, will be ^-sHifl^fi-/)' • • • (21) The number of revolutions that can be obtained by a given power will be found when the dimensions of the propeller and slip are given; or n= ■=— (22) When the dimensions of the cylinders and eflfectual pressure on the piston are given, the number of revo- lutions will be ^^ ^ps{l-f) (23) 51 Lf VPS The effectual pressure which is required to give a propeller so many revolutions per minute will be obtained when the dimensions of the cylinder are given; or bin D^ VPS (24) Cut off the steam at J the stroke, and c being the 6 74 A TREATISE ON cubic feet of steam for each number of revolutions in botb tbe cylinders, and h = volume of steam com- pared with water at the given pressure. 2463 X 12 X 4 ^^ . ^„ i.- /• + c = — — = d8.4, say 70 cubic feet. If 1 pound of anthracite coal evaporates 8 pounds of water per hour, the consumption of coal will be, in tons per hour ^^^^ = 4X1' (^^^ coal = -— — — - = 0.918 tons per hour. 4.8 X 770 ^ The speed of the vessel will be in statute miles per hour, Jf=5LlJ55(l_0.40) = 12 mUes. oo The distance from New York to Liverpool is about 3100 miles; the vessel will run that in a time of To Boston in 10 days, 8 hours. To Halifax in 8 days, 22 hours. "When the power of the steam-engine, dimension, and slip of the propeller are given, to find the speed of the vessel in statute miles per hour. M 1.76 Z> VPS ' (27) ^=1-S^. (28) 'n SCREW-PROPELLERS. 75 The engines are working on a double crank, repre- sented in Fig. 9. The scale is I inch to a foot. The cranks are opposite to each other; therefore, the steam- engines do not work at rigid angles, but, if it is thought more desirable that they should, the crank can be set in any angle. This is a matter of small consequence, compared with the frictions in the bearings a and d, which the opposite cranks prevent, and it is the rea- son why the stroke of the piston can be so short in proportion to the diameter. /•/. M Now for the strength of this crank, which, at first sight, promises little for durability. Let the end a be the continuation of the propeller-shaft, and the steam-engines be applied at & and c. Let the force which is applied at c be denoted by the letter c. The force which acts to twist the crank at h will be the pro- duct of c s; but, as there is a bearing at d, the resist- ance in h acts to twist the crank at c with a force - = h s. If the crank should break, the force c must break it both in c and b at the same time; which will require twice the power that would break it only at 76 A TREATISE ON 6, or -- will be the force by which the crank should be twisted at h, which is exactly the same as if the force c was applied at b to twist the shaft at a; or, if the diameters at a, h, and c be equal, the strength of the opposite crank will be the same as a common single crank. Let D be the diameter of the propeller-shaft in inches, and of wrought-iron, D = 18 v/PD, (29) D = 18 v/35 X 14 = 12 inches. The diameter at d will be the same as the diameter 5 of a crank-pin of a common or single crank, 5 = v/D^-M.2 6-^ — 1.1s, . . . (30) 8= n/12^+ 1.2 X 3^ — 1.1 X 3 = 8.5 inches; 5 is expressed in feet in this equation. This kind of crank was tried and tested by a Swedish engineer, 0. E. Carlsund, and the results were most satisfactory. The engines were 30 inches in diameter, and 18 inches stroke. They made from 105 to 110 revolutions per minute. The air-pumps were direct action, working at the same speed with 18 inches stroke. Pressure in the boiler was 30 pounds per square inch. Cut off the steam at i the stroke, the air-pumps and valves worked silently. In direci^action propeller-engines, the arrangement of the air-pumps and their valves is of the greatest importance. Many engineers are of the opinion that SCREW-PROPELLERS. 77 the more vacuum the better; there is, however, an exception with propeller-engines, when the engines and air-pumps are direct action. As an example, in an experiment made by the same engineer, 0. E. Carlsund, he placed a small cock on the condenser; while the steam-engine was working with a good vacuum, he opened the cock, letting in a little air to the condenser; the number of revolutions increased from 15 to 20 per cent. When an air-pump makes more than n double- strokes per minute, it does not work well, or rather works to a disadvantage. This number n will be found by the formula n = lliv/r .... (31) in which a denotes the area of the valves and '^ = area of air-pump piston. Q = stroke of the air-pump piston, p = pressure in the condenser in pounds per square inch. When the number of double-strokes per minute are given, the area of the valves will be a=^^L^ (32) 12 ^/p When we obtain from this formula a = ^or a> % the air-pump will work with disadvantage; or, when the valve a is applied in the air-pump piston, we only 3 geto= -%; then and p are the only quantities 8 which can be modified, and will be 78 A TREATISE ON 0=^v/p (33) In the steam-engine represented on Plate VIII., the arrangement of the valves is the same as those represented on Plate XII. ; the stroke of the air-pump piston is the same as the stroke of the cylinder-piston, and the diameter of the air-pump is 14 inches. In that case, the diameter of the valves must be calcu- lated, and will be as follows : — i,= ^^^ (34) 3.464 4'p 3.464 V5 which should be the diameter of the air-pump valves. (See Plate XIII.) These formulae undergo a little modification with different arrangements of condens- ers, positions of valves and air-pumps, &c. The inclination of those engines represented on Plate VIII. is about 16°. It is of no consequence what angle the steam-engines make with each other, but rather make it to suit the bottom of the vessel. There will always be angle enough that one engine will help the other over the centre, and the more it varies from 90° the less friction it will be in the bear- ings. With that kind of engine, there is sufficient room in the breadth of any vessel to get the connect- ing-rod twice the stroke. The entire engine comes below the load line, and makes an arrangement with every convenience. SCREW-PROPELLERS. 79 Under full motion, all the parts of the machinery are accessible to the engineer. The weight of both the engines, including air-pumps and condensers, will be — stationary parts 22 tons, moving or working parts 8 tons; total weight 30 tons: occupying a space in the length of the vessel of only 8 feet 4 inches. Lower hold is 12 feet; on the first deck 7 feet; on the second deck 8 feet. 80 A TREATISE ON A Table of I^uatimis, collected for convenience and reference. 40 n B" ^yps cot. r= — . TtD P—cot.V7i D . „ hDL p= e H D L Vv> + 1/ L = sin. Vcos. Y m ^ sin. V COS. Y , 2.5 B^ S m rtl) •^P'+ H^JD^ _®+2?r§« 16 J. -PS Drv' &> 102.4 :!^(2- 2 ^ 360 Z S). (2 + S). 0.3 Qw—^vi). 360 i P = 360Z 2500 (1) (2) (3) (4) (5) (6) 0) (8) (9) (10) (11) (12) (13) (1|) (15) (16) (17) (18) d _^ ViQuiy^lPS ^ ?'(i-/) ■ 5-=-^£i^-i--(l — n 33000 X 12 ^ •' ^ 50 Vm P- BVPS 51 D^ \/PS 51 w z> v^p;^ ■ dU{l—f) • CoaZ 1.76 D'>/PS S = I- 88 M Pn -- 18 'ypD. 5_x/D3+1.2s»— 1.1, ._10a^- 210 n%Q P- lOv/p S=4f^P- 3 Vp 360 i V + q—r) {^w—''w) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) SCREW-PROPELLEKS. 81 A = acting area of the propeller in square feet. I) = diameter -n P = pitch V of the propeller in feet. L — length ) S = slip in decimal fractions. H= Actual horse-power. M= Statute miles per hour. Q = displacement in tons. gj2 _ greatest inverse section area in square feet. a = area of the steam-cylinder in square inches. b = extreme breadth of the propeller-blades, in feet. c = cubic feet of steam for each revolution. d = diameter of the steam-cylinder in inches. e = length of the circle-arc in the angle v in feet. /= friction and working-pumps per cent. Jc = volume of steam compared with water. m = number of blades in the propeller. n = number of revolutions per minute. _p = effectual pressure in pounds per square inch. s = stroke of the piston in inches. °w, 'w, ^w, and v = angle of a circle in degrees. ^ = area of the air-pump piston. a = area of the valves. jOD = diameter of the air-pump. h = diameter of the air-pump valves. = stroke of air-pump piston in inches. p = pressure in the condenser in pound per square inch. D = Diameter of the propeller-shaft. 8 = Diameter of the crank-pin. 82 A TREATISE ON Description of the Arrangement of the Steam-Engine and the Vessel. Plate VIII. represents a section of a vessel of the aforesaid dimensions. On the bottom lie two inclined, direct-action, condensing steam-engines. A. The engineer's room. B. First cabin. G. Ventilators and skylight. D. State-rooms. E. Store-rooms. F. Passages. Q. House on deck. H. Berths. The engine is manoeuvred in the engine-room, A, by the three rods, a, h, and c, which run through the columns in the engineer's room. The four gauges on each side are steam-gauge, vacuum-gauge, water-gauge for the boiler, and salinometer; in the front, is a clock and counter. a is for regulating the injection- water connected with the injection-cock Jc. h for reversing the eccentrics, to back and go ahead. Connected with a conical Cog-wheel. c. for regulating the steam to the engine. By the same rod steam can be given to full stroke of the piston. d. Cylinder-head. e. Injection water-pipe. Fhii-t mi. ■ S'//f/f/r.y/-/(/7i.y ff/7- f/nY//u/f//u'//t of Pjvpfllfj' .S'T/v/j/t Hiityims. by T.W.JTY STROM. v/7/7;77?///WZi==__,A> ., .^.,.. gr;\'\T\TmTty^^^ ,ii| jirNvioviiv ,ifi\ SOREW-PEOPELLEES. 83 /. Steam-pipe. g. Steam-chest. 7i. Exhaust-pipe. i. Discharge-pipe. Ti. Injection-cock. I. Condenser. m. Air-pump, double and direct action. n. Feed and force-pumps, double and direct action. o. Air-vessels. p. Eccentrics for the steam-valves. q. Frames and guides. r. Steam-cylinders. A Table for finding the pitch of Propellers. By this table, represented on Plate IX., the pitch of propellers can be found in an instant with any dia- meter, and angles of the propeller-blades, between 45 and 70 degrees. The degrees will be found on the cir- cular arc, which is divided as a diagonal scale, so that where the arm crosses the diagonals at a it shows every tenth minute of a degree. The pitch will be found on the pitch-arm, where the diameter lines cross the same, when set on a given angle at a. If the angle is to be found when the diameter and pitch are given, move the pitch-arm until the given pitch crosses the given diameter line, the corresponding angle will be found at a. Propeller-makers, not fully acquainted with the 84 A TREATISE ON theory of propellers, will sometimes find it difficult to lay out a propeller with a certain pitch, or when a few diameters of a propeller are given to ascertain the pitch or angle, then, a reference to this table and formulae will meet their approbation, for convenience in practical use. ri.ii,- ix. TABLE TO FIND THE PITCH OF PROPELLERS J.W.NYST110M. SCREW-PROPELLERS. 85 DESCKIPTION or A PROPELLEE-ENGINE, PATENTED BY K. F. LOPER AND J. W. NYSTROM. Philadelphia, Apeil 15, 1851. The Engine is represented on the Plates X., XI., XII., and XIII. rig. 1, side elevation. '' Fig. 2, transverse elevation. Fig. 3, transverse section through the air-pump and condenser. Fig. 4, the slide-valve motion. Fig. 6, the air-pump and valves. Our improvements are particularly applicable to that class of marine steam-engines which are employed to drive screw-propellers, and which it is desirable should occupy the smallest possible space compatible with a due degree of strength and efficient action, and a free access to the several parts for manoeuvring adjust- ments and repairs, where also it is desirable that the engine should work at a high speed, in order to drive the screw-propeller at a sufficient velocity without the 86 A TREATISE ON employment of gearing, and, at the same time, to obtain a large amount of duty from an engine of comparatively small size and light weight. The improvements consist, first, in constructing and arranging the condenser and air-pump, which ap- pertain to each steam-cylinder, in such a manner that they shall constitute the columns by which the cylin- ders are supported. Second, in the construction and arrangement of the valves to the steam-cylinders working in connection, which is such that the cut-oflE" valve of one cylinder is actuated by the valve-rod, or valve of the other cylinder. Third, in the construction and arrangement of the air-pump valves, which is peculiarly applicable to en- gines running at a high velocity, where it is essential that these valves should close quickly. In the double cylinder propeller-engine, represented in the accompanying plates, it is the bed-frame which is cast hollow, as known in the sections. Figs. 3 and 5, and is bolted fast to the kelsons or timbers of the vessel. The condensers of the two cylinders are mounted upon the one side of the bed-frame, while the air- pumps are mounted upon the opposite side. The condensers D have the form of a hollow cylinder open at the top and bottom ; at their lower extremities they are screwed to the bed-frame, and SCEEW-PROPELLEES. 87 at their upper end are screwed or bolted to the steam- cylinder. These condensers are each fitted with an adjustage a, through which the injection-water is introduced. The force-pumps B are fastened on each side of the condensers, and their plunges are connected di- rect to the crosshead E, and the steam piston-rod F. The air-pump column O, valve-chest E, and side-pipes /, are cast in one piece; which is prolonged above the air-pump head, as represented at h, formed and fastened to the cylinder and bed-frame, as the condensers on the opposite side. The condenser and air-pumps thus constructed form the columns, on which the cylinders J are supported, while the hollow bed-frames form the cistern or well in which the water of injection and condensation collects, and by which they are conducted to the air-pump. In the bed-frame are cast partitions and canals, so that, when the vessel is running in a hard sea, the condensing water will still obtain a regular motion to the air-pump. Owing to the air-pumps being double and direct action, the condensing water will be forced out through the discharge-pipe Q as uniformly/ as it injects through the adjustage a, and thereby render the condensation and operations even. The form of the bed-frame will, however, depend on the form of and room in the vessel. The steam-cylinders J are mounted on the columns D and Q; they have also cast fast to them an exhaust- 88 A TREATISE ON pipe V, through which the steam passes, from one cylinder through the exhaust-pipe e in the other cylinder, so that the exhaust steam from each cylinder passes to both the condensers, which make the con- densation more even, and an easy passage for the steam from the cylinders. Fig. 3, the steam approaches the injection-water at c, above which is placed a cock, by which the steam can be let out through the pipe U, and the engine will work with high pressure. Fig. 4, each cylinder is fitted with a piston M, piston-rod F, and crosshead E, and connecting-rod L, which latter takes hold of a pin of a crank N, secured to a crank-shaft Xbeneath. The crosshead is steadied by guides secured to the inner faces of the condenser and air-pump. Each cylinder is fitted with an appropriate steam slide-valve P, which is moved by an appropriate ec- centric, secured to the crank-shaft of the engine, and traverses upon the face in which the steam and ex- haust parts/, e, are formed. In each steam- valve are formed the suitable passages d, d, g, which, by the motion of the valves, are made to move over the steam and exhaust part of the valve- seat. The face of each valve slides upon a valve-seat, while the back of each valve forms a seat, to which a supplementary slide-valve h is fitted, by whose movement the steam passages d, d, of the steam slide- PLcLteX FlaUXl. Fig3- 3M.fTy!^traw.. Fldte M. ( J^Ny Strom . rcateXm. SCREW-PEOPELLERS. 89 valves are closed at the proper moment, to cut off the admission of steam to the cylinder. The supplement- ary or cut-off valve of each cylinder is connected by a link, and moved with the steam slide-valve of the opposite cylinder; and as, in this example, the cranks and eccentrics of the two cylinders are set at right angles with each other, the cut-off valve is moved with the greatest speed, while its appropriate slide-valve is moving with less speed, or is almost stationary. The eccentrics are fastened to the cog-wheel, which is geared by tJfj'e screw S (see Manoeuvring the Engine) in such a jfosition that the mean cut-off will be about one-third of the stroke ; but, if it is desirous to admit more steam, or to cut-off at a greater part of the stroke, it can be regulated by fastening the eccentrics to the cog-wheel in such a position as the cut-off re- quires, which can easily be regulated between id and fths of the stroke. This method of arranging and working the valves is extremely advantageous in cutting off the steam, as the operation is effected with rapidity, while it is peculiarly applicable to fast-moving engines, where it is desirable to employ the least complex and the most simple machinery. When the engine is to be manoeuvred to go back or ahead, and the steam is cut off at J of the stroke, it may happen that the engines stand in such a position that no steam can be admitted to the cylinders, which obstructs the starting of the vessel; for that purpose, 7 90 A TREATISE ON there are apertures, o, o, o, o, in the side of the steam- valve, which can be opened and closed at pleasure, with a suitable slide-valve connected with a rod which runs through a stuflBng-box on the steam-chest, and thereby the steam can be let into the cylinders at full stroke, which renders the starting safe at any position of the engine. When the engine is to be worked at full-stroke of steam, these supplementary valves are adjusted by the engineer in such a position, that the apertures o, o are uncovered throughout the whole stroke, thus permitting the steam to enter through these apertures when those at the back of the steam- valves are closed by the cu1>off valve 6. Each valve-chest contains two valves, h and I, and their appropriate valve-seal m; each communicates with the adjacent air-pump barrel by a passage n, which leads from the space between the two valves. The lower valve-chest is connected directly with the hollow bed-frame, while the upper is connected there- with by means of a pipe I, through which the water passes to the lower valve of the chest. The valves are conical or trumpet shaped, in order that they may open without a shock when the water strikes them ; and each is secured to the stem, which traverses upon a stationary spindle j?. The prompt closing of each valve is insured by a spiral spring r, coiled upon the stationary spindle, and concealed within the tubular stem, which also protects it from injury. As the air-pump piston is moved up and SCREW-PROPELLERS. 91 down in its barrel, the lower valve I, of the valve-chest rises alternately, to admit the water to pass into the air-pump barrel, while the opposite upper valve of each chest rises correspondingly, to allow the water to pass to the discharge-pipe Q. As the space in which the spring is closed is al- ternately diminished and increased in capacity by the opening and closing of the valves, it is essential that free passage should be given, to allow the water, which may be drawn therein when the valve drops, to pass out freely as the valve opens ; this is effected by perforating the tubular stems with small holes re- presented at s, Fig. 5. As the crosshead has the air-pump to work on one end, it is of importance to arrange it so that the fric- tion in the guides will be on the opposite end, where also the force-pumps are connected, so that the work on both ends of the crosshead will be nearly equal; this is obtained by moving the engine in the direction of the arrow. Fig. 3, which will be understood by examining the figures 2 and 3, Manoeuvring the Engine. To reverse the eccentrics, is operated by an endless screw S, geared into a wheel, partly movable on the shaft X, to which wheel the eccentrics are fastened. The screw S has a F tread, and so arranged that, if 92 A TREATISE ON the engine should start while the screw is in gear, it will throw itself out of gear. The steam is regulated bj the wheel * (Fig. 1), connected with the valve o' (Fig. 4). The injection- water is regulated by the handle a', connected with the rod a, injection-cock, and adjustage a. The air-pump of the engine, represented in the ac- companying Plate XIII., is double and direct ac- tion. The piston-rod j is connected direct to the extremities of the crosshead of the steam piston-rod, so that, when the latter moves up and down, the air- pump piston is moved in a corresponding manner. Each air-pump barrel is fitted with two valve-chests H H, one near the bed-frame, the other at the same height as the injection-cock. See Fig. 1. Economy of space is one of the most important re- quisites in steamers ; and it is believed that an engine of this arrangement and construction attains this re- quisition in a higher degree than any upright condens- ing engine now in use. While it excels in this respect, it is evident that such an engine will not weigh as much as others whose members are arranged in the usual manner ; for in this engine the ordinary framing for supporting the steam-cylinders is dispensed with, and the con- denser and air-pumps are made to supply their place, as well as to their own peculiar duty. In the conden- sation of steam, and the production of vacuum, this economy of weight, in connection with the correspond- SCEEW-PROPELLERS. 93 ing economy of space, is all important; as any saving in the weight and space occupied by the machinery increases in a corresponding manner the capacity of the vessel for carrying freight. What we claim as our invention, and secure by letter patent, is. First,- the construction and arrange- ment of the columns, by which the steam-cylinder is connected with the bed-frame in such a manner that they constitute the air-pump and condenser, substan- tially as herein set forth. Second, the method herein described, of actuating the cut-off valve of one steam cylinder, by a motion derived from the valve or valve-rod of the other cylinder, substantially as herein set forth. Third, the adjustable supplementary valve o, in con- nection with the aperture o, in the steam-valves, by means of which the steam can be worked at full pres- sure, throughout the whole length of the stroke with- out disengaging the cut>off valve. 94 A TREATISE ON LOPER^S PRQPELLEE. Plate XIV. represents the one known as Loper's Propeller. Its peculiarity consists in that the pro- peller-blades form an angle with the centre line in the centre of the same, and, therefore, is no screw. Loper's rule for this angle is to make it from 25° to 45° at the hub, independent of the diameiter of the hub, or proportion of pitch and diameter of the pro- peller. That this angle really constitutes a novelty of the instrument is thoroughly tested. The one represented on Plate XIV. is the original propeller on the steamer S. S. Lewis, and is a true representation of Captain Lopes's own design, with the following dimensions : — Diameter . . . . .13 feet. Length 3' 8" inches. Mean pitch . . . . .33 feet. Angle of the blades of the extremity 50° " " hub . 27° The propeller was geared to run . li turn, while the steam-engine made . 1 " SCREW-PROPELLERS. 95 Dimensions of the Steam-Engine. Diameter of cylinders . . .60 inches. Stroke of the piston . . . 40 " Steam pressure per square inch . 20 pounds. Cut off the steam at one-half the stroke, vacuum about . . .10 " Leaving an effectual pressure per square inch on the piston . . 26 " which gave the propeller revolutions per minute . . . . .44 Dimensions of the Vessel. Length on deck . . . 210 feet. Beam 30 " Draft of water to the 7th water-line 15 " Greatest immerse section area . 400 sqr. feet. Area of 7th water-line . . 2250 " Tonnage of displacement . 1293 tons. The steamer S. S. Lewis left Philadelphia for Boston, September 13, 1851. Passed the Philadelphia navy yard at lOh. 24' " Brandywine light-house . 6h. 25' a distance of 90 miles in . . . 8h. 1' in which time the steam-engine made 12,270 revolu- tions; multiplied by 1.75, will be 214,745 revolutions of the propeller, and travelling 96 A TREATISE ON 214,745 X 33 = 184 miles, 5280 ' of which will be a slip = Mill^ = 32.85. per cent. ^ 134 ^ The wind was south-west, and the sails were set in about f ths of the distance, which diminishes the slip somewhat. By the formulae 8 and 9, the slip would be 2.5 aS' = ^ ^/ 3.14 ^x 13^ + 33' (400 + ^1293^) 2.5 X 52.45 X 5164 pn c = 13-3 =30.8, which was the slip first calculated. Her speed in statute miles per hour wiU be M = J^ = 11.2 miles. 8h 1' Her speed, first calculated, was from the formula (26) M=z 1^-^^ (1—0.808) = 10.33 miles per hour ; 88 with the supposition that the steam-engine should be large enough to transmit a power of 350 horses, clear to the propeller, by which we calculated the number of revolutions of the propeller by the formula (22) : or 50 v/350 50 X 18.7 ,^ , ^. n = — — = -— — --— - ^40 revolutions 13 V38 X 0.308 13 x 1.78 nearly. The S. S. Lewis was intended to run between Boston and Liverpool, but made only one trip. In a PUte~W. Plate ISr SCEEW-PROPELLERS. 97 distance of 2600 miles from Boston, which she made in 11 days, the propeller broke, and she was obliged to finish the remaining 300 miles under sail. In Liver- pool, another propeller was put on, which differed from Loper's propeller — ^first, in that it was a regular screw; second, it had 39 per cent, less pitch than the former one. Plate XV. is a true representation of the propeller made in Liverpool. Its pitch is only 20 feet. 98 A TREATISE ON A CENTEIPETAL PEOPELLEE. Plate XVI. represents a centripetal propeller con- structed on the principle mentioned on page 60, that the curvature of the generatrix is independent of the slip, or velocity of the propeller; but that the inclined generatrix has an inclination to the centre line about 45°, and the angle w^, calculated from the formula (11) page 61, and the mean pitch at the extremity of the propeller-blades being about 2.5, the diameter, we have w° = ^f f = 72°. . . . (11) At the same time the propeller is constructed with an expanding pitch in two directions, as mentioned on page 60. The expansion in the direction parallel to the centre line, is measured by the angle Y of the screw helix to the centre line, in the point where the pitch is to be calculated. In Fig. 2, Plate XVI., the line a h represents the helix of the expanding screw, at the extremity of the propeller-blades. It will be found that the angle F= 64° at the point a at the fore-edge of the propeller, which will be a pitch of P= cot. 64° X 3.14 D = 1.48 Z>, SCEEW-PEOPELLEES. 99 but in the after-edge, at the point h, the angle F= 46°, and pitch P= cot. 46° x 3.14 D = 2.93 D ; then, the expansion of pitch at the extremity of the blades will be as 1.48 : 2.93 = 11 : 21 nearly as 1 : 2 ; but this is not the true expansion of the screw, which depends on the second expansion from the centre line, to the extremity of the blades, measured by the dif- ference of the angles lo and w'. See page 53, about those angles. It is evident that the angles w and w' must be proportioned by the mean pitch in the fore and after- edge of the propeller, taken at a diameter = 0.7 D. Let us, without any calculation or formula, deter- mine the angles w = 58° w' = 78, which is in the neighborhood of 72 degrees given by the formula (11) page 61 ; then, at any distance = r from the centre, the mean pitch will be found by the formula (35), P=-—^^^, — V • • ■ (^^) V + (1 — r) (w' — w) in which L = length of the propeller, which, in this instance, is J of the diameter. V = the angle in which the propeller-blades are pro- jected at the periphery, which, in this instance, is 52°. r = a fraction of the radii of the propeller. 100 A TREATISE ON" Suppose the centre of effort of the propeller-blades is at 0.7 from the centre, we have the mean pitch for the propeller P = 360^^-P =2071). 52+(l — 0.7) (78—58) The pitch in the centre of the propeller will be 52+ (1 — 0) (78 — 58) The true expansion of the mean pitch will be cal- culated as follows : — in the fore-edge mean pitch P= (1.666— 1.48) (1—0.7) -H.482=1.537, in the after-edge mean pitch P= (2.93 — 1.66) 0.7+1.66 = 2.55; of which we obtain the true expansion within the screw to be as 1.537 : 2.55 = 6 : 10 nearly. Suppose this propeller to be applied on a vessel with similar dimensions as the S. S. Lewis, and hav- ing a diameter = 13 feet; then, from the foregoing, we will obtain the following pitches : — P= 19.25 feet in the comer a. P= 38 feet in the corner h. P= 21.55 feet in the centre. P=i 26.9 feet, the mean pitch of the propeller. P=20 " " " fore-edge. P=33 " « « after-edge. A propeller constructed on this principle, with such arrangement of expanding pitch, will act to propel Tlazt- XK. / '^' i ^^'^^^^ * /.v-^ r/i N ^ / /a\ 7; yi''^ ^^- '^fc^'-.Zfx • ^^S^kM-' / %;, \ "'^"'' iJA.Si\ \-.. \ / N? ^/^■•-. i // '. »'-'''. 1 ■ ; Jir' \ \^, ..) i„_ >5 \ ;{. /l \ '''^(j "--y ^^', 1 '; \ \ y ,y t<(^ I i i \^^<--^ f^ffs-=/«-' K- A- ^ '^\~~^^\ / / ^-- -Vi^ 1— 1 1 — -V''^— ^''^^^ /^ '7 \ Vb^'' '//" \ V. P-,„<'^'(5§^ '"-^-^r-tii^^ ^S^""\>^ a. ft/ L.. J.WJTysli-om.. SCREW-PEOPELLERS. 101 the vessel with the greatest effect within the limits of 70 and 20 per cent, slip; and the propeller making from 40 to 60 revolutions per minute. That is to say, it will act to propel the vessel with its greatest effect within the limit of 15 and 3 miles per hour, caused by fair and head wind, of which the 3 miles should be against a gale. We do not mean to say that those (15 and 3) are standard points, but do say that it is impossible to reach that limit by a propeller with a uniform pitch. It is often found that propellers with more pitch, even in a moderate head-wind, lose some of their power; and, on the other hand — as the English propellers — with a very narrow pitch, when running in a fair wind, the propeller is often of no use for propelling, and sometimes has a negative slip, which keeps the vessel back. The English propellers now running between the United States and England, do not expect to make what is termed a very good time, something like 9 or 10 days ; but, even if they have head-wind the whole trip, they will come surely in not far from their usual time. That the average time for Atlantic propellers of more or less pitch will probably be in favor of the latter, provided they are constructed as usual into the day. 102 A TREATISE ON IMPROVEMENTS STEAM-ENGINES AND PEOPELLERS.* 0. E. CARLSUND. The inventions and improvements granted by this letter patent consist in, that by means of propellers, and thereto suitable machinery, propel as well mer- chant vessels as men-of-war, principally with those direct-action steam-engines, obtaining the greatest possi- ble effect with the greatest economy of fuel, space, and expense of labor. Description. Plate XVII., Fig. 1, shows a vertical section of a steam-vessel, with its (in angle) direct-action steam- engine. Plate XVIII., Fig. 2, shows the plane of the same engine. Figs. 3 and 4 are details of the same. Same letters refer to the same parts. The principle on which the machinery acts is as follows, viz. : a is * Translated from 0. E. Carlsund's Patent (Swedish).— JST. SCREW-PROPELLERS. 103 the propeller-shaft on which the crank h is applied. On this crank are applied two direct-action steam- engines, which form an angle with each other, so that, when one of them stands on the centre, the other one acts to turn the crank, and thereby obtain the rotary motion. The proper angle which the machinery ought to form with each other, I have found to be between 90° and 128°; but it can without detriment be made more or less. The advantage which is gained by machinery of this description is, that it makes a simple and compact engine, which occupies the least possible space in the length of the vessel. It also allows a sharp section area of the vessel, and thereby diminishes the resistance area of the same. It is in this manner to place machinery in vessels in which my principal invention consists, which was accom- plished by me in 1843 — and I have since then, made several improvements, that have been applied to a great number of propel- vessels; which detailed im- provements I will here describe. To work this engine direct, and without gearing, with the velocity which is required for propellers, it has been of great importance to diminish the moving parts, weight, and dynamic momentum, for which I have applied a new sort of piston, c, Fig. 1, which consists of a strong plate of a concave form, with iron packing rings. This piston is fastened to a piston-rod d, which outer end is guided in a bearing and a con- necting-rod e, combine the piston-rod direct with the 104 A TREATISE ON crank. The air-pump /is directly combined with the piston-rod by the crosshead g, also the force-pump h, and thereby simplifies the motion, and, by the air and force-pump's resistance, diminishes and balances the machine's momentum and friction in the bearings. In consequence of the velocity and great number of revolutions per minute which this direct-acting ma- chinery must make, it demands a quite different con- struction of the air-pumps and their valves than those which have been applied in engines for paddle-wheels or propeller-engines with gearing, because otherwise the valves could not sustain the violence of the water caused by its inertia. It has, therefore, been of im- portance to construct these pumps and valves to over- come the mentioned difficulties, and has, by several experiments, so succeeded that there is machinery in this country whose air-pumps make a velocity from 300 to 400 feet per minute, and empty the condens- ing water from 70 to 120 times per minute. The manner in which I have succeeded to produce such a result is as follows : — The air-pump / (Plate XVII., Fig. 1) is single and downwards acting. The water and air which this pump, while it goes up, has taken in from the con- denser Ic, through the bottom valve I, presses while it goes down through the top valve m, in the time piston presses against the layer of air, which, when going up, has interspaces; which layer of air forms an elastic packing between the air-pump and the water, which PlateJm. Piateisim. f.^.JSfys'broin.. viat^ :xzr. 'BJ^-te.lK . JW-Ny^tram- SCEEW-PROPELLEKS. 105 is to pass through the top valve m, and thereby damp the shock of the piston against the bottom. Further, that the valves I and m may obtain a slow motion, it has a conic or parabolic form, whereby the water presses on it increasing, until it lifts up the same; this form is also suitable to the free motion of the water. Notwithstanding this form is suitable for the quick motion of the valves, there is applied a spring of brass, or any other suitable material, around the spindle n on which the valves move. This spring prevents the violent shock which otherwise would occur; and, when the piston returns, the spring presses the valve down quicker than it would do of its own weight and acceleration. Further, I have applied a reservoir p over the top valve with a discharge-pipe q. This discharging- pipe has a crooked form by which the water will meet an elastic medium, and thereby uniform the violent motion. This is the principle which constitutes my inven- tion of air-pumps and their valves, and has been ap- plied on a number of vessels. The machinery is manoeuvred as well on the deck as in the engine-room. Plates XVII. and XVIII. show in detail the manner in which this engine and air-pumps were built in Motala machine-shop (Sweden) in 1843. In the propeller-shaft, a is a hole in which is a round bolt 1, through which inner edge is a square rod 2, which latter can, when the bolt 1 moves in or out, 8 106 A TREATISE ON move itself into a groove 3, on each side of the shaft a. This rod 2 projects over the shaft a, and forms a screw, corresponding to the screw-nuts 4 and 5, on which the eccentrics 6 and 7 are fastened. The bolt 1 obtains an alternative motion from the cog- wheel 8, and the screws give the eccentrics a rotary motion on the shaft a, thereby regulating the motion of the steam-engine to stop, hack, and ahead. This cog-wheel 8 gets its motion by the rod 9, which can be manoeuvred by the engineer on deck, or in the engine-room by the handle 10. In the same manner, the engineer can manoeuvre the steam and injection-water by the rods 14 and 19, as follows : — The rod 14 is combined with the steam-valve 15, by which the steam can be regulated at pleasure, to the steam-chest through the canals 17 and 15, through which the steam can be admitted at fuU stroke or cut-oiF. The rod 19 combines the injection- cock 20, and regulates by the handle 18. This sim- ple mechanism answers all the purposes that are ne- cessary to manoeuvre the engine, and, when the steam is worked with great expansion, it makes the ma- noeuvring sure at any position of the engine, as the steam can be let through the canal 15 direct to the inner steam-valve, and, when the engine is started by moving the handle 15 a little, the canal 15 will be shut, and the engine work as usual with cut-off. Plate XIX. represents a vertical section of a steam- boat and engine (in angle) direct action. Plate XX. shows the plane of the same engine. This engine is SCREW-PROPELLEES. 107 a later improvement on the one before described. It was accomplished in 1848, and will be described as follows : — The propeller-shaft a is formed of opposite cranks h h, on which the connecting-rods are applied, each on its crank, so that when the crank stands, say verti- cally, the one engine pulls or pushes on its crank just as the other one pulls or pushes in the opposite di- rection on its crank, and thereby balance each other, from which arise the following important advantages (see Tables on pages 32, 33, &c.). 1. TTie two engines balance each other so that no counterbalance is required, and thereby a greater regularity in the moving system is obtained. 2. The powers applied to the crank eoitnteract each other, so that the greater part of the friction in the bearing is dispensed with, which is inevitable with the single crank. 3. That the two engines' dynamic momentv/m sup- press each oth&r, that the shaking and side motion of the vessel is thereby damped. The consequence of these advantages is that machinery of this construction can, without any increased friction, have a considerable shorter stroke in proportion to the diameter, and thereby ob- tain a greater number of revolutions than heretofore has been possible. Also, a greater expansion of the steam can be used, from which a greater economy of fuel is obtained. It is not necessary that these cranks 108 A TREATISE ON should be precisely opposite each other; they can be set in any angle. Also, the machinery; it is not necessary it should have the same angle as on the drawing. The condenser h also constitutes the framing and bed-plate for the engine, which makes a strong, com- pact, and simple machinery, in which are saved weight and expense. The air-pumps and also the force-pumps on this engine are of a different construction than the one before described; but their principle is the same, because they are single and downward acting, and accompanied with the same parabolical valves as before described. The pumps press the water and air through the discharge-pipe q, which is fastened to the bottom of the vessel. The machinery of this con- struction has proved, to the most favorable advantage, that it makes from 105 to 110 revolutions per minute. The air-pumps and their valves, and the whole ma- chinery work silently and even; the condensation uncommonly good. Machinery of this construction occupies the least possible room in proportion to its power. Plate XXI. Fig. 1 is a sectional drawing of a mer- chant steamrsMp, of a larger size, ivitJi a horizontal direct^action steam-engine. Fig. 2 shows the plane of the same. Fig. 3 is a section of details. This kind of engines cannot be placed in so sharp vessels as those before described, but are more suita- ble for vessels of larger size. The principle of the SCREW-PROPELLERS. 109 engine, and its operation will be described as fol- lows : — To the piston c are annexed two piston-rods d, which run over the shaft a, on both sides of the cranks h, h, to the crosshead g, which is guided in the frames. To this crosshead is directly attached the air-pump /, force-pump A, and connecting-rod i, to the cranks h h. The condenser Tc, reservoir 2^, and air-pumps, also the framing and bearings, are all cast in one pieqe. The air-pumps and force-pumps are douile action. Their valves are of the same construction as before described ; the water is forced out through the discharge^pipe q, which runs through the bottom of the vessel. (See Plate.) The one end qf the force-pumps feeds the boiler with water, and the other end for launching the vessel, and other purposes. JHarizontal direct-action engines have in other coun- tries before been used, but the air-pumps and force- pumps have not been attached direct to the cross- head, but have been geared from cog-wheels to obtain the slow motion. It is in this manner to apply air- and force-pumps direct from the crosshead, and to combine the condenser air-pumps and framing in one piece, which constitute my invention. One engine of this kind has been accomplished by me. Plate XXII. Fig. 1 is a section of a man-of-war with a horizontal direct-action steam-engine of 300 horse-power. Fig. 4 shows the plane of the same. This kind of engines are more suitable for men-of- 110 A TREATISE ON war, because the engine comes entirely below the load- line, and, therefore, inexplorable for hostile shots. This engine differs from the former one in that it has four piston-rods d d d d, of which one lays over and one under the propeller-shaft on each side of the cranks i h. The crosshead c, which combines the four piston-rods, moves in the same plane as the pro- peller-shaft, which was not the case in the former one with two piston-rods, where the connecting-rod formed a greater angle to the centre line in its lower, than in its upper, position. In this engine, those angles will be equal. The condenser, air-pump, and framing are all cast in one piece, as described in the former one, but the framing on one side of the engine contains the double-acting force-pumps cast in one piece. In the former engine, the cylinders laid on one side of the vessel, but in this one, the cylinders lay zigzag. Fig. 2 shows the double-acting force-pump h, with its valves I and m, canals for the water, and air-vessels. Fig. 3 shows the air-pump f, with its valves I and m, cistern p, and discharge-pipe q, in the framing and condenser h. The valves here described will be seen on Plate XIII. oh a larger scale.* Fig. 4 shows a new stuffing-box for horizontal and inclined piston-rods or shafts, &c. This stuffing dif- fers from the common by the ring S, which separates the packing into two parts. This ring has two * These valves are as near Carlsund as I can remember them. — N. SCREW-PROPELLERS. Hi grooves turned, on the in and outside, to leave room for the tallow and oil which is let into the cock. In the grooves are bored a number of holes, through which the oil or tallow runs into the piston-rod or shaft, and they being constantly oiled and clean. (The stuffing-box is shown on a larger scale, on Plate XII.) Plate XVIII. Fig. 6 shows the plane and eleva- tion of my invented propellers. By experience and calculations it has been confirmed that the common propellers of Ericsson, Smith, Hunter, &c., also the several different kinds of propellers which I have tried* in this country, all labor under two essential disadvantages, viz. : — 1st. That the water, by the centrifugal force, is thrown out to the periphery in the direction of the radii, in proportion as the velocity of the propeller increases, from which results a loss of effect in pro- pelling the vessel. This loss of effect increases as the resistance of the vessel increases, as in head-winds and towing, especially when the propeller has a large pitch in proportion to the diameter. And 2d. That the propeller-blades on a common propeller are too much exposed to braching, when the vessel is running in harbors and shallow water, or in narrow canals. These essential defects are remedied by the propel- ler which is here described. Fig. 6, a, the propeller-shaft ; b, its centre, to which a * Carlsund did not try any with curved generatrix. — N. 112 A TREATISE ON number of blades, c c c c c c, are fastened. These blades can be placed to form a regular screw, or ratber with an expanding pitch, so that the fore-edge of the blades only cut the water while the vessel is running its uniform speed. This expanding pitch gives the propeller-blades a parabolical form, so that, when operating, it gives the water an accelerative motion backwards, until its resistance balances the velocity and resistance of the vessel forwards. The ring or band d which circumscribes and combines the propeller-blades, I have generally made the same breadth as the length of the propeller in the direction of its centre line, but it can, without detriment, be made more or less. It is not necessary that this band shall be cylindrical; it can have a slight conical form. Also, it is not necessary that this band shall be fast- ened to the propeller-blades; it can be stationary to the vessel, and the propeller revolves in it.* This outer band has the advantage of counteracting the centrifugal force of the water, but throws the water backwards parallel to the centre line of the propeller, so that, when the propeller runs in shallow water on canals, it does not interrupt the bottom or sides of the same. That this is a fact is thoroughly tested in the canal between Stockholm and Gothen- burg. The vibration and shaking which the common propeller gives to the vessel are, by this band, entirely * Inventions of this kind have been patented in this country (America), but have not succeeded. — ^N. Plate W[. 0. E. CAKLSUND'S IMPROVEMENTS Fi^./. J.WN:fli- -LtttC-XXZT- O.E.CARLSUND'5. IMPROVEMEOTS. Fig. I. SCRE-W-PROPELLERS. 113 avoided; and the band acts as a fly-wheel to govern the motion of the machinery, which is of importance when the engine works with great expansion of the steam. This propeller has been applied on several vessels, and has proved the above-described advantages, that I hereupon declare the following claims to be my in- vention : — GlaiTns. 1st. The in angle direct-action steam-engine, work- ing on the same crank and shaft. 2d. The in angle direct-action steam-engine work- ing on two opposite cranks on the same shaft. 3d. To apply direct to the crosshead the air-pumps and force-pumps, and to use the conical or parabolical valves moving on stationary spindles with spiral springs in the closed stem, and the downwards-run- ning discharge-pipe for the condensing water. 4th. To manoeuvre the engine on deck, or in the engine room, by means of revolving the eccentrix as herein described. 5th. To use the bed-plate as condenser. 6th. The horizontal direct-action steam-engines with one or more piston-rods attach the air-pumps and force-pumps direct to the crosshead of the engine, and to use the condenser as framing for the engine. 7th. To use the outer band around the propeller- blades. 114 A TREATISE ON A TEEATISE BODIES IN MOTION IN FLUID; PRACTICAL RULES, AND EXAMPLES HOW TO CALCULATE THE RESISTANCE FOR ANY DESCRIPTION OF BODIES. Resistance to bodies in motion in fluid is a subject which has, from an early period, received attention by experiments, but the philosophy of it has not yet been presented in a well-established theory. The latest and most extensive experiments have been accom- plished by MoRiN, in France, Lageehjelm, in Sweden, and Beaufot, in England. Those experiments have been very extensive, but incomplete for the purpose they were intended ; that, even at the present time, comparatively little light has been thrown on it from experiments or theory. In the years 1811-12-13-14 and 15, extensive ex- periments were made in Sweden, conducted under the sanction, and at the expense of the Society of Iron- Masters at Stockholm, by Messrs. Lagerhjelm, Foe- SELLES, and Kallstenius. The experiments were BODIES IN MOTION IN FLUID. 115 accomplished at the Fahlu Mine, and published in two volumes, of which Assessor Lagerhjelm sent some copies of the first volume to Colonel Beaupot (in the year 1819), who was deeply interested in the subject; Colonel Beaufoy experimented, and published a work at his own expense, entitled "Nautical and Hydraulic Experiments," London, 1834. It is not to be expected that, in so small a compass as this book, will be found a complete work on a sub- ject embracing the amount of matter that bodies in motion in fluid does; and it is a subject which par- ticularly belongs to ship-building. This is a book devoted to screw-propellers, and the arrangement of their steam-engines ; yet, the two are so connected in navigatioti, that it will not be deemed out of place here to glance at it. A plane A immersed in fluid (Fig. 1, Plate XXIII.) will sustain a hydraulic pressure on each side equal to the weight of a column of the fluid, with the same base as the plane A, and an altitude equal to the depth d of the centre of the plane under the surface of the fluid. Letters will denote : — A = area of the plane immersed ; in square feet. d = depth of the centre of gravity of the plane be- low the surface of the fluid ; in feet. e = the weight of one cubic foot of the fluid in which the plane is immersed. 116 A TREATISE ON P= hydrostatic pressure in pounds, on one side of the plane, or on the fore side if the plane is in motion. J) = hydrostatic pressure in pounds on the opposite side of P. If the plane is stationary, it will be Pz=p, and the hydrostatic pressure P=Aed, (1) Example 1. Fig. 1. Suppose the plane . J. = 4 square feet, immersed to a depth . (Z = 3 feet in fresh water, we have . . . e ^ 63 pounds. Kequire the hydrostatic pressure : — P=4x63x3 = 756 pounds on each side. Fig. 2. — If the plane A moves in a direction at right angle to itself, the pressure Pwill increase on the fore-side of the plane, but the pressure p on the oppo- site side will be diminished. Suppose a plane = J., at a depth = d, moves at a velocity = v feet per second. From its centre extend two tubes, a and h, vertical over the surface of the fluid. The tubes to be open at both the ends, but, in the immersed ends, bent at right angle, so that the aperture in the -tube a is turned apposite the direction of motion, and the aperture of the tube 6 with the direction of motion, as seen in Fig. 2. Now, if the plane A moves in the direction of the arrow, the water will rise in the tube a and descend in the tube h, so that the ascension = descension over and under the water line. FlauXXUL. BOblES m MOTION IN FLUID. 117 Call tlie ascension = c, and descension = c'. The height of the column of water which resists the motion of the plane will consequently he P=d + c, and the height of the column which acts with the motion will be p = c? — d; then we have for P and p the pressures P= Ae{d + c), (2) 'p = Ae{d—c'), (3) The force R, which gives the plane the motion, will consequently be the difference of P and p, or B^P-P, (4) Example 2. Fig. 2. — Suppose the plane A moves at a velocity = v, so that the ascension c ^ 1 foot, and, also c' = 1 foot, -4. = 3 square feet, d = h feet, e = 63 pounds. What will be the two pressures Pandj)? P= 3 X 63 (5 + 1) = 3 X 63 X 6 = 830 pounds, 23 = 3 X 63 (5 — 1) == 3 X 63 X 4 = 756 pounds, and the force R = 830 — 756 = 74 pounds, which is the force which gives the plane the motion. By insertions of the formulae (2) and (3) in (4), we will obtain R = Ae{d + c)—Ae {d — c'), but it is evident that c = d, therefore B = Ae1c, (5) According to the force of gravity, the ascension c is such that if a body falls freely through the space c, it will obtain a velocity = v, which is equal , to the velocity of the' force Pand the plane A, and known by the formula 118 A TREATISE ON c = f, (6) in which ^ = 16.08 feet, the space which a body falls in the jfirst second, and v = velocity in feet per second. Example 3. Fig. 2. — The plane A moves at a ve- locity of 6 feet per second. What will the ascension cbe? c = — = -1^ = 0.56 feet. 4 X 16.08 67.32 By insertion of the formula (6) in (5) we obtain B = ^, (7) Example 4. Fig. 2. — What will be the resistance to a plane J. = 4.5 square feet, moves at a velocity v = 8 feet per second, in fresh water e =: 62.5 pounds? „ 4.5 X 62.5 X 8*^ ..n a E = — - — =-r-7r7; — = 560 pounds. 2 X 16.08 ^ This is a true calculation with neglected circum- stances, namely : If there was no atmospheric pres- sure on the surface of the water, or if there were a number of tubes that would let down air to every part of the back side of the plane. As such is not the case, we must suit our formulae and calculations to the circumstances, and find how the atmospheric pressure will affect them. It is evident that, if the velocity is -y = o, the pres- sure P andp will be as described in Fig. 1 ; but, if the velocity is = v, the pressure B and ascension c will BODIES IN MOTION IN FLUID. 119 still follow the same law as described in Fig. 2 ; but the pressure p will differ so that when the velocity reaches v = V4:g {d + 32.92), ... (8) (in which d = depth, and 33.92 = the height of column of water which balances the atmosphere) the pressure p will be equal to o. Example 5. — At a depth of 6 feet, what velocity is required to make the pressure p ^ o ? V = v/4 X 16.08 (6 + 32.92) = 50 feet, or about 34.4 miles per hour, a velocity not often ex- ceeded by bodies in motion in fluid. In the following formulae, we will suppose that the velocity v does not exceed the formula (8), then we have the extremities of the pressure j) at 1 and o, just as the velocity is o and v. Whatever the pressure p may be, it will balance a column of water less than the depth d, if the velocity is greater than o, ov v>o. If the tube h be turned so that the aperture stands parallel with the motion and close to the back side of the plane, it is probable it would indicate a true de- scension c' for the pressure r, so that d -.c^d: {d+ 32.92), of which '''-^T32:92' (^^ Example 6. Fig. 2. — What will be the descension d, at a depth cZ = 8 feet, ascension c = 6 feet? 6x8 48 8 + 32.92 "40.92 = 1.175 feet. 120 A TREATISE ON By insertion of the formula (9) in (3), we obtain rrom the formula (6) we have c — — and ^-^'<^n,(.:'32.92))'- ■ '"^ P.Ae{d + £),. . . . (12) and E = P—p, we have B = Ae(d + —\ — A ed(l + - - V \ 4.g) \ 4^ ((^+ 32.92)/' that is R = Aev^ /-, . d i^^JT^)' ■ ■ (") Ag \ d + B2.92' This should be the true formula for calculating the resistance to planes in motion in fluid, but it is found that the resistance will be a little more owing to some air and friction in the water which diminishes the pressure p. Then, to make the formula durable in practical use, the term 32.92 must differ, and is found to be about 28. In the accompanpng table, the term s d ~d + 2S is calculated at different values of d. BODIES IN MOTION IN FLUID. 121 Table I. Depth of the ^ = .- d +■28 moving body be- low the surface of the water. Feet. 3 inches. 6 inches. 9 inches. 0.0000 0.00885 0.01755 0.02603 1 0.0345 0.04275 0.0507 0.0587 2 0.0666 0.0744. 0.0819 0.0894 3 0.0968 0.1041 0.1113 0.1184 4 0.1255 0.1321 0.1388 0.1460 5 0.1520 0.1583 0.1645 0.1760 6 0.1770 0.1880 0.1885 0.1945 7 0.2000 0.2057 0.2116 0.2170 8 0.2226 0.2280 0.2331 0.2384 9 0.2436 0.2486 0.2536 0.2585 10 0.2638 0.2683 0.2730 0.2774 U 0.2828 0.2870 0.2914 0.2361 12 0.3000 0.3041 0.3085 0.3130 13 0.3175 0.3215 0.3252 0.3290 14 0.3338 0.3373 0.3416 0.3452 15 0.3490 0.3530 0.357 0.3600 16 0.3645 0.3674 0.3713 0.3750 17 0.3780 0.3813 0.3850 0.3885 18 0.3922 0.3948 0.3982 0.4015 19 0.4050 0.4075 0.4110 0.4140 20 0.4175 0.4200 0.4230 0.426 21 0.429 0.432 0.435 0.438 9.9. 0.441 0.443 0.446 0.449 23 0.452 0.454 0.457 0.460 24 0.462 0.464 0.467 0.470 25 0.473 0.475 0.477 0.479 26 0.482 0.485 0.488 0.490 27 0.492 0.495 0.497 0.499 28 0.501 0.503 0.506 0.508 29 0.510 0.512 0.514 0.516 30 0.5185 0.520 0.522 0.524 31 0.526 0.528 0.530 0.532 32 0.534 0.536 0.538 0.540 33 0.542 0.544 0.546 0.548 34 0.550 0.552 0.554 0.556 35 0.556 0.559 0.560 0.562 36 0.563 0.565 0.567 0.568 122 A TREATISE' ON In the following formulge we will substitute the quantities 8, instead of the term --. Also, a co- efficient k instead of the symbols e and g, so that e f 1 for salt water, k = -— := 4: g I 0.97 for fresh water, then, when the plane or body moves in salt water, the co-efficient k will not be seen, because then ^ = 1, and the formula for the resistance will be simply Bz=Av^k{l + 8), . . . (14) Example 7. Fig. 2. The -plane Moves at a velocity . At a depth of 6 feet In fresh water . Require the resistance -4 = 7 square feet V = 16 feet per second 5 = 0.231 ^ = 0.97 -B = ? in pounds. ig = 7 X 16** X 0.97 (1 + 0.231) = 213.5 pounds. Example 8. Fig. 2. — What will be the resistance to the plane A when the dimensions are Resistant area . . . J. ^ 1 square foot Moves at a velocity . . v = 10 feet per second At a depth of 3 feet . . h = 0.0968 In salt water . . . kz=\ Require the resistance . i2 = ? in pounds. i?= 1 X 10' (1 + 0.0968) = 109.68 pounds. The same plane, moving at the same depth, with velocities ■u = 5 the resistance will be i2 = 27.3 pounds. v = 2 " " i2 = 4.37 " v = l " " i? = 1.09 " BODIES IN MOTION IN FLUID. 123 Example 9. Fig. 2. — Dimensions of the plane being Kesistant area . . . J. = 3 square feet Moves at a velocity . . v = 12 feet per second At a depth of 6 feet . . S = 0.177 In fresh water . , . A = 0.97. Eequire the resistance . i? = ? in pounds. iB = 3 X 12^ X 0.97 (1 + 0.177) = 493 pounds. Fig. 3. — If the plane has an inclination to the direction of motion of an angle = $ degrees, as shown in the figure, when not in motion, the pressure at right angle to the plane will be the same as described in Fig. 1, but the pressure in the direction of motion will be the total pressure multiplied by sin.^, which is the same as the area projected parallel with motion, was the acting area for the resistance. In the follow- ing we will call A = area projected parallel with the motion. 4> = angle of resistance to the motion. ^ = angle of incident /rom the motion. The ascension and descension in the tubes a and h will also be as the sines for their corresponding angles of resistance and incident, so that the pressures Pand p will be r, A fj , v^sin.^\ /^j;^ P=Ae{d + -~-j, . . . (15) , /, iy^sin.d)8\ ,-,g^ p = Ae{d T^h • • • (^^) 124 A TREATISE ON When the two angles are resistance will be B^Ahv' sin.^ (1 + 5), . . . Example 10. Fig. 3. — Suppose the plane has An inclination of . . ^ = 42° ^, the formula for • • (17) Projected area . Having a velocity At a depth of 8 feet In fresh water . Kequire the resistance B A = 6.45 square feet w = 5 feet per second h = 0.222 in the table h = 0.97. ? in pounds. R = 6.45 X 5' X 0.97 x sm.42° (1 + 0.222) = 127.7. Example 11. Fig. 3. — The plane has An inclination of . . = 30° Projected area . Having a velocity At a depth of 6 feet In salt water J. = 2.97 square feet v = 6 feet per second h = 0.177 in the table Jc = l Require the resistance i2 = ? in pounds. B = 2.97 X Q^sin.SO° (1 + 0.177) = 62.7 pounds. B= sin.20° =42.8 " B = sinM° = 108.6 " Fig. 4. — If the plane has different angles of resist- ance and incident, the formula for resistance will be B==- AJcv' (sm.O + sin.

= 18° Moves at a velocity . . w = 9.5 feet per second At a depth of 8.25 feet . h = 0.228, Table I. In fresh water . . . ^ = 0.97 BODIES IN MOTION IN FLUID. 129 Eequire the resistance . i2 ■= ? in pounds. B = ix 0.97 X 9.5^ {sm.lS° + 0.228— sm.*18°cos.l8°) = 184.5 pounds. Exam'ple 15. Fig. 6. — "When the dimensions are Angle of resistance . . v/i>. i^:-. '•;.-«■--■■.■• ■ , , ; ,.^.>-. '^"- .1 •i -..ft. ;; , ; jir%. /A Fiy./7. JWmstron. BODIES IN MOTION IN FLUID. 133 of all those roughnesses a a a, or -; but the number s of roughnesses in the direction of motion comes in contact with the same particles of water (fluid) which have obtained a velocity by the foregoing roughness, and do not act in full as resistant on the latter ones; therefore, the square root must be extracted from the number of roughnesses in the direction of motion; so that the total resistance for the plane, or what in this case is called the friction, should be measured by the formula f=^iB + c), .... (26) s VI or, by insertion of the symbols e and g, we have f= V' ih + c), . . . (27) VI s 4, g To ascertain the two quantities s and c only from theory is rather a difficult matter; we will, therefore, allude to experiments and approximate those values only for water, which is the principal fluid for which this is intended, and is found to be about s = 580, when the surface of friction is wood smooth planed, and been in the water for some time, and slime has collected on the same. For the same kind of friction surface, the cohesion is about c = 2 feet, the altitude of a column of water that would balance 134 A TREATISE ON the cohesion. Then / will be expressed in pounds by the formula /=-?-^ (5 + 2), . . . . (28) •^ 580 s/r ^ ' which formula gives the friction as near as may be desirable for practical purposes. a = friction area, which is parallel to the motion, dead area. Fig. 14 The plane having an inclination io and from the direction of motion. When the plane has an inclination to the direction of motion, the angle of resistance of the roughness will increase as the angle of inclination; but, when the inclination is from the direction of motion, the angle of resistance of the roughness will be diminished, say /= 1 when the friction plane is parallel to the motion. Then /= 1 + sin.^^ cos.^ friction on the plane inclined to the direction of motion, and /= 1 — sin?^ cos.^ will be the friction on the plane inclined from the direction of motion. Example 21. Fig. 7. — Suppose the plane a to be 1 2 feet 6 inches long, and breadth 1 foot 9 inches, the edges being so sharp that no resistance to them exists. Friction area a=2 x 12.5 x 1.75 = 43.7 square feet Moves at a depth . . d! := 5.5 feet Co-efficient . . . 5 = 0.164 BODIES IN MOTION IN FLUID. 135 With a velocity . . ■w = 7.66 feet per second Kequire the friction . /= ? in pounds. /= 43.7 X 7.66^ ,Q -^g^ + 2) = 0.27 pounds. 580 v/12.5 ^ Example 22. Fig. 16. — What will be the friction on a board of the same dimensions as in Example 21, but moves in the direction of its breadth, as repre- sented by Fig. 16 ? /= 43.7 X 7.66^ ,Q_-^g^ + 2) = 0.72 pounds. 580v'1.75 Example 23. Fig. 17. — What will be the friction on a board moving in a direction as represented by Fig. 17. Dimensions: — Length . . . . =10 feet 3 inches Breadth . . . . =1 foot 8 inches Friction area a^l0.25x 1.666x2 = 34 square feet Moves at a velocity . . -y = 9 feet per second At a depth . . . . (Z = 10 feet Co-efficient .... 5 = 0.264 The angle being . . . ^ = 38° Kequire the friction . . /= ? in pounds. 3_4_x_9^264^J} _ ^_^ ^^^^^ 580v/1.666 X sec.38° Example 24. Fig. 6. — To find the friction on the dead areas of the two triangles a e g and fih, the figure having the same dimensions as in the Example 14, and moves at the same depth and velocity. ag = 2.675 feet me = 4.15 " 136 A TREATISE ON Friction area a = 4.15 x 2.675 = 11.1 square feet. The length for friction on a triangle will be only half its length in the direction of motion, or ^ _ m6 _ 4^ _ 2.075 feet. Kequire the friction/^ ? in pounds. /= jLi-iil^ (0.228 + 2) = 2.68 pounds. 580 v/2.075 Fig. 6. — When the friction area has an inclination to the direction of motion, as represented by Fig. 6, the formula for its friction will be f a v^ (1 + sm.® coss^) (2 + h) /o(^^ ■ ~ 580^^7! ' in which a = area of the whole inclined plane multi- plied by cos.'^, the angle of inclination to the direction of motion, which, in this Fig. 6 will be a = cos.<^ (aeif+egJii) =eiom = meX e i. Example 25. Fig. 6. — To find the friction on the inchned planes a e 'i/ and eg hi. The figure has the same dimensions as in Example 14, which will be The length . . l = m e = 4.15 feet The height . . . e i = 1.5 feet Friction areaa = m exeix2 = 4.15 xl. 5x2 = 12.45 square feet At a depth of 8.25 feet . 8 = 0.228 Moves at a velocity . . v = 9.5 feet per second Angle of resistance . . <> = 38° Require the friction . . / = ? in pounds. BODIES IN MOTIOK IN FLUID. 137 .^ 12.45 X 9.5^ X (1 + sin.' BS x cos.SS) (2 + 0.228) _ ~ 580 v/ilS "~ 2.75 pounds. Then the actual resistance to the Fig. 6 will be the sum of the resistance B, and the two frictions on the dead and inclined friction areas, from Example 14 resistance il = 184 pounds " 24 friction /= 2.68 " " 25 " /==2.75 " Actual resistance i2 = 189.93 pounds. Fig. 7. — The inclined friction area having its incli- nation from the' direction of motion, the formula for its friction will be 580 N/r > - ^ ) Example 26. Fig. 7. — To find the friction on the planes ah c d and a d e f, the figure moves in the direction m n, shown by the arrow. The dimensions of the figure to be the same as in the Example 16, viz: — Angle of incident . . ^ = 25° The length mn . . . Z = 4.29 feet The height . . . . ad = % feet Friction area a' = 4.29 x 3 x 2 = 25.74 square feet At a depth of 8 feet . . 5 = 0.222 Velocity . . . . t; = 6 feet per second Require the friction . . /== ? in pounds. 10 138 A TREATISE ON . _ 25. 74 X 6\l—dn.^25°XGos.25 ) (2 + 0.222) ^i^^ 580 v'i:29 Fig. 10. — The moving figure having friction areas to and from the direction of motion. The formula for friction on both the inclined planes will be the sum of the two formulae (30) and (31), which will appear in one formula, as /= — ^ J- a (1 + sin.^^ cos.<^) 580 W L ' a + (1 — sin.^^ cos.^)~\, . . . (32) in which I = length of the whole figure in feet, a = friction area to the direction of motion, a' = friction area/rom the direction of motion. It will not be correct to calculate the friction from the two formulae (30) and (31) separately and add them together, owing to the quantity I. To simplify the calculations and setting up of the formula (32), it will be best to separate the two terms within the parenthesis as follows : — J = a (1 + sin.^c^ cos.^), . . . (33) g'' = a'(l — sin?^cos.^, . . . (34) 580 W ^ ^ Example 27. Fig. 11.— What will be the frictions on the inclined planes, the figure having the same dimensions as in the Example 19? BODIES IN MOTION IN FLUID. 139 The height Length. . da = 0.75 feet ae = 1.87 feet el=. 2.35 feet ?= 1.87 + 2.35 = 4.22 feet Friction area a'=2.35 x 0.75 x 2 Friction area a=1.87x 0.75x2 : Angle of resistance . . : Angle of incident Moves at a velocity At a depth of 6 feet Require the friction : 3.52 square feet : 2.28 square feet 15° V, the actual resistance will be L 180 \^7 ^7'/ J /' = : a -y P, (2 + h), (37) 180 ^/Z Example 30. Fig. 18. — What will be the actual re- sistance to this figure at the four diflferent lengths ? Example 1. Z= 5feetZ' " 4. Z=20 " >V The figure to be moved in salt water at A velocity . . . ■« = 10 feet per second A depth . . . . d=<6 feet Its co-efiicient . . . 5 ^ 0.177 The side . . , . s = 1 foot square Resistant area . . . J. = 1 square foot. Ex. 1. Friction area a = 4x5 = 20 square feet. i2 = 1 X 10^ (1 + 0.177) = 117.7 pounds. . j,^ 20 X 10^ ^Q .^^^ ^ ^^ gg ^^^^^^ 180 n/5 B = 117.7 — 10.85 = 106.95 pounds. Ex. 2. Friction area a = 4 x 10 = 40 square feet. B = 117.7, same as Ex. 1. ^^ ^ 1£ (0.177 + 2) = 15.35 pounds. 180 n/10 i2 = 117.7 — 15.35 = 102.35 pounds BODIES IN MOTION IN FLUID. 145 Ex. 3. Friction area a = 4 x 15 = 60 square feet Proper friction area 'a=4x 10.16 =40.64 square feet Proper length V = 10.16 feet. Require the actual resistance from the formula (36) ? i2=10f 1 + (2:lII±^(iL_i^^)1=103.32. L 180 \^r5 N/iole^-l Ex. 4. Friction area . . a = 4 x 20 = 80 Proper area . . . . a' = 40.64 Proper length . . . .^ Z' = 10.16 Require the actual resistance R ? i2=10f 1 + (^:l!I±^(^_i£;Et)l=106.16. L 180 Vv20 n/10.16^-' Example 31. — What will be the actual resistance to a parallelepiped of dimensions Length . . . . Z = 12 feet Breadth . . - . & = 1.5 feet Thick t= 0.75 feet Resistance area A = 1.5 x 0.75 = 1.125 square feet 6 Z= 2 X 1.5 X 12 = 36 tl=2x 0.75 X 12 = 18 Friction area . . . a = 54 square feet Move at a velocity . . v = 9 feet per second At a depth . . . . d= S feet In salt water . . . 7c = l. ?L = JL = 48. A 1.125 Proper area at 8 feet a' = 41.4 x 1.125 = 46.6 < 48, 146 A TREATISE ON consequently, the actual resistance to be calculated from the formula (37), -o 1 ^u 7/ 10.35(1.5 + 0.75) TT .- Proper length V = i— '- = ll.oo. Require the actual resistance = ? in pounds. i^ = 9f 1.125+ (M^(ii__i££-)1 L 180 V^12 y/llM'-^ = 95.6 pounds. Example 32. — What will be the resistance to a cylinder with dimensions Length . . ' . . 1 = 17.5 feet Diameter . . . . Z) = 1.33 feet Resistant area^ = 1.33^x0.785 = 1.39 square feet Friction areaa=1.33x 3.14x17.5 = 73.1 square feet Moves at a velocity . . ■?; = 12 feet per second At a depth . . . d = i feet Proper area at 4 feet a'=36.2 x 1.39 = 50.3 square feet a 73.1 A 1.39 = 52.6> 50.3, Proper length V = — ' , „„ = 12 feet ^ ^ 3.14 X 1.33 In fresh water . . . h= 0.97. Require the actual resistance i? = ? in pounds. i2=12^x0.97ri.39 + MII±l).(_IM__^3v-| = 200 pounds. Fig. 17. Plate XXVIII. — A cube moving in a direction as shown by the arrow, with one of its edges foremost, its angle of resistance is evidently $ = 45°. BODIES IN MOTION IN FLUID. 147 Example 33. Fig. 17. — Suppose the side of the cube to be 2 feet, we have the Area of resistance J. =2>/2 + 2^ = 5.656 square feet y, and the column w is the circle arc a 6 in degrees. When the centre for the circle arc a 6 is on the line x, its corresponding angle of resistance will be found in the column x. In the. figures 19, 20, and 21, the circle arc a 6 is 90°, and its centre on both the lines x and y. Oppo- site 90°, in the column w°, will be found 39° 34', in the columns x and O y, which is the mean angle of resistance for a circle. PlatcXXVm Fig.n Fig. IB. Fig.tt. Fig.S.3. -<*--—; ■^r-Jt' 'f.\e_ ^<-' f J.W.JfystroBK BODIES IN MOTION IN FLUID. 149 Table III. X w° y (|) X w° y^ 0° 58' 1 89° 6' 22° 48' 46 63° 36' 1 28 2 88 12 23 16 47 63 02 1 58 3 87 41 23 43 48 62 28 2 28 4 87 09 24 12 49 61 54 2 58 5 86 40 24 38 50 61 10 3 27 6 86 08 25 4 51 60 36 3 58 7 85 33 25 32 52 60 02 4 27 8 85 00 25 57 53 59 28 4 57 9 84 27 26 24 54 58 54 5 26 10 83 44 26 50 55 58 20 5 57 11 83 10 27 16 56 57 48 6 26 12 82 36 27 42 57 57 14 6 57 13 82 02 28 58 58 56 40 7 26 14 81 88 28 34 59 56 16 7 57 15 80 54 28 59 60 55 31 8 26 16 80 20 29 25 61 54 57 8 57 17 79 46 29 50 62 54 23 9 26 18 79 12 30 15 63 53 49 9 56 19 78 38 30 40 64 53 16 10 25 20 78 03 31 4 65 52 45 10 55 21 77 29 31 28 66 52 12 11 25 22 76 55 31 52 67 51 39 11 54 23 76 21 82 15 68 51 06 12 23 24 75 47 32 32 69 50 32 12 53 25 75 13 38 02 70 49 58 13 22 26 74 41 38 25 71 49 25 13 51 27 74 06 83 49 72 48 52 14 20 28 73 35 44 13 73 48 19 14 49 29 72 00 34 34 74 47 46 15 18 30 72 25 34 57 75 47 12 15 46 31 71 52 85 18 76 46 40 16 16 32 71 19 35 40 77 46 08 16 45 33 70 47 36 00 78 45 36 17 14 34 70 15 36 20 79 45 04 17 42 35 69 43 36 40 80 44 32 18 11 36 69 11 37 00 81 44 00 18 40 37 68 37 37 19 82 43 29 19 8 38 68 17 37 38 83 42 58 19 35 39 67 35 37 55 84 42 28 20 3 40 66 58 38 12 85 41 85 20 29 41 66 25 38 29 86 41 28 20 58 42 66 52 38 46 87 40 59 21 26 43 65 23 39 06 88 40 30 21 53 44 64 44 39 20 89 40 02 22 20 45 64 10 39 34 90 39 34 150 A TREATISE ON Fig. 19. — A cylinder moving in a direction parallel to its bases (which are supposed to be at right angle to its axis), angle of resistance is only $ = 39° 34'. Example 35. Fig. 19. — Suppose the cylinder to have the length equal to the side of the cube in the Examples 33 and 34, and a diameter equal to circum- scribe a square of the cube, then its resistant area will be the same as in the Example 33, Which is . . . . A = 5.656 square feet Angle of resistance Moves at a velocity At a depth of 6 feet Require the resistance $ = 39° 34' V = 8 feet per second 5=0.177 i2 = ? in pounds. E = 5.656 X 8^ X sm.39° 34' (1 + 0.177— sm.^39° 34' X cos.39° 34') = 225.3 pounds. Figs. 20 and 21. — The cylinder being cut in two equal parts, parallel through its axis, the one part to be moved with the square side forward, and the other with its hemisphere forwards. Example 36. Fig. 20. — The half-cylinder to have the same dimensions as the cylinder in the preceding example, moving at the same depth and velocity in the direction of the arrow. Require the resistance B = t i2 = 5.656 X 8' (1 -f sin. 39°34' x 0.177) = 401 pounds. Example 37. Fig. 21. — Require the resistance when moving with the hemisphere forward, and having the same dimensions as before. BODIES IN MOTION IN FLUID. 151 R = 5.656x8^ (s^•J^.39° 34' + 0.177 — siWSQ" 34 x cos.39° 34') = 248.5 pounds. Fig. 22. — The centre c for the arc a h being on the line 9/, and its angle to° = a c b', its angle of resistance in the column y . Example 38. Fig. 22. — Suppose the Angle of the arc is . Then angle of resistance Area of resistance . Moves at a velocity At a depth of 5 feet In salt water . Require the resistance w° = 40° . Exam'ple 39. Fig. 23. — Suppose the figure moves in the direction shown by the arrow, and the Angle w° = 43° resistance . $ = 65° 23', Table III. Angle w = 32° incident Area of resistance Moves at a velocity . At a depth of 6 feet 6 in. In salt water Require the resistance 4) = 71° 19', " J. = 5 square feet «? = 10 feet per second 5=0.1885 h= 1 i2 = ? in pounds. 152 A TREATISE ON i2 = 5 X 10' (sm.65° 23' +sm.71° 19' x 0.1885 — sira.*65° 23' X cos.65° 23') = 401.5 pounds. Eocample 40. Fig. 23. — The figure to be moved in an opposite direction of the arrow, and having the same dimensions as in the Example 39, and moves at the same depth and velocity, then Angle of resistance . . . . Example 41. Fig. 24. — The figure to be moved in the direction of the arrow vith dimensions — The angle of the arc . Angle of resistance and in- cident . Resistant area . Moves at a velocity At a depth of 6 feet In fresh water . Require the resistance w ^ 28° O = 14° 20' -A = 14 square feet ■y = 12 feet per second h = 0.177 h = 0.97. -B = ? in pounds, from the formula (23). i2 = 14 x 12' X 0.97 X 5m.l4° 20' (1 + 0.177 — sm.''14° 20' X cos.l4° 20') = 559 pounds. Fig. 25. — The centres for the arcs a h and a h' being both on the line x, but different angles w". The FlaielXIX. J\N ^t/firom- BODIES IN MOTION IN FLUID. 153 angle of resistance and incident to be foUnd in the column X <^. The resistance from the formula (21). Example 42. Fig. 25. — The figure moves in the direction of the arrow with dimensions — Angle w° = 35° resistance . ^ = 17° 42' Angle w = 28° incident . 4) = 14° 20' Resistance area . . . J. = 10 square feet Moves at a velocity . . ■« = 15 feet per second At a depth of 6 feet . . 5 = 0.177 In fresh water . . . lc= 1 Eequire the resistance . ^ = ? in pounds. i2== 10 X 15^ {dnl1° 42' + 8in.U° 20' x 0.177 — otV17° 42' X cos.l7° 42') = 763 pounds. Example 43. Fig. 26. — The figure moves in an opposite direction to the arrow, and having the same dimensions as in the Example 42, and moves at the same depth and velocity, then Angle of resistance , . .:= 14° 20' Angle of incident . . . 4) = 17° 42' Require the resistance . . iE = ? in pounds. i2 = 10 X 15** (sin.l4° 20' + sml1° 42' x 0.177 — sm.*14° 20' X cos.l4° 20') = 669 pounds. The length a a' can be found by the angle w and breadth h V. aa' :=l length of the figure, J 6' = 5 breadth of the figure, ae=zTi height of the figure, ac=zr radii of the circle arc. 11 154 A TREATISE ON Then A = b h, and s^n.w = ^=.^-, . . . (39) By solving this formula in favor of l, we have dn.w {P + W) = 2l h, sin.w r + sin.w W = 2 lb, sin.w V — 2 Z 6 = — miM W, 2lh sin.w 111 . W sin.w sin?w dn?w \ sin.w/ \sm.w / sin.w \ sin.w lastly Z = 5(J_+ M 1\ . . (40) \sin.w \ sin.w I Example 44, Fig. 25. — Suppose The angle of the curve a & is . . vP = 35° And the breadth hV . . . 6 = 4 feet Require the length . , . . a a' = ? in feet ^ '^ = 9 (--^o +J • L.o — l) = 6.34 feet, The angle of the arc 5 a' . . . w = 28° BODIES m MOTION IN FLUID. 155 Breadth h b' 6 = 4 feet, da'=-( ^ + I i iWsfeet. The whole length a a' = 6.34 + 8 = 14.34 feet. Example 45. Fig. 25. — The figure is to have the same dimensions as in the preceding examples. What will be the friction on the cuiVed sides. The height a e = —=-— = 2.5 feet. 6 4 Length .... Z = 14.34 Friction area a = 2x6. 34 x2. 5 = 31. 7 square feet Friction area a' = 2x8x2. 5= 40 square feet Angle of resistance . . O = 17° 42' Angle of incident Moves at a velocity At a depth Co-eflScient Eequire the friction ^ = 14° 20' v==16 feet per second d==& feet 5 = 0.177 /== ? in pounds. q = 31.7 (1 + 8in.m° 42' X eos.l7° 42') = 34.4, q' = 40 {l—sinJ'U^ 20' x cos.l4° 20') = 37.6, 15- (2 + 0.177)_ (34.4 + 37.6) ^ ^g ^^^^^ 580 n/14.34 Example 46 . Fig. 26. — The figure having the same dimensions as in the preceding example, but move in an opposite direction. Friction area . . . a = 40 square feet Friction area . . . a' = 31.7 square feet Angle of resistance . . O = 14° 20' Angle of incident . . 4) = 17° 42' Fig. 25 Fig. 26 156 A TREATISE ON Require the friction . . /= ? in pounds. ^ = 40 (1 + sinni° 20' x cos.U° 20') = 42.3, 2'= 31.7 {l — 8in.n7° 42' X cos.l7° 42') = 28.9, j^ 15^ (1 + 0.177) (42.3 + 28.9) ^ ^^ g 580 s/UM r Example 42 i2=763 Example 45 /= 16 .Actual resistance E = 779 pounds. Example 43 E = 669 Example 46 /= 15.9 .Actual resistance E-= 684.9 pounds. Fig. 27. — A force R being applied between two planes A and M immersed in water at a depth d. It is evident the planes will obtain velocities so that their resistance will be equal to the applied force E, or V = velocity of the plane A, M= velocity of the plane N. We have E = Ak v" {1 + S) = Mk M" {1 + S), . (41) and Example 4 7. Fig. 27. — Dimensions of the planes are J. = 6 square feet JS'= 11 square feet The force applied . . , ^ = 483 pounds At a depth in salt water . . tZ = 6 feet BODIES IN MOTION IN FLUID. 157 Co-efficient 5 = 0.177 Eequire the velocities . . v = 1 and M= 1 = 4; = 8.3 feet per second, 16 (1 + 0.177) M= 8.3 — = 6.1 feet per second. Fig. 28.^ — A force B being applied between the plane A and tlie figure j^, whose angles of resistance = h—dn.^^cos.(^) , S \/A (1 + 5) = ■>/ JS' {dn.^ + dn.^ 8 — dn.*^ cas.4>)- S -K^ISl {dn.^ 4- sm.^S — dn.*^ cos.^), /]Sl {dn.<^ + dn.^ S — dn}^ cos.cE>) y/A (1 + 5) + VM{dn.^ + dn.^ h — dn^^cos.iS?) (46) For convenience, in setting up this formula and calculating, it will be better to substitute other cha- l6s A TEEATISE ON racters for the quantities under the root marks; for instance — = ^/^ (sin.<|) + sin.^h — «*».*$ cos.^), (47) 0=v/2~(r+iy, .... (48) Then the formula for the slip will appear as S= e (49) + ©' ' Example 48. Fig. 28. — The figures having dimen- sions — Projected area . . , M = 39 square feet Area of the plane Angle of resistance Angle of incident Moves at a depth Co-efficient Kequire the slip J. = 18 square feet 4) = 2P ^ = 18° cZ = 6 feet 5 = 0.177 8=1 per cent. © == n/39 {sin.21 + sm.l8 x 0.177 — 5^*21 x cos.21) = 3.93, O = %/18 (1 + 0.177) = 4.52, 3.93 S= = 0.462, or 46.2 per cent. 4.57 + 3.93 Fig. 28. — Suppose the figure to have the angle of resistance $ = ^ incident, and the curved side a' }> a being an arc of a circle with a radii V . I' r = — - 4- — (50) Suppose the figure to be immersed so that the plane ah a' h' is level with the surface of the water, and Ir = draft of water, BODIES IN MOTION IN FLUID. 159 d = depth of the centre of gravity of N under the surface of the water. In this figure, the projected area ® is a parallelo- gram, and the draft of water N and ^=v Another imaginary breadth for the angle w° is This value of b is to be inserted in the formula (50). ^ = 01-^ 2b' • • • • (^'^ The area of the load-line aba' b', will be found by the formula a = .Z(r-|), . . . (52) 90 V 2> This, multipUed by the draft of water, will be the cubic contents of the figure; but, before inserting the correct draft of water, we must prepare it to suit our next coming irregular figures, and will be i'-?^, w This, multiplied by the formula (52), will be the cubic contents of the figure 90 b 6 b V 2/' ■ ^ ^ By solving this formula in favor of w°, we obtain 160 A TREATISE ON the angle of resistance from the cubic contents, and the greatest immerse section, and 2dWnr'w^fj_^2dmi (-!> 90 i 6 ~ lr6 2^^7t?^«J°=C90b& + 90x 2dm(r—^, (7 90ir6 90 Z/ h'\ This is the formula by which the angle w° for a vesser should be calculated, with the supposition that the cubic contents of the displacement should be near the formula (53). As such cannot always be the case, we must make the formula more dependent on the displacement. Formula (54). — Multiply the terms by the factor 90 5 we have 90Cb6 90Z 90(^6? qij — _L. ^^_ of which _ 90_& / (7b _ <^\ , 90 ^ Ttr'yidM b / 7t r The first term of this formula is only a small fraction of the angle w°, which, without detriment, can be taken out, and the remainder will be 90 Z ,, w= , (c) PlateMX. BODIES IN MOTION IN FLUID. 161 Here the length I must be dependent on the displace* ment exceeding the formula (53). In the common reasonable proportions of steamboats, the displacement is generally in the neighborhood of the formula (53) ; then the angle w will be very near 200 Cd ,... "=-nrF' (^^) In which are contained the following three supposi- tions : — 1st. The vessel to have no hollavo lines. 2d. The greatest immerse section to be in or near the middle of the length I. 3d. The lines to be about equal fore and aft. If the displacement of the vessel is given in tons (of 2240 pounds) multiply the ^ , (■ 34.9 for salt water 1 ^ -,. , tons = Q by { ^^ ^ ^^^ ^^^^^ ^^^^^ | = C displace- ment in cubic feet. The formulae and calculations heretofore treated are based on that JResistance to iodies in motion in fluid is direct as the sine for the angle of resistance. This gives the resistance but little more than actually occurs in practice. When the figures, or resistant-planes, are of a curved or round form, at right angle to the motion, as a cone moving with its vertex forward, the water being thrown out in every direction, which again diminishes the resistance nearly as the sine for the angle of inclination, or resistance to a cone will be 162 A TREATISE ON nearer as the square of the sine for the angle of inclinor tion. A well-constructed vessel can be compared with a cone, as it is expressly suited to the motion of the water. Then the formula (44), (45), (46), and (47) will be — E=Akv' {l+8) = ]SJcM'' sin?^ (1 + h—sin.^^ cos.^) , (56) B=A8^{1 + 5)= JSf (1—^) sin?^ {l+h—sin?c^cos.^), (57) » „_ sin.O s/M {\ + h — sin}<^ cos.^) /ggs ~ '/A{l-\-h)+dn.^ VM{l + h— dn?^ cos.^) ' and © = dn.f^ x/Mil + h — dn?i^ cos.^), . (59) To this formula (§9) must be added the friction. We have before supposed the vessel to have the angles of resistance and incident about equal, then the fric- tion areas a and a' will also be equal, and the for- mula (32) will appear as /= A?^ (S + 2 + t^J±^_^\ (60) 580 s/7^ ^g /' ^ -' The term within the parentheses, v^ dn^^ cos.^ is only a small fraction of the friction f, which, with- out detriment, can be relinquished, and the friction /=^ 1^(5 + 2), .... (28) 580 s/Z in which the area 2 a should be calculated from BODIES IN MOTION IN FLUID. 163 2a = 2^(^+4b% (61) This value of 2 a inserted in the formula (28) will be ■ (62) ^~ 1160 b ' which should be the friction of the vessel in pounds, and (63) m_ pZ(5 + 2)(jSr+4b«) ^-S Il60"b ' • is the value which is to be added to the formula (59 ) before it is inserted in the formula (49). = v/XTr+l), .... (48) S=- e . , (49) + © Example 49, F:^. 29. — This is a vessel for which the angle of resistance is to be calculated. Her dimensions are Q = 882 tons G= 30782 cubic feet Z= 180 feet 6 = 26 feet b = 13 feet d ^ 5.75 feet JSr= 300 square feet Tonnage of displacement Capacity Length in the load line Beam .... Draft of water Depth of centre of gravity Greatest immerse section Require the angle of resistance ^ = ? 180*^ X 13 5.75 X 26 „.^ok e ^ T = ._-__-_ + -2^j3- = 357.25 feet, 8 X 5.75 X 26 200 X 30782 x 5.75 qn z^ 300 X 13 X 357.25 = 25.4= 164 A TREATISE ON See column . . X and w' From . . 13° 22' « 26 ^ Table III Subtract . 12° 53' « 26j Multiply . 0.4 X 0° 29' = 11.6' To . . 12° 53' Add . . 0° 11' Angle of resistance 4> = 13° 4' the answer. Example 50. Fig. 29. — The same vessel to have a propeller of dimensions, Diameter . . . . jD = 11 feet Pitch P = 27 feet Require the acting area . J. = ? in square feet. From the formula (81), page 80. 2.5 X Y\? A = = 113.4 square feet. ^/ 27* + 3.14* X IP Example 51. Fig. 29. — The vessel to have the same dimensions as in the example 49. The Vessel Depth of centre of gravity Co-efficient . Greatest immerse section Draft of water Length in the load-line . Require the two values . c? = 5.75 feet 8 = 0.17 M = 300 square feet b = 13 feet I = 180 feet e = ? ® = sm.l3°4'^/300 (1 + 0.17-^m.n3°4'x cos.l3°4^) = 4.13, BODIES IN MOTION IN FLUID. 165 ffi = I ^ 180 (0.17 + 2) (300 + 4 X 13^) ^ j 3^ •'^ 1160 X 13 ' ' = 4.13 + 1.37 = 5.5. The F^opeUer. Depth of centre of gravity . (Z = 6.5 feet Co-efficient . . . . 5 = 0.188 feet Acting area . . . J. = 113.4 square feet Eequire the value . . O = ? O =^ 113.4 (1 + 0.188) = 11.56, Eequire the slip S=^ S= ,^ ,P , , = 0.32 or 32 per cent. 11.56 + 5.5 ^ Figs. 30, 31, and 32. — These are three different vessels, supposed to have equal length, beam, and draft of water, and propellers of equal diameter. In the following examples, we will calculate the angle of resistance for the vessels and slip of their propellers. Eomnvple 52. Fig. 30. — Dimensions of the vessel being, Length in the load-line . Beam .... Draft of water Depth of centre of gravity Greater immerse section Tonnage of displacement Capacity Kequire the angle of resistance 4> = ? I = 100 feet 6= 20 feet li = 10 feet d = 2.8 feet ®= 100 square feet q = 143 tons 0= 5000 cubic feet 166 A TREATISE ON 100^ X 10 8 + 2.8 X 20 ' 2 X 10 200 X 5000 X 2.8 + ^A?J = 226 feet, w = See column From . Subtract Multiply To . Add . 1^ = 12.4, 100 X 10 X 226 X $ and w go ^f, « 130 6° 26' « 12° Table III. 0.4 X 0° 29' = 11.6' . 6° 25' . 0° 55'.6 Angle of resistance $ = 6° 55'.6 the answer. Example 53. Fig. 30. — Dimensions of the propeller being . D = 8.33 feet . P= 21.5 feet . J. = ? in square feet. , = = 42.6 square feet. -^ 21.5^ + 3.14^ X 8.33^ Example 54. Fig. 30. — The dimensions being the same as in the example 52. Diameter Pitch .... Require the acting area 2.5 X 8.333 The Vessel. Depth of centre of gravity Co-eflBcient . Greatest immerse section Angle of resistance Require the values d = 2.8 feet ir = 0.091; Mz=. 100 square feet = 6° 55' e=? BODIES IN MOTION IN FLUID. 167 © = sm.6° 55' N/l00(l+0.091-^n.''6°55'xcos.6°55') = 1.2. ffi== I ^IQQ (0-1515 + 2) (100 +Tx 10^) _ ^ Qg ^ 1160 X 10 e = 1.2 + 0.96 = 2.16. The Propeller. Depth of centre of gravity Co-efficient . Acting area . Require the value . d= b feet 5 = 0.1515 J. = 42.6 square feet o = ? O = ^/ 42.6 (1 + 0.1515) Require the slip 8=1 2.16 S = - = 0.23 or 23 per cent. 7 + 2.16 Example 55. Fig. 31. — Dimensions of the vessel being Length in the load-line . Beam .... Draft of water Depth of centre of gravity Greatest immerse section Tonnage of displacement Capacity "... I = 100 feet 6 = 20 feet b = 10 feet d = 3.8 feet JS'= 157 square feet Q = 258 tons C= 9000 cubic feet Require the angle of resistance $ = ? r = WxlO + 3.8x20 ^ 8 X 3.8 X 20 2 X 10 ^"'•°^'^*'''' 168 A TREATISE ON ^ 200 X 9000 X 3.8 ^ 26° ** 157 X 10 X 167.8 ' See column . . a; $ and w | Table Angle of resistance * = 13° 22' the answer 26° j III. Example 56. Fig. 31. — Dimensions of the propeller being Diameter . . . . D= 8.33 feet Pitch P= 18.5 feet Require the acting area . . J. = ? in square feet. A ^-^ ^ ^■^— = 44.7 square feet. ^ 18.5' + 3.14' X 8.33' Example 57. Fig. 31. — Dimensions of the vessel being the same as in Example 55. The Vessel. Depth of centre of gravity Co-e£&cient Greatest immerse section Angle of resistance Require the values d = 3.8 feet S = 0.12 JS' = 157 square feet * = 13° 22' e = ? © = sinl%° 22' -^157 (1 + 0.12 — sin.'13° 22' x cos.l3° 22') = 2.39, ^ _ pIOO (2 + 0.1515) (157 + 4 X 10*) , , . ® - < mo^-To = ■^•^^' = 2.39 + 1.15 = 3.54. BODIES IN MOTION IN FLUID. 1^9 P)'(ypeller. Depth of centre of gravity Co-efficient . Acting area . Eequire the value c? = 5 feet 5 = 0.1515 J. = 44.7 square feet O = n/44.7 (1 + 0.1515) = 7.2 Require the slip S=1 3.54 ;S'= 0.33, or 33 per cent. ■Dimensions of the vessel 7.2 + 3.54 Example 58. Fig. 32.— being Length in the load-line Beam in the load-line . Draft of water Depth of centre of gravity Greatest immerse section Tonnage of displacement Capacity of displacement Eequire the angle of resistance $ = ? ^^00^ x_10_ i66j = 22° 6' e = ? e = si«.22° 6' v/183(l+0.143— ^ra.*22°6'xcos.22°6') = 7. -. \v\m (2 + 0.1515) (183 + 4 X lO**) , r,, ® - N moino • ^ ■^'"^' © = 7 + 1.04 = 8.04. BODIES IN MOTION IN FLUID. 171 Ths Propeller. Depth of centre of gravity . d = ^ feet Co-efficient .... 5 = 0.1515 feet Acting area . . . . -4. = 50 square feet Require the value . . . O = ? O = v/50 (1 + 0.1515) = 7.6. Eequire the slip aS'= ? g S = -=^75 ^ = 0.51 or 51 per cent. 7.6 + 8 ^ When the dimensions of the vessel are given, the number of horse-power 5" which is required to drive the vessel if miles per hour, will be when B = Mi^dn.^(^ (1 + 8 — sinJ^<^ cos.^), and 60 X 60 « V of which and and ^~ 5280 ~ 1.465' «; = 1.465 J^ v^ = 2.15M% (e) ^_ BQOv _Bv "■ 33000 "~ 550' of which P_550 5'_ 550 5" _ 375g ,., ^-""^-1.465 if W' ' ^^^ 172 A TEEATISE ON Those values of «^ and B, from (e) and (/), inserted in the formula (56) will be 375 g ^ 2.15 IP Msm.'4> (1 + 5 — dn.^^ cos.^), and E= ?:1^^' Msm.'$ (1 + l — s%n?^cos. $), o7o lately 174.3 ^ This will give the horse-power less than what ac- tually occurs in practice, owing to the square of the sine. A broken exponent would be more correct, but the formulae in connection with the ordy Table III. are incomplete; therefore, we will rather retain the exponent square of the sine. When the formulae are accompanied with the tables before spoken of, the trigonometrical signs and exponents will disappear; then the calculations will be simple and the results correct; although it will be seen, in the following ex- amples, that the present calculation gives the result very near the fact. To the formula (64) must be added the friction in horse-power, which will be ob- tained by multiplying the co-eflficient 1160 formula (62) by 174.3, and insert M^ instead oiv"; as ^_ if^^/7(^+2)(g -^-4y) 202500b ' • • ^^^^ Example 61. Fig. 30. — The vessel to have the same dimensions as in the Example 52. BODIES IN MOTION IN FLUID. 173 Greatest immerse section . JS'== 100 square feet Angle of resistance . . <|) = 6° 55' Co-efficient . . . 5 = 0:091 Speed of the vessel to be . If = 15 miles per hour Require the number of horse-power fi"= ? „ 15* X 100 X sin.Q° 55' .-, , n nm • a«o kk/ s. rL = -^-s-r-r; (1 + 0.091 ■ — sin.%° 55 X 174.3 ^ cos.6° 55') = 30.25 horses. Friction ^^ 15* X y/lQO (+ 2)0.1 51 5 (100 -h 4 X 10^) _ ~" 202500 X 10 "" 17.9 horses. Required horses 5"= 30.25 + 17.9 = 53.1 5, the answer. Example 62. Fig. 31. — The vessel to have the same dimensions as in the Example 55. Greatest immerse section . W= 157 square feet Angle of resistance . . O = 13° 22' Co-efficient . . . 5 = 0.12 Speed of the vessel to be . if = 12 miles per hour Require the number of horse-power H =t ^ 12*xl57xsm.^l3°22' ^ 0.12 — sm.''13° 22' x 174.3 ^ cos.l3° 22') = 87 horses. Friction _ 12*v/100(2+0.1515)(157+4xl0^) _^^ 2horse« "" 202500 X 10 . - Required horses 11= 87 + 10.2 = 97.2, the answer. Example 63. Fig. 32. — The vessel to have the same dimensions as in the Example 58. 174 A TREATISE ON Greatest immerse section . ^=183 square feet Angle of resistance . . $ = 22° 6' Co-eflBcient . . . 5 = 0.143 Speed of the vessel to be . if = 10 miles per hour Kequire the number of horse-power ^= ? ^^ 10^xl83x^W22°6- ^ ^ 0.143 -«*».^22° 6' x 174.3 ^ cos.22° 6') = 150 horses. Friction ^_ 10^ ^/^00 (2 + 0.1515) (183 + 4 x 10^) ^ „ " 202500 x 10 Required horses £"= 150 + 6.2 = 156.2, the answer. Number of revolutions of the propeller will be per minute 88 if n = p{i-sy Tonnage. The United States Custom House measurement for tonnage of vessels, is expressed by the formulae. T=ll{l-ii) BODIES m MOTION IN FLUID. 175 in which T= Tonnage of the vessel. h = Extreme beam in feet, taken above the mean vales. i( = Depth of the vessel in feet. In double-decked vessels, half the beam h is taken as the depth. For single-decked vessels, the depth is taken from the under side of the deck-plank, to the ceiling of the hold. I = Length of the vessel in feet, from the fore-part of the stem to the after-part of the stern-post, measured on the upper deck. By this rule, the tonnage of the vessels represented and exemplified by the Figs. 30, 31, and 32, will be equal, and T=z 20 X 10 (l06 — f 10) = 210 Tons; 95 this appears to be an incomplete rule. Machinists require the tonnage more accurately, when making contracts for capability of engines and propellers, in reference to the speed of the vessels. The tonnage of the displacement will be found very near by, f 0.015 very sharp vessels, ^ ~ ^ I 0.020 very full vessels, in which Q = Tonnage of the displacement. M:=: Greatest immerse section in square feet. I = Length in the load line in feet. For ordinary vessels, take the co-eflSicient = 0.017. 176 A TREATISE ON The results from the last examples for the figures on Plate XXXI. are collected in this Table IV. VESSEL. Fig. 30. Fig. 31. Fig. 32. Length at the load line I 100 100 100 feet. Beam at the load line h 20 20 20 (( Draft of water tr 10 10 10 it Depth of centre of gravity d 2.8 3.8 4.66 a Greatest imTnerse section M 100 157 183 square feet. Custom-house measurem. T 210 210 210 tons. Tonnage of displacement Q 143 258 350 u Capacity of displacement c 5000 9000 12200 cubic feet. Kadii of the circle arc r 226 167.8 138.66 feet. Angle of the circle arc w° 12° 4' 26° 45' 44° 75' degrees. Angle of resistance * 6° 55' 13° 22' 22° 6' u Speed of the vessels M 15 12 10 miles per h. Power required H 53.15 97.2 158.2 horses. PROPELLER. Diameter D 8.38 8.33 8.33 feet. Pitch P 21.5 18.5 16 it Acting area A 42.6 44.7 50 square feet. Depth of centre of gravity d 5 5 5 feet. The values © 7.2 2.39 7 velocities. The values O 7 7.2 7.6 ti Slip s 23 33 51 per cent. The Steamer S. S. Lewis. Example 53. — The dimensions on the S. S. Lewis will be found on page 95, viz. : — Length in the load line . I = 200 feet Beam . . . . . 6 = 30 feet Loaded to a draft of water . it = 15 feet Depth of centre of gravity . cZ = 6.8 feet -PiattXXXI. BODIES IN MOTION IN FLUID. 177 Greatest immerse section . M= 400 square feet Tonnage of displacement . Q = 1293 tons Capacity of displacement . C = 45100 cubic feet Require the angle of resistance 4> = ? 8 X 6.8 X 30 2 X 15 ,0 200 X 45100 X 6.8 07 30 ^ 400 X 15 X 374.3 ' * See column . . . x ^ and w ' From . . . .14° 20' " 28° - Table III. Subtract . . .13° 57' " 27° Multiply . . 0.3 X 0° 23' = 7 nearly To . . . .13° 57' Add . . . .0° 7' Angle of resistance $ = 14° 4' the answer. Example 54. — Dimensions of the S. S. Lewis pro- peller are Diameter Pitch . . . . Require the acting area 2.5 X 13* A = Z) = 13 feet P=: 33 feet J. = ? in square feet. = 104.5 square feet. v/ 33"^ + 3.14^ X 13' Example 55. — The steamer S. S. Lewis and her propeller brought into action. The Vessel. Depth of centre of gravity Co-efficient . d = 6.8 feet 5 = 0.196 feet 178 A TREATISE ON BODIES IN MOTION IN FLUID. Greatest immerse section . ^=z 400 square feet Angle of resistance . . O = 14° 4' Require the. values . . © = ? © = sm.l4° 4' n/400 (1 + 0.196— sm.'14° 4' x cos.l4° 4') = 5.2, ^_ x/200 (2 + 0.196) (400 + 4 x 15^) , . 1160 X 15 © = 5.2 + 1.5 = 6.7. The Propeller. Depth of centre of gravity . d= 7.5 feet Co-efficient . . . . 5 = 0.211 Acting area . . . ^ = 104.5 square feet Require the value . • O = ? O = n/104.5 (1 + 0.211) = 11.2. Require the slip /S = ? DESCRIPTION OF A CALCULATING MACHINE. 179 DESCRIPTION CALCULATING MACHINE PATENTED BY J. W. NYSTROM. Philadelphia, Mabch 4, 1851. Plate XXXII. This is a calculating machine, by which all the calculations in this work are computed. It is a round plate, on which are fixed two movable arms; by moving those arms, the calculations are computed. The machine has been in use four years by the author of this book, who is the inventor, and has thoroughly tested its practical utility. On the round plate are engraved a number of curved lines, in such a form and divisions, that, in the construction or manufacturing of it, a diflBculty represented itself, viz. : the need of a correct instru- ment to draw the curved lines, which are laid out in progressive divisions. Such an instrument has been completed, and one which can be relied on for its accuracy. The first division was done by the dividing-machine, in the office of the Coast Survey, at Washington, which is perhaps the best in the 180 DESCRIPTION OF world. The facility with which the calculating machine now can be manufactured, enables the patentee to furnish them at greatly reduced prices. The calculating machine was exhibited at the Franklin Institute Exhibition in the year 1849. Since then, it has been patented, and several improve- ments made upon it. DESCRIPTION. The Calculating Machine consists of a round disk of metal or any other suitable material. It has two graduated arms A and B, extending from the centre G, to the periphery. On the outer end of each arm is a screw e, for the purpose of fastening the arms in any particular place on the disk. In the centre is a screw G, to clamp the two arms A and B together; when clamped, they can be moved freely around the centre G. The operation of calculation is performed by mov- ing these two arms together and independently ; for each operation, the arm B must be set in one particu- lar place on the disk called zero. In order to be cor- rect and save time, and to facilitate the setting of the arm B on zero, there is placed a nail E, which is to be operated when the arm B approaches zero; this is done by keeping one finger on a spring attached to the nail E, under the dish ; pinch the spring with the finger, the nail E will project over the periphery, and FlatelZmj:. 182 DESCRIPTION" OF Before going further, it may be well to define what is meant by logarithm. Logarithm is an exponent of ajoower, to which 10 must be raised to give a certain number. This expo- nent is called the logarithm for the number, which will be understood by, ■c logarithm^or 100 = 2 because 10^ = 100. log. 1000 = 3 " 10' = 1000. log. 10000 = 4 " 10* =10000. log. 616600 = 5.79 « 10""= 616600. The unit of the logarithm is called the character- istic or index, and the decimal part is called the manr tissa; the sum of the index and mantissa is the loga- rithm, as, index 616600 = 5 mant. 616600 = 0.79 log. 616600 = 5.79 The invariable number 10 is called the base for the system of logarithm. It is not necessary that the base should be 10; it can be any number; but as our system of arithmetic has 10 to the base, it is more convenient to have the same base for the system of logarithms. All the tables of logarithm now in common use are calculated with 10 to the base. But whatever the base may be, the nature of the loga- A CALCULATING MACHINE. 183 ritlims remains the same; which is, the logarithm for the base = 1, and logarithm for 1 = 0. Logarithms are principally used in Multiplication, Division, Powers, and Extracting of Roots, which is executed by the following four simple rules. MULTIPLICATION. KuLE 1. Add together the logarithms for the factors; the sum is the logarithm for the jproduct. Example. 3456 X 50. log. 3456 = 3.53857 hg. 50 = 1.69897 sum = 5.23754 = log. 172800. the product. DIVISION. KuLE 2. Subtract the logarithm for the divisor, from the logarithm far the dividend ; the difference is the loga- rithm for the quotient. Example. 172800 : 50. hg. 172800 = 5.23754 log. 50 = 1.69897 difference = 3.53857 = log. 3456. the quotient. 184 DESCRIPTION OF POWEES. EuLE 3. Multiply the logarithm for the number, hy the exponent of the power; the -product is the logarithm for the power of the numher. Example.— im^ log. LSO^* = 2.176091 x 3 = 6.528273 = log. 3375000 = 150^ EXTRACTING OF EOOTS. Rule 4. Divide the logarithm for the numher, hy the index, for the root; the quotient is the logarithm for the root of tlie number. Example.— V~WmW. log. 3375000 = 6.528273 : 3 = 2.176091 = log. 150 = ^"33750M This is the principle upon which the calculating machine is based, and the addition and subtraction of the logarithm for the numbers, are computed without noticing the logarithm ; only by moving the arms A and B, together and independently, by the intersec- tions of the curved lines the result is obtained. It is not necessary to understand the nature of logar rithms to use the instrument, it only requires atten- tion to the simple rules which here will be given. Circle a. — In the circle a are two sizes of figures, of which the larger one represents the first figure of a number, and the small one the second; for instance A CALCULATING MACHINE. 185 28 is a number composed of two figures ; 2 is the^irs^ figure, and 8 the second. Example. — To set the arm A on the number 28 (circle a). Set the arm A on the large 2, move the arm further, until it comes to the small 8 (the eighth line from the large 2), fasten the arm A with the screw e, then the arm A is set on 28 ; but this 28 can represent any multiple or parts by 10, for instance 0.0028, 0.28, 2.8, 28, 280, 28000, &c. When the number contains more than two figures, as 2835, the third and fourth figures are to be found on the arm, where it intersects the curved line; the third figure is the small figure on the arm counted from a, and the fourth figure is represented by the small divisions between the figures on the arms. When the arm stands on 28, move it further, until it intersects the eighth curved line at 3 on the arm, then the arm stands on 283 ; move it a little further until the same curved line intersects the fifth division be- tween 3 and 4 on the arm, then the arm stands on 2835. In the same manner the arm is to be set on any other number, which, by a little practice, is done instantly. Oircle log. — This circle contains the logarithm for the numbers in the circle a. When the arm is set on a number circle a, at the same time the arm inter- sects a curved line belonging to the circle log., this latter intersection is the decimal part (mantissa) of the logarithm for the number in the circle a. 13 186 DESCRIPTION OP Example. Log. 7 = 0.845. — Set the arm A on 7, circle a, the mantissa in the circle log. is 845 ; the first figure 8, is the 8 curved line, numbered in circle log. which intersects the arm, the second figure 4 is the fourth figure on the arm, counted from log., and the third figure 5, is the fifth division between 4 and 5 on the arm. The nature of logarithms in connection with their numbers is such, that the index for the logarithm is always one less than the number of figures in the number for which the logarithm is to be found (when the base of the logarithm is 10). Index 60 = 1 because 60 is two figures and 2 — 1=1 616 = 2 " 616 " three " 3 — 1 = 2 6166 =3 " 6166 " four " 4 — 1 = 3 61668000 = 7 " 61663000 " seven " 8 — 1 = 7 In difiicult calculations, combined with powers and roots, a correct account must be kept on these indices, in order to be certain of the number of figures in the result. For that purpose there is a small hand on the screw G, which is to be operated separately for each operation by the arms. For multiplication, add the indices hy the small hand. For division, subtract the index for the divisor, from the index for the dividend, then when the operations are finished, the small ha7id shows the index for the result ; adding one to it gives the number of figures in the result. If the index becomes negative, the result is a corresponding A CALCULATING MACHINE. 187 decimal fraction, and the hand shows how many it is before the figures, including the index 0. Example. — If the arm shows 28, on circle a, and the hand shows the index + or — , as Index — 3 the decimal fraction is 0.0028 — 2 " " 0.028 — 1 " " 0.28 = the corresponding number is 2.8 + 1 " " 28. + 2 " " 280. + 4 " " 28000. +7 « « 28000000. &c. &c. Abhr&oiations. Set A on, means set the arm A on. Set B on, " set the arm B on, Fasten Ae, " fasten the arm A with the screw e. Fasten Be, " fasten the arm B with the screw e. Clamp C, " clamp the two arms with the screw G. MULTIPLICATION. Multiplication is to be computed on the circle a, and without any exception follows this simple rule 1. Multiply two numbers together. Rule 1. Set A on one of the nvmbers, fasten Ae. Su B cm zero, clamp C, hose Ae. Move the arms waiil B comes to the other number, then the arm A shows the product. 188 DESCRIPTION OF Example 1. — Multiply the number 3 by 2. Set A on 3, fasten Ae ; set B on zero ; clamp G ; loose Ae; move tbe arms until B comes to 2 ; then A shows the product 6. Example 2.— Multiply 436000 by 12500. Set A on 436000, fasten Ae; set B on zero ; clamp C; loose Ae; move the arms until B comes to 12500; then A shows the product 5450000000. When the arm A is set on 436, set the hand on its index 5 ; when the arms are moved until B comes to 125, add its index 4 to 5, which will be 9, the index for the result. These operations are accomplished instantly. When there are more than two factors to be multi- plied together, consider the product of two factors as one factor, and continue the multiplication with the next factor, as before described. It matters not how many factors there may be, the multiplication can be continued to any extent, and <^^ 25.7 feet. V 42° Set J. on 360, ^ on 42, clamp C. Move the arms 196 DESCRIPTION OF until B comes to 3, then A shows the pitch P= 25.7 feet. When there are a number of factors in both the numerator and denominator, it is most convenient to separate them by the characters described on page 190, and operate in the same manner. If there are any roots to extract, place the root marked in the first divisions as follows : — Example 2. Formula 24, page 73. — The values of those quantities are, say Number of revolutions Diameter of propeller Pitch of propeller . Slip of propeller Diameter of cylinders Stroke of piston Friction Require the effectual pressure 9.25 s/ps.mnjy « = 63 per minute Z) = 9 feet 3 inches P= 24 feet /S' = 32 per cent. (Z = 29J inches s = 25 inches /= 30 per cent. p = 1 in pounds. v/24 X 0T32r 50 r_63 r9.25( r9.25r cPs{l—f) (1 — 0.30)^^29.5*^29.5-' 25 = 49.2 pounds. Multiply 24 by 0.32, mantissa is 885 divided by 2 = 4425. Set A on mantissa 4425. Subtract 0.30 from 1 = 0.70. Set B on 0.70, clamp G, move the arms until 5 comes to 50, and proceed as described on page 190, until B arrives at the last factor 9.25, then A shows the effectual pressure p = 49.2 pounds per square inch. A CALCULATING MACHINE. 197 Example 3. Formula 20, page 73. — The values of the quantities being the same as in the preceding example. Require the diameter d = 1 m inches. , _^ I n/M50T^^ __3 1 ^/24x^:32 /. 50 r63r9!25^ — \ (1—/)^ "N (1 — 0.30) J 49.2*^ T-' 1 = 27.8 inches. Set A on the mantissa 4425 (from the preceding example corresponding to VPS). Set B on 0.70, clamp Cj move the arms until B comes to 50, and proceed as before described. Keep account of the indexes, and remember to make two operations by the factor 9.25. At the last operation the index will be 4 and mantissa 336. As the cube root is to be ex- tracted, divide 4.336 by 3 = 1.445. Set A on the mantissa 445, the answer on circle a\s> d = 27.8 inches. This example is computed in less than one minute. TEIGONOMETEY. Trigonometry is that part of geometry which treats of triangles. It is divided into two parts, viz., plane and spherical. Plane trigonometry treats of the triangles which are (or imagined to be) drawn on a plane. Spherical trigonometry treats on the triangles which are (or imagined to be) drawn on a sphere. A triangle contains seven quantities, namely, three sides, three angles, and the surface. When any three 198 DESCRIPTION OF of these quantities are given, the four remaining ones can by them be ascertained (one side or the area must be one of the given quantities), and the operation is called solving the triangle, which is only an application of arithmetic or geometrical objects. For the foundation of the above-mentioned solution, there are assumed eight help quantities, which are called trigonometrical functions, which are the following in names and number, corresponding with the Fig. 1. 1. Sinus, 2. Cosinus, 3. Sinus-versus, 4. Cosinus-versus, 5. Tangent, 6. Cotangent, 7. Secant, 8. Cosecant, i 3 abbreviated sin.C cos.C sinv. C cosv.C tan.C cot.C ■ sec.C cosec. C r = Radius of the circle, which is the unit by which the functions are measured. A CALCULATINa MACHINE. 199 In the accompanying tables, these functions are cal- culated at every 10 minutes per degree in the quad- rant of the circle represented by Fig. 1. The circle arc between the two lines 8 and 2, 3, is a measure of the angle for which the functions are mentioned. This angle is denoted by the letter C, and the expres- sion sin. G means the line 1 compared with the radii r at a given angle G. Say the angle G to be 60°. In the first column of the table for the sin., 60° corresponds with 0.86602 in the next column, which is the length of the sin. for 60° compared with the radii r as a unit, and the ex- pression sin.m° X 36 means 0.86602 x 36 = 31.17672, and likewise with all the other trigonometrical ex- pressions. In a triangle, these functions have a certain relation to the opposite side of the angle; it is this relation- ship which enables us to solve the triangle only by the application of simple arithmetic. In a triangle, the sides are denoted by the letters ah c; their respective opposite angles are denoted by A, B, G, and the area by Q. The sides a, h, and c bear the following relation to the trigonometrical functions : — 200 DESCRIPTION OF Plane Trigonometry. Fig. 2. A right-angled triangle. a : c=zl : sin. C, of which a = — — - , . (1) sm. a:h=l :cos.C, " b = a cos.C, . (2) 6 : c == 1 : tan. O, " c = 6 tan. G, . (3) sin. G : COS. G = tan. G : r. Formula (1). — If the side c and the angle G are given, the side a will be found simply by dividing the side c by sin. G. Suppose the side c is 36 feet and the angle G= 30°. In the accompanying table for sine, 30° in the first column corresponds with 0.500t3. in the next one, which is the length of the sine for 30°, and 36 36 == — = 72 feet, the length of the side a. sin.m° 0.5000 ' ^ Formula (2). — The length of the side a and the angle G are given. Multiply the length a by cos. G, the product is the length of the side 6. Suppose the side a is 325 feet long, and the angle G = 42°, in the table for cosine, marked on the bottom of the table, 42° in the last column corresponds with 0.74314 in the next column, which is the length of the cosin. for 42°, and A CALCULATING MACHINE. 201 325 X cosA2° = 325 x 0.74314 = 241.52 feet, the length of the side h. Formula (3). Fig. 2. — Let the line c represent the height of a steeple. From the centre at A is measured the horizontal line, 6 = 285 feet, to a point C, in which the angle to the top of the steeple is measured to be 38°. What will be the height of the steeple? See table for the tangent. In the first column, 38° cor- responds with 0.78128 in the next column, which is the length of the tangent for 38°, and 285 X tan.B8° = 285 x 0.78128 = 222.5 feet, the height of the steeple. An oblique-angled triangle. . a :b = sin.A : sin.B, of which a = — ^-'vj-, (4) b : c = sin.B : sin. G, of which b = —. — '—-, (5) sm. U a : c= sin.A : sin. C, of which c = —. — '-r-, (6) . ^ c sin.B /wx sm.U = — = — , \i) b Examples for those formulae will be computed by the machine, and without the tables. Without the machine, tables of this construction will be competent and useful for all mechanical calculations. 14 202 DESCRIPTION OF NATURAL SINE. Degrees. 0' 10' 20' 30' 40' 50' 60' .00000 .00291 .00581 .00872 .01163 .01454 .01745 89 1 .01745 .02036 .02326 .02617 .02908 .03199 .03489 88 2 .03489 .03780 .04071 .04361 .04652 .04943 .05233 87 3 .05233 .05524 .05814 .06104 .06395 .06685 .06975 86 4 .06975 .07265 .07555 .07845 .08135 .08425 .08715 85 5 .08715 .09005 .09294 .09584 .09874 .10163 .10452 84 6 .10452 .10742 .11031 .11320 .11609 .11898 .12186 83 7 .12186 .12475 .12764 .13052 .13340 .13629 .13917 82 8 .13917 .14205 .14493 .14780 .15068 .15356 .15643 81 9 .15643 .15930 .16217 .16504 .16791 .17078 .17364 80 10 .17364 .17651 .17937 .18223 .18509 .18795 .19080 79 11 .19080 .19366 .19651 .19936 .20221 .20506 .20791 78 12 .20791 .21075 .21359 .21643 .21927 .22211 .22495 77 13 .22495 .22778 .23061 .23344 .23627 .23909 .24192 76 14 .24192 .24474 .24756 .25038 .25319 .25600 .25881 75 15 .25881 .26162 .26443 .26723 .27004 .27284 .27563 74 16 .27563 .27843 .28122 .28401 .28680 .28958 .29237 73 17 .29237 .29515 .29793 .30070 .30347 .80624 .30901 72 18 .30901 .31178 .31454 .31730 .32006 .32281 .32556 71 19 .32556 ,32831 .33106 .33380 .33654 .33928 .34202 70 20 .34202 .34475 .34748 .35020 .35293 .35565 .3,5836 69 21 .35836 .36108 .36379 .36650 .36920 .37190 .37460 68 22 .37460 .37730 .37999 .38268 .38536 .38805 .39073 67 23 .39073 .39340 .39607 .39874 .40141 .40407 .40673 66 24 .40673 .40939 .41204 .41469 .41733 .41998 .42261 65 25 .42261 .42525 .42788 .43051 .43313 .43575 .43837 64 26 .43837 .44098 .44359 .44619 .44879 .45139 .45399 63 27 .45399 .45658 .45916 .46174 .46432 .46690 .46947 62 28 .46947 .47203 .47460 .47715 .47971 .48226 .48480 61 29 .48480 .48735 .48988 .49242 .49495 .49747 .50000 60 30 .50000 .50251 .50502 .50753 .51004 .51254 .51503 59 31 .51503 .51752 .52001 .52249 .52497 .52745 .52991 58 32 .52991 .53238 .53484 .53729 .53975 .54219 .54463 57 33 .54463 .54707 .54950 .55193 .55436 .55677 .55919 56 34 .55919 .56160 .56400 .56640 .56880 .57119 .57357 55 35 .57357 .57595 .57833 .58070 .58306 .58542 .58778 54 36 .58778 .59013 .59248 .59482 .59715 .59948 .60181 53 37 .60181 .60413 .60645 .60876 .61106 .61336 .61566 52 38 .61566 .61795 .62023 .62251 .62478 .62705 .62932 51 39 .62932 .63157 .63383 .63607 .63832 .64055 .64278 50 40 .64278 .64501 .64723 .64944 .65165 .65386 .65605 49 41 .65605 .65825 .66043 .66262 .66479 .66696 .66913 48 42 .66913 .67128 .67344 .67559 .67773 .67986 .68199 47 43 .68199 .68412 .68624 .68835 .69046 .69256 .69465 46 44 .69465 .69674 .69883 .70090 .70298 .70504 .70710 45 60' 50' 40' 30' 20' 10' 0' Degrees. NATURAL COSINE. A CALCULATING MACHINE. 203 NATURAL SINE . Degrees. 0' 10' 20' 30' 40' 50' 60' 45 .70710 .70916 .71120 .71325 .71528 .71731 .71933 44 46 .71933 .72135 .72336 .72537 .72737 .72936 .73135 43 47 .73135 .73338 .73530 .73727 .73923 .74119 •74314 42 48 .74814 .74508 .74702 .74895 .75088 .75279 .75470 41 49 .75470 .75661 .75851 .76040 .76229 .76417 .76604 40 50 .76604 .76791 .76977 .77162 .77347 .77531 .77714 39 51 .77714 .77897 .78079 .78260 .78441 .78621 .78801 38 52 .78801 .78979 .79157 .79335 .79512 .79688 .79863 37 53 .79863 .80038 .80212 .80885 .80558 .80780 .80901 36 54 .80901 .81072 .81242 .81411 .81580 .81748 .81915 35 55 .81915 .82081 .82247 .82412 .82577 .82740 •82903 34 56 .82903 .88066 .83227 .83888 .88548 .88708 •83867 33 57 .83867 .84025 .84182 .84339 .84495 .84650 •84804 32 58 .84804 .84958 .85111 .85264 .85415 .85566 •85716 31 59 .85716 .85866 .86014 .86162 .86310 .86456 •86602 30 60 .86602 .86747 .86891 .87035 .87178 .87520 •87461 29 61 .87461 .87602 .87742 .87881 .88020 .88157 •88294 28 62 .88294 .88430 .88566 .88701 .88835 .88968 •89100 27 63 .89100 .89232 .89363 .89493 .89622 .89751 •89879 26 64 .89879 .90006 .90132 .90258 .90383 .90507 •90630 25 65 .90680 .90753 .90875 .90996 .91116 .91235 •91354 24 66 .91354 .91472 .91589 .91706 .91811 .91986 •92050 23 67 .92050 .92163 .92276 .92387 .92498 .92609 •92718 22 68 .92718 .92826 .92934 .93041 .98147 .93253 •93358 21 69 .93358 .93461 .93564 .93667 .93768 .93869 •93969 20 70 .93969 .94068 .94166 .94264 .94360 .94456 ■94551 19 71 .94551 .94646 .94739 .94832 .94924 .95015 •95105 18 72 .95105 .95195 .95283 .95371 .95458 .95545 •95680 17 73 .95630 .95715 .95798 .95881 .95964 .96045 •96126 16 74 .96126 .96205 .96284 .96363 .96440 .96516 •96592 15 75 .96592 .96667 .96741 .96814 .96887 .96958 •97029 14 76 .97029 .97099 .97168 .97236 .97304 .97371 •97487 13 77 .97487 .97402 .97566 .97629 .97692 .97753 •97814 12 78 .97814 .97874 .97984 .97992 .98050 .98106 •98162 11 79 .98162 .98217 .98272 .98325 .98378 .98429 •98480 10 80 .98480 .98530 .98580 .98628 .98676 .98722 98768 9 81 .98768 .98813 .98858 .98901 .98944 .98985 •99026 8 82 .99026 .99066 .99106 .99144 .99182 .99218 •99254 7 83 .99254 .99289 .99323 .99357 .99389 .99421 •99452 6 84 .99452 .99482 .99511 .99539 .99567 .99593 •99619 5 85 .99619 .99644 .99668 .99691 .99714 .99735 •99756 4 86 .99756 .99776 99795 .99813 .99830 .99847 •99862 3 87 .99862 .99877 99891 .99904 99917 .99928 •99939 2 88 .99939 .99948 99957 .99965 99972 .99979 •99984 1 89 .99984 .99989 99993 .99996 99998 .99999 LOOOO 60' 50' 40' 30' 20' 10' 0' Degrees. NATURAL COSINE. 204 DESCEIPTIOIir OF TANGENT. Degrees. 0' 10' 20' 30' 40' 50' 60' 00000 . 00290 . 00581 . 00872 . 01163 01454 01745 89 1 01745 . 02086 . 02327 . 02618 . 02909 03200 08492 88 2 03492 . 03783 . 04074 . 04866 . 04657 04949 05240 87 3 05240 . 05582 . 05824 06116 06408 06700 06992 86 4 06992 . 07285 . 07577 07870 08162 08455 08748 85 5 08748 . 09042 09335 09628 09922 10216 10510 84 6 10510 10804 11098 11893 11688 11983 12278 83 7 12278 12573 12869 13165 13461 18757 14054 82 8 .14054 14350 14647 14945 15242 15540 .15838 81 9 .15838 16186 16435 16734 17033 17332 .17632 80 10 .17632 17932 18233 18533 18884 .19136 .19488 79 11 .19438 19740 20042 20345 20648 .20951 .21255 78 12 .21255 21559 21864 22169 22474 .22780 .23086 77 13 .23086 23393 23700 24207 .24315 .24624 .24932 76 14 .24932 25242 25551 25861 .26172 .26483 .26794 75 15 .26794 .27106 27419 .27782 .28045 .28859 .28674 74 16 .28674 .28989 .29305 .29621 .29988 .80255 .30573 73 n .30573 .30891 .81210 .31529 .81849 .32170 .32491 72 18 .32491 .32813 .33186 .38459 .38783 .34107 .34432 71 19 .34432 .34758 .35084 .85411 .35739 .36067 .86397 70 20 .36397 .86726 .87057 .37388 .87720 .88053 .38886 69 21 .38386 .38720 .39055 .39391 .39727 .40064 .40402 68 22 .40402 .40741 .41080 .41421 .41762 .42104 .42447 67 23 .42447 .42791 .43135 .48481 .43827 .44174 .44522 66 24 .44522 .44871 .45221 .45572 .45924 .46277 .46630 65 25 .46630 .46985 .47340 .47697 .48055 .48413 .48773 64 26 .48773 .49188 .49495 .49858 .60221 .50586 .50962 63 27 .50952 .51819 .51687 .52056 .52426 .52798 .53170 62 28 .53170 .53544 .53919 .54295 .64672 .55051 .55430 61 29 .55430 .55811 .56193 .56577 .56961 .57347 .67785 60 30 .57735 .58123 .58513 .58904 .59296 .59690 .60086 59 31 .60086 .60482 .60880 .61280 .61680 .62083 .62486 58 32 .62486 .62892 .63298 .63707 .64116 .64527 .64940 57 33 .64940 .65365 .65771 .66188 .66607 .67028 .67460 56 34 .67450 .67874 .68300 .68728 .69157 .69588 .70020 55 35 .70020 .70455 .70891 .71329 .71769 .72210 .72654 54 36 .72654 .73099 .73546 .73996 .74447 .74900 .75355 53 37 .75355 .75812 .76271 .76732 .77195 .77661 .78128 52 38 .78128 .78598 .79069 .79543 .80019 .80497 .80978 51 39 .80978 .81461 .81946 .82433 .82923 .88416 .83909 50 40 .83909 .84406 .84906 .85408 .85912 .86414 .86928 49 41 .86928 .87440 .87955 .88472 .88992 .89515 .90040 48 42 .90040 .90568 .91099 .91638 .92169 .92704 .98251 47 43 .93251 .93796 .94345 .94896 .95450 .96008 .96568 46 44 .96568 .97532 .97699 .98269 .98848 .99419 1.0000 45 60' 50' 40' 30' 20' 10' 0' Degrees. COTANGENT. A CALCULATING MACHINE. 205 TANGENT. Degrees. 0' 10' 20' 30' 40' 50' 60' 45 1.0000 1.0058 1.0117 1.0176 1.0235 1.0295 1.0355 44 46 1.0355 1.0415 1.0476 1.0537 1.0599 1.0661 1.0723 43 47 1.0723 1.0786 1.0849 1.0913 1.0977 1.1041 1.1106 42 48 1.1106 1.1171 1.1236 1.1302 1.1369 1.1436 1.1503 41 49 1.1503 1.1571 1.1639 1.1708 1.1777 1.1847 1.1917 40 50 1.1917 1.1988 1.2059 1.2130 1.2203 1.2275 1.2348 39 51 1.2348 1.2422 1.2496 1.2571 1.2647 1.2722 1.2799 38 52 1.2799 1.2876 1.2954 1.3032 1.3111 1.3190 1.3270 37 53 1.3270 1.3351 1.3432 1.3514 1.3596 1.3679 1.3763 36 54 1.3763 1.3848 1.3933 1.4019 1.4106 1.4193 1.4281 35 55 1.4281 1.4370 1.4459 1.4550 1.4641 1.4732 1.4825 34 56 1.4825 1.4919 1.5013 1.5108 1.5204 1.5301 1.5398 33 57 1.5398 1.5497 1.5596 1.5696 1.5798 1.5900 1.6003 32 58 1.6003 1.6107 1.6212 1.6318 1.6425 1.6533 1.6642 31 59 1.6642 1.6752 1.6864 1.6976 1.7090 1.7204 1.7320 30 60 1.7320 1.7437 1.7555 1.7674 1.7795 1.7917 1.8040 29 61 1.8040 1.8164 1.8290 1.8417 1.8546 1.8676 1.8807 28 62 1.8807 1.8939 1.9074 1.9209 1.9347 1.9485 1.9626 27 63 1.9626 1.9768 1.9911 2.0056 2.0203 2.0352 2.0503 26 64 2.0503 2.0655 2.0809 2.0965. 2.1123 2.1283 2.1445 25 65 2.1445 2.1608 2.1774 2.1942 2.2113 2.2285 2.2460 24 66 2.2460 2.2637 2.2816 2.2998 2.3182 2.3369 2.3558 23 67 2.3558 2.3750 2.3944 2.4142 2.4342 2.4545 2.4750 29. 68 2.4750 2.4959 2.5171 2.5386 2.5604 2.5826 2.6050 21 69 2.6050 2.6279 2.6510 2.6746 2.6985 2.7228 2.7474 20 70 2.7474 2.7725 2.7980 2.8239 2.8502 2.8769 2.9042 19 71 2.9042 2.9318 2.9600 2.9886 3.0178 3.0474 3.0776 18 72 3.0776 3.1084 3.1397 3.1715 3.2040 3.2371 3.2708 17 73 3.2708 3.3052 3.3402 3.3759 3.4123 3.4495 3.4874 16 74 3.4874 3.5260 3.5655 3.6058 3.6470 3.6890 3.7320 15 75 3.7320 3.7759 3.8208 3.8667 3.9136 3.9616 4.0107 14 76 4.0107 4.0610 4.1125 4.1652 4.2193 4.2747 4.3314 13 77 4.3314 4.3896 4.4494 4.5107 4.5736 4.6382 4.7046 12 78 4.7046 4.7728 4.8430 4.9151 4.9894 5.0658 5.1445 11 79 5.1445 5.2256 5.3092 5.3955 5.4845 5.5763 5.6712 10 80 5.6712 5.7693 5.8708 5.9757 6.0844 6.1970 6.3137 9 81 6.3137 6.4348 6.5605 6.6011 6.8269 6.9682 7.1153 < 8 82 7.1153 7.2687 7.4287 7.5957 7.7703 7.9530 8.1443 7 83 8.1443 8.3449 8.5555 8.7768 9.0098 9.2553 9.5143 6 84 9.5143 9.7881 10.078 10.385 10.711 11.059 11.430 5 85 11.430 11.826 12.250 12.760 13.196 13.726 14.300 4 86 14.300 14.924 15.604 16.349 17.169 18.074 19.081 3 87 19.081 20.205 21.470 22.003 24.541 26.431 28.636 2 88 28.636 31.241 34.367 38.188 42.964 49.103 57.289 1 89 57.289 68.750 85.939 114.58 171.88 343.77 00 60' 50' 40' 30' 20' 10' 0' Degrees. COTANGENT. 206 DESCRIPTION OF SECANT. Degrees. C 10' 20' 30' 40' 50' 60' 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001 89 1 1.0001 1.0002 1.0002 1.0003 1.0004 1.0005 1.0006 88 2 1.0006 1.0007 1.0008 1.0009 1.0010 1.0012 1.0013 87 3 1.0013 1.0015 1.0016 1.0018 1.0020 1.0022 1.0024 86 4 1.0024 1.0026 1.0028 1.0031 1.0033 1.0035 1.0038 85 5 1.0038 1.0040 1.0043 1.0046 1.0049 1.0052 1.0055 84 6 1.0055 1.0058 1.0061 1.0064 1.0068 1.0071 1.0075 83 7 1.0075 1.0078 1.0082 1.0086 1.0090 1.0094 1.0098 82 8 1.0098 1.0102 1.0106 1.0111 1.0115 1.0120 1.0124 81 9 1.0124 1.0129 1.0134 1.0139 1.0144 1.0149 1.0154 80 10 1.0154 1.0159 1.0164 1.0170 1.0175 1.0181 1.0187 79 11 1.0187 1.0192 1.0198 1.0204 1.0210 1.0217 1.0223 78 12 1.0223 1.0229 1.0236 1.0242 1.0249 1.0256 1.0268 77 13 1.0263 1.0269 1.0277 1.0284 1.0291 1.0298 1.0306 76 14 1.0306 1.0313 1.0321 1.0329 1.0336 1.0344 1.0852 75 15 1.0352 1.0360 1.0369 1.0377 1.0385 1.0394 1.0403 74 16 1.0403 1.0411 1.0420 1.0429 1.0438 1.0447 1.0456 73 17 1.0456 1.0466 1.0475 1.0485 1.0494 1.0504 1.0514 72 18 1.0514 1.0524 1.0534 1.0544 1.0555 1.0565 1.0576 71 19 1.0576 1.0586 1.0597 1.0608 1.0619 1.0630 1.0641 70 20 1.0641 1.0653 1.0664 1.0676 1.0687 1.0699 1.0711 69 21 1.0711 1.0723 1.0735 1.0747 1.0760 1.0772 1.0785 68 22 1.0785 1.0798 1.0810 1.0823 1.0837 1.0850 1.0863 67 23 1.0863 1.0877 1.0890 1.0904 1.0918 1.0932 1.0946 66 24 1.0946 1.0960 1.0974 1.0989 1.1004 1.1018 1.1083 65 25 1.1033 1.1048 1.1063 1.1079 1.1094 1.1110 1.1126 64 26 1.1126 1.1141 1.1167 1.1174 1.1190 1.1206 1.1223 63 27 1.1223 1.1239 1.1256 1.1273 1.1290 1.1308 1.1325 62 28 1.1325 1.1343 1.1361 1.1378 1.1396 1.1415 1.1488 61 29 1.1433 1.1452 1.1470. 1.1489 1.1508 1.1527 1.1547 60 30 1.1547 1.1566 1.1586 1.1605 1.1625 1.1646 1.1666 59 31 1.1666 1.1686 1.1707 1.1728 1.1749 1.1770 1.1791 58 32 1.1791 1.1813 1.1835 1.1856 1.1878 1.1901 1.1923 57 33 1.1923 1.1946 1.1969 1.1992 1.2015 1.2038 1.2062 56 34 1.2062 1.2085 1.2109 1.2134 1.2158 1.2182 1.2207 55 35 1.2207 1.2232 1.2257 1.2283 1.2308 1.2334 1.2360 54 36 1.2360 1.2386 1.2413 1.2440 1.2466 1.2494 1.2521 53 37 1.2521 1.2548 1.2576 1.2604 1.2632 1.2661 1.2690 52 38 1.2690 1.2719 1.2748 1.2777 1.2807 1.2837 1.2867 51 39 1.2867 1.2898 1.2928 1.2959 1.2990 1.3022 1.8054 50 40 1.8054 1.3086 1.3118 1.3150 1.3188 1.3216 1.3250 49 41 1.3250 1.3283 1.3317 1.3351 1.3386 1.3421 1.3456 48 42 1.3456 1.3491 1.3527 1.3563 1.3599 1.3636 1.3678 47 43 1.3673 1.3710 1.3748 1.3785 1.8824 1.3862 1.3901 46 44 1.3901 1.3940 1.3980 1.4020 1.4060 1.4101 1.4142 45 60' 50' 40' 30' 20' 10' 0' Degrees COSECANT. A CALCULATING MACHINE, 207 SECANT. Degree^. 0' 10' 20' 30' 40' 50' 60' 45 1.4142 1.4183 1.4225 1.4267 1.4809 1.4352 1.4395 44 46 1.4395 1.4439 1.4483 1.4527 1.4572 1.4617 1.4662 43 47 1.4662 1.4708 1.4755 1.4801 1.4849 1.4896 1.4944 42 48 1.4944 1.4993 1.5042 1.5091 1.5141 1.5191 1.5242 41 49 1.5242 1.5293 1.5345 1.5897 1.5450 1.5503 1.5557 40 50 1.5557 1.5611 1.5666 1.5721 1.5777- 1.5888 1.5890 39 51 1.5890 1.5947 1.6005 1.6063 1.6122 1.6182 1.6242 38 52 1.6242 1.6303 1.6364 1.6426 1.6489 1.6552 1.6616 37 53 1.6616 1.6680 1.6745 1.6811 1.6878 1.6945 1.7013 36 54 1.7013 1.7081 1.7150 1.7220 1.7291 1.7362 1.7434 35 55 1.7434 1.7507 1.7580 1.7655 1.7780 1.7806 1.7882 34 56 1.7882 1.7960 1.8088 1.8118 1.8198 1.8278 1.8360 33 57 1.8360 1.8443 1.8527 1.8611 1.8697 1.8783 1.8870 32 58 1.8870 1.8959 1.9048 1.9138 1.9280 1.9322 1.9416 31 59 1.9416 1.9510 1.9606 1.9702 1.9800 1.9899 2.0000 30 60 2.0000 2.0101 2.0203 2.0307 2.0412 2.0519 2.0626 29 61 2.0626 2.0735 2.0845 2.0957 2.1070 2.1184 2.1800 28 62 2.1300 2.1417 2.1586 2.1656 2.1778 2.1901 2.2026 27 63 2.2026 2.2153 2.2281 2.2411 2.2543 2.2676 2.2811 26 64 2.2811 2.2948 2.3087 2.3228 2.3870 2.3515 2.3662 25 65 2.3662 2.3810 2.3961 2.4114 2.4269 2.4426 2.4585 24 66 2.4585 2.4747 2.4911 2.5078 2.5247 2.5418 2.5593 23 67 2.5593 2.5769 2.5949 2.6131 2.6316 2.6503 2.6694 22 68 2.6694 2.6888 2.7085 2.7285 2.7488 2.7694 2.7904 21 69 2.7904 2.8117 2.8334 2.8554 2.8778 2.9006 2.9238 20 70 2.9238 2.9473 2.9713 2.9957 8.0205 3.0458 3.0715 19 71 3.0715 3.0977 3.1243 3.1515 3.1791 8.2073 8.2360 18 72 3.2360 3.2653 3.2951 3.3255 3.3564 3.3880 3.4203 17 73 3.4203 3.4531 3.4867 8.5209 8.5558 3.5915 3.6279 16 74 3.6279 3.6651 8.7031- 3.7419 3.7816 3.8222 3.8637 15 75 3.8637 3.9061 3.9495 8.9939 4.0893 4.0859 4.1335 14 76 4.1335 4.1823 4.2328 4.2836 4.3362 4.3900 4.4454 13 77 4.4454 4.5021 4.5604 4.6202 4.6816 4.7448 4.8097 12 78 4.8097 4.8764 4.9451 5.0158 5.0886 5.1635 5.2408 11 79 5.2408 5.3204 5.4026 5.4874 5.5749 5.6653 5.7587 10 80 5.7587 5.8553 5.9558 6.0588 6.1660 6.2771 6.3924 9 81 6.3924 6.5120 6.6363 6.7654 6.8997 7.0396 7.1852 8 82 7.1852 7.3371 7.4957 7.6612 7.8844 8.0156 8.2055 7 83 8.2055 8.4046 8.6137 8.8336 9.0651 9.3091 9.5667 6 84 9.5667 9.8391 10.127 10.437 10.758 11.104 11.473 5 85 11.473 11.868 12.291 12.745 13.234 18.763 14.335 4 86 14.835 14.957 15.636 16.880 17.198 18.102 19.107 3 87 19.107 20.230 21.493 22.925 24.562 26.450 28.653 2 88 28.653 31.257 34.382 38.201 42.975 49.114 57.298 1 89 57.298 68.757 85.945 114.59 171.88 343.77 00 60' 50' 40' 30' 20' 10' 0' Degrees. COSECANT. 208 DKSCEIPTION OF DESCRIPTION OP THE TRIGONOMETRICAL SCALE ON THE CALCULATING MACHINE. In the precediug treatment of triangles, tables of the trigonometrical lines are required for the solutions. When accompanied with one of these simple calcula- tors, all those tables are dispensed with, and it is so arranged that it is not necessary to notice the func- tion, only operating by the angles themselves, ex- pressed in degrees and minutes. This makes the trigonometrical solutions so simple that any one who understands the simple arithmetic will be able to solve the trigonometrical questions. Explanation. — The inner scale between sin. and cos., marked on the arms, is for trigonometrical calculations. The numbers in the circles sin. and cos. represent the angles in degrees, and the divisions on the arms show the exceeded minutes where the lines intersect the arms. When the arm is set on an angle in the circle sin., the circle a shows the length of the sinus for that angle. When set on an angle in the circle COS., the circle a shows the length of the cosinus. By this arrangement, any of the trigonometrical functions can be found. Example 1. — To find the length of the tangent for the angle G= 54°. . ^ sin.C sm.54° i oh7/.o tan.U = - = — = 1.3763. cos. U cos.ol° A CALCULATING MACHINE. 209 Set A on STO.54, B on cos. 54, clamp G. Move the arms until B comes to zero, then J. shows the length of the tangent = 1.3767 circle a. Example 2. — To find the secant for an angle (7=35°. sec.35° = — — = 1.2207. COS. 3 5 Set Aon 1, B on cos. 3 5°, clamp C. Move the arms until B comes to zero, then A shows the length of the secant = 12207. Example 3. Fig. 2. — An inclined plane a is to be built to a height c = 42 feet; the angle C is 39°. What wUl be the length of the plane a? a = -; — —. = 66.8 feet. sin.Sd Set A on 42, B on sin.S9°, clamp C. Move the arms until B comes to zero, then A shows the answer a = 66.8 feet. Example 4. — An inclined plane is to be built for a railroad. The highest point is 264.5 feet above the lower or foot of the plane; the whole plane is 1463 feet long. What is the angle of inclination ? ■sin.G=^^=lQ°2h'. 1463 Set A on 264.5, B on 1463, clamp G. Move the arms until B comes to zero, then A shows the angle C= 10° 25' on circle sin. 210 DESCEIPTION OF Example 5. Fig. 3. Formula 4. — The side 6 is 436 feet long; the angle A = 42° 25', B = 21° 46'. What will be the length of the side at 436 X 8inA2° 25' a ■■ = 793.1 feet. s^?^.21° 46' Set A on 436, B on sm.21° 46', clamp C. Move the arms until B comes to smA2° 25', then the arm A shows the answer 793.1 feet. Example 6. Fig. 3. Formula 7. — The angle B^ 46° 38'; c = 33430 feet, h = 83190 feet. Require the angle (7=? . ^ 33430 X sinAQ° 38' ..„ .„, ''''■^= 33^90 16° 59'. Set A on sm.46° 38', B on 83190, clamp C. Move the arms until B comes to 33430, then A shows the answer C = 16° 59' on the circle sin. SPHEEICAL TEIGONOMETEY. Spherical trigonometry treats of the triangles which are (or imagined to be) drawn on the surface of a sphere. The sides are arcs of the great circle of the sphere, and measured by the angle of the arc. In spherical trigonometry the functions bear quite a different relation to the sides than in plane triangles, which here will be seen. A CALCULATING MACHINE. 211 Fig. 4. — A right-angled spherical triangle; A being the right angle. sin.a : 1 ■ = sin.c : sin. O cos.a : 1 = cot.B : tan. G » tan.a : 1 = tan.b : cos.O sin.c : 1 := tan.h : tan.B Fig. 5. — An oblique-angled triangle. sin.a : sin.h = sin.A : sin.B sin.h : sin.c = sin.B : sin. G Fig. 6. — A line drawn from the angle B perpendicu- lar to the side b ; the two parts of b are denoted by the letters m and n. J? cos.a : 1 = cot.m : tan.G tan.a : 1 = tan.m : cos.G COS.A : tan.c = sin.m : sin. (b — m) cos.a : cos.c = cos.m : cos. (6 — m) tan.a : tan.c = cos.B : cos.m 212 DESCRIPTION OF By spherical trigonometry, we ascertain distances and angles on the surface of the earth. It is princi- pally used in navigation; but, even on shore, persons knowing places on the earth, desire to know their exact distance, and in what direction or course the one place is from the other. For that purpose we will prepare the above formulae. By places being known, we mean their latitudes and longitudes are known; then, in the above formulae, tloB quantities A, B, G, and a, b, c, require to be expressed in latitude and longitude. When the difference in latitude and longitude is given', we obtain, in the difference of longitude, circle planes, which are drawn at a distance from the centre of the earth, equal to the sine for the latitude, with a radii = cosine for the latitude; then, in the parallel, the distance between two points will be the differ- ence in longitude multiplied by cosine for th.e lati- tude; but this will not be the shortest distance between the two points ; but the arc of a circle plane drawn through the two points, and the centre of the earth, will be the shortest distance. Letters will denote, L = difference in longitude between two places, in time expressed in degrees of the great circle. I = latitude in degrees. d' = distance between two places on the arc of a parallel, in degrees of the great circle. then d' = L cos. I, A CALCULATING MACHINE. 213 Example 1. — What will be the distance between Boston and Cape Creaux, (South Spain,) their differ- ence in longitude L = 74° 20' latitude I = 42° 20'. d'= 74.33 COS. 42° 20' = 54.95° multiplied by 60 will be geographical miles 3297, the distance on the parallel. d = distance on the arc of the great circle in de- grees, or shortest distance between two points, then sin.i d = sin.i L cos.l .... (1) TAP 90' Ji= ii_^=37°10' sin.i d = sin.2,1° 10' x cos.42° 20' = 26° 32' d = 26° 32' X 2 = 53° 4' multiplied by 60 will be a distance 3184 geographical miles, the shortest dis- tance. The difference between the two distances will be, d' = 3297 d = 3184 113 miles. When the vessel sails the distance d', she always keeps the same course east or west, but in the dis- tance d, the vessel will always be in a higher latitude than the starting-points, and the course from the first point will be, . ^ sin.L cos.l /n\ sin. (J = : — ^ — \^)- s%n.d G = course in degrees from the meridian, require the course from Boston to Cape Creaux. 214 DESCRIPTION OF . ^ sm.74:° 20' X cosA2° 20' ^^o ka/ sin.C= . ^„, , = 62° 56' sinM° 4' To operate this by the calculator, will be the same as before described, viz. : Set A on sm.74° 20', B on dnM° 4', clamp C; move the arms, until B comes to cos.42° 20' ; then A shows the course = 62° 56' or 51 points nearly. When the vessel has sailed half the distance, she is in the highest latitude, which will be found by the formulae. tan.l' = J^^^, (3) cos.i L tan.1' = ^° 20^-, = 48° 50' cos.37° 4' ' When the two places lie in different latitudes, and' tlieir difference in longitude is given, to find the near- est distance and course. Letters will denote, I = lower latitude. V = highest latitude. C = course from the latitude I. C = course from the latitude I', d = shortest distance between I and V in degrees of the great circle. L = diJ0ference in longitude in degrees tan.m = cot. I' cos.L. ?i = 90 + Z — m.. — I, when I and V are on one side of the equator. + I, when V is on one side and I on the other. A CALCULATING MACHINE. 215 Then the formulaB for calculating the shortest dis- tances and courses, to and from any hnown points on the earth, will be simply 7 dn.V cos.n /a \ cos.d = , .... (4) cos.m si^.C=^^', .... (5) mn.d dn.G'^'^-^, .... (6) sin.d Example 2. — Require the shortest distance and course from New York to Liverpool? I == 40° 42' N. latitude ) ^^^ ^ork, 74° " W. longitude j r = 53° 22' N. latitude Uiverpool, 2° 52' W. longitude J L = 71° 8' difference in longitude. tan.m =co<.53° 22' x cos.71° 8' = 13° 31', ^ = 90 _ 13° 31' — 40° 42' = 36° 47', , s^•?^.53° 22' x cos.35° 47' _ .70 ko, cos.d = 47 bS , cos.\6° ol Shortest distance = 47 x 60 + 58 = 2878 geo- graphical miles. dn (7= g^^-71° 8' X cos.53° 22' _ ^go 2%'= 41 points. 8inA1° 58' ^ Course from New York N. E. f E. 216 DESCRIPTION OF Example 3. — "What will be the distance and course from San Francisco to Port Jackson, Australia? V = 37° 47' N. latitude ) g^^ j,^^^^^ 122° 21' W. longitude i I = 33° 50' S. latitude 1 p^^, ^^^ 151° 25' E. longitude J Difference longitude i = 360 — 122° 50' — 151° 25' =86° 14'. tan.m = cot.B7° 47' x cosM° 14' = 4° 50', n = 90 + 33° 50' — 4° 50' = 119°, cos.d= ^-^-S?" 47' X C0. .119 _ 1070 20', cos.4° 50' ' Shortest distance = 107 x 60 + 20 = 6440 geo. miles. .m.(7'= ^^^-^6°14'x cggj3_°_50: ^g^o ig' = 5J points. sm.l07° 20' ^ Course from San Francisco S. W. h. W.i "W. nearly. When computing those examples by the calculator, there will be no more figuring on paper, than those shown in the print; the machine brings out the answer expressed in degrees and minutes. Improvement^ on the Calculating Machine by the addi- tion of two more Scales. The additional scales do not appear on the accom- panying Plate ; they are placed one on the inside, and one on the outside of the two scales. The outer one is divided into 90 equal parts around the whole circle, representing the 90 degrees in a quadrant; each of A CALCULATING MACHINE. 217 tiiose divisions is divided into 6 equal parts, repre- senting every 10 minutes per degree. This scale is accompanied with, a vernier, which is divided into 10 equal parts corresponding with 9 divisions on the scale ; by this arrangement, every minute on a degree can be distinctly read, as the ten divisions on the vernier are in a space about half an inch. The outer edge of the scale is numbered in hours, minutes, and seconds, in time; corresponding to degrees and minutes of the circle. The object of this scale is, for adding and subtracting degrees and minutes, and to turn de- grees and minutes into time. The inner scale (called the compass) is laid out from the mariner's compass, in paints and fractions thereof, which corresponds with courses, distances, and differ- ence in latitudes, longitude, &c., on the outer scales. The two additional scales are principally for navi- gation ; and the combination of the four scales will cffoer aU calculations at sea, so that navigators will be enabled to make quick and correct calculations, with- out reference to any tables. 15 218 DESCRIPTION OF NAVIGATION. To navigate a vessel upon the supposition that the earth is a level plane, on which the meridians are drawn north and south parallel with each other; and the parallels east and west, at right angles to the former. Kg. 7. N w- a- Fig. 7. — The line N. S. represents a meridian north and south; the line E. W. represents a parallel east and west. A ship in I, sailing in the direction I, V, and having reached I', it is required to know her position to the point I, which is measured by the line I, I, and the angle N. Z Z' ; and imagined by the lines I a and al't These four quantities bear the following names : — d = l V, distance from I to V. C = N. Z V, course or points from the meridian. 13 = 1 a, departure or difference in longitude. f=a I', difference in latitude. When any two of those four quantities are given, the other two can be ascertained by them, which ope- ration will be illustrated by the formulae. A CALCULATING MACHINE. 219 Departwe ? Biff, latitvde? Distance? {'ii = d sin. C ftan.G f=d COS. G b tan.G {;= d= d- sin. C' / cos.G^ Gourse? COS. G = _/ d- svn. G = -^, a tan.G = -J, (1) (2) (3) (4) (5) (6) (7) (8) (9) When the course is expressed in points from the meridian, we will change the characters for G, to distinguish them from the expression of degrees; namely, f lat.G=cos.G ) „ . , G m pomts < ^ . ^ > (7 in degrees. I. dep. G = sm. G J d, Ir, and I, are geographical miles. The characters represent lat. = difference in latitude ; dep. = departure. These characters are marked on the arms, over the compass on the machine. While the vessel is running from I to I', the dis- tance is measured by the log and time ; and the course is measured by the compass, commonly expressed in points. 220 DESCRIPTION OP B::ample 1. — A vessel sails ecist northreast (6 points) 236 miles. Require her departure and difference in latitude ? Set J. on 236, B on zero, clamp C. Move the arms until lat.B comes to 6 points; then A shows the differ- ence in latitude = 90.3 miles. Set dep.B on 6 points, then A shows the depar- ture = 218 mile^. Example 2. — A ship sails between north and west 173 miles, and found her difference in latitude was 74 miles. Eequire her course and departure? Set A on the distance 173 miles, B on 74, clamp G. Move the arm until A comes to zero; then lat.B shows the course = 51 points. Set A on 173, B on zero, clamp C. Move the arms until dep.B comes to 51 points; then A shows the de- parture = 156 miles. In the same manner, any question in plane sailing is computed. The supposition that the earth is a level plane will suffice wheu navigating a ship near the equator; but, in higher latitudes and long dis- tances, it wUl be necessary to partly relinquish this supposition, to find the difference in longitude of the meridians in degrees or in time at the equator. At the equator, there are 60 geographical miles per degree, but in higher latitudes there are so many miles less in a degree as the cosine for the latitude is less than the radii. Say the departure II to be the difference in longitude in miles at the latitude I, and A CALCULATIKG MACHINE. L the difference in longitude in degrees at tor, we have i : ib == 1 : 60 cos.l, which will be in the form of equations, f b = 60 Gos.l L, 60 cos.l L 221 the equa- Departure? Biff, latitude? f= tan.O 60 cos.l L Distarwe? d = . sin.G Course ? d=y/V-\-f, . 60 cos.l L Latitude? sin.G := ian.G= cos.l = cos.l = d ' 60 cos.l L Biff, longitude? cos. L= L = 60 r d.sin. G ~WL' 1 f tan. G ■^="6rr' 60 cos.l d.si/n. G L=z 60 cos.l / tan.G 60 cos.V (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) 222 DESCRIPTION" OF Now for any question of departure, difference in lati- tude, distance, course, latitude, difference in longitude, there is one formula which contains the given quan- tities and gives the answer. Example 3. — A ship sails in north latitude in a course E. S. E. i E. = 61 points. In a distance of 132 miles, she made a difference in longitude of 3° 34'. What latitude is she in? This question will be answered by the formula (19), which contains the given quantities course, distance, and differertce in longitude. First multiply 60 by 3°, and add 34' = 214' miles. cos.l = 132 xdep.Qi ^ 530 -^^, ^^^ latitude. 214 ' Set A on 214, B on 132, clamp C. Move A to dep. 6i points; the co8.B shows the latitude = 53° 15'. In the same simple manner, all the questions are solved by the machine, which gives a positive answer instantly. Persons who do not possess this machine will still find the formulae convenient. Engineers on steam- boats often wish to know their positions at sea, for which purpose the accompanying table is inserted. A CALCULATING MACHINE. 223 North. South. Points. Degrees. ain.C. dep.C. COS. C. lat.C. tan. 0. lat. N. J- f 2° 49' 5 37 8 26 .0491 .0979 .1544 .9988 .9952 .9880 .0492 .0983 .1982 N. by E. and N. by W. S. by E. and S. by W. V 1 If 11 15 14 4 16 52 19 41 .1936 .2430 .2901 .3368 .9811 .9700 .9570 .9416 .1989 .2505 .3032 .3577 N. N. E. and N. N. W. S. S. E. and S. S. W. 2 2i 2J 2i 22 30 25 19 28 7 30 56 .3827 .4276 .4713 .5140 .9239 .9039 .8820 .8577 .4142 .4730 .53^3 .5993 N. E. by N. and N.W.byN. S. E.byS. r and S. W. by S. 3 3i 3J 3f 33 45 36 44 39 22 42 11 .5555 .5981 .6343 .6715 .8314 .8014 .7731 .7410 .6883 .7463 .8204 .9062 N. E. and N. W. S. E. and S. W. 4 4J 4i 45 47 49 50 37 53 26 .7071 .7410 .7731 .8014 .7071 .6715 .6345 .5981 1.000 1.103 1.218 1.348 N. E. by E. and N.W.byW. S. E. by E. and J S.W. byW. ( 5 bi bi 5f 56 15 59 4 61 52 64 41 .8314 .8577 .8820 .9039 .5555 .5140 .4713 .4276 1.496 1.668 1.870. 2.114 E. N. E. and W. N. W. E. s. E. r and -j W. S. W. 6 6i 6f 67 30 70 19 73 7 75 56 .9239 .9416 .9570 .9700 .3827 .3368 .2901 .2430 2.414 2.795 3.295 3.991 E. by N. and W. by N. E. by S. C and ■{ W. by S. 7 71 78 45 81 34 84 22 87 11 .9811 .9880 .9952 .9988 .1936 .1544 .0979 .0491 5.027 6.744 11.14 20.32 East or west . . . 8 90° 1.000 0.000 ■ c» 224 DESCEIPTION OF Example 4. — A ship sails from Sandy Hook (New York), latitude 40° 27', in a course S. E. by S. \ E. = 3i points, until her latitude is 32° 54'. Require her distance and diflFerence in longitude from Sandy Hook? From . . . .43° 27' Subtract . . .32° 54' Difference in latitude /= 7° 33' x 60 = 453 miles. From the formula (6) we have the distance , 453 453 KQK K ., d = — — - := TT-s— — = 585.5 miles. cosM 0.7731 From the formula (23) we have ^_ 453 X tanM _ 4:5B x 0.8204 _,, ofi_^o 216' ~ 60 X eos.3254 "~ 60 x OM ' ' ' the difference in longitude. Longitude of Sandy Hook . . . 74° 00.5' Subtract ...... 7° 21.6' Longitude in 66° 39' From this point, require the course and distance to Cape Florida? Latitude . . Z = 25° 41' ) Longitude . . = 80° 05' P "?" ^^""'^^ Subtract . . 66° 39' Difference in longitude L = 13° 26' From ... 32° 54' Subtract . . 25° 41' Difference in latitude /= 7° 13' x 60 = 433 miles. From the formula (17) we have A CALCULATING MACHINE. 225 y^^ n 60xcos.25°41'xl3°26' , «,,„ ., . , tan. U = — = 1.676^54 points, or course = S. W. by W. 4 W. From the formula (6) we have distance d = — — - =: — _ - = 841 miles. COS.64 0.514 When the course and distance are required very accurate^ calculate them from the formulae on p. 215. To find the Trigonometrical lines for any minute, hj the accompanying tables. Example 1. — Find the length of s*ra.35° 34', from s^?^.35° 40' = 0.58306 subtract sm.85° 30' = 0.58070 proportional part . 236 multiply by . . 0.4 94.4 add ... 58070 sm.35° 34' = 0.581644, the answer. Emmple 2.— Find the length of the tan.Q8° 47'? from tora.68° 50' = 2.5826 subtract tom.68° 40' = 2.5604 proportional part . multiply by add . 222 0.7 154.4 2.5604 tan.68° 47' = 2.57584, the answer. APPENDIX. A few simple rules to calculate the time and position on the earth, from the motion of the heavenly bodies. Letters mil denote, A == meridian altitude of the sun or any other hea- venly body. a = any other observed altitude of the same. (A and a = correct centre altitudes.) jD = declination of the sun, or any other heavenly body. I = latitude in the place of observation. L = longitude between the meridian in the place of observation, and the meridian where the heavenly body passes under the observation. V = the angle at which the earth rotates, before or after six o'clock, when the heavenly bodies set or rise, viewed from the latitude I. t = apparent solar time in tours. t' = apparent mxmi time in hours. X = time in minutes, when the moon passes from one meridian to another, and the longitude between the two meridians is L. 228 APPENDIX. COS. l = 90 — A±D, . A^90 — l±D, . D= + 90 ±A±l, sin.v = tan.l tan.D . J sinM (1 + sin.v) — sin.A sin.v tan.l = sin. .a' cos.L — sin.a cos.L' tan.D = tan.D (+ sin.a + sin.a') sin.a' COS.L — sin.a cos.L' tan.l (+ si/n.a + sin.a') t^Ii . . . . 15 (1) (2) (3) (4) (5) (6) (7) (8) t' = 0.965 t (9) X = 0.14 L (10) Where the quantities have the double signs plus and minus, use the top signs when the latitude and declination are of equal names ; use the bottom signs when the latitude and decHnation are of different names. Example 1. Formula 1. — On the 21st day of Octo- ber, 1852, the meridian altitude of the sun's lower limb was observed to be 36° 27'. Eequire the latitude? Observed altitude . . . .36° 27' Co»rection, semid. parallax, &c Correct centre altitude . Subtracted from True zenith distance Sun's declination, subtract Latitude in . add 10' ^ = 46 37' . 90 . 43 23 Z> = 11 0' S Z= 32 23 i\r APPENDIX. 229 Example 2. Formula 4. — What time does the sun set and rise, on the 27th day of May, 1853, in latitude 43° 19'? mi V -I T ,. r in the morning D = 21° 22' The sun s decimation { { in the afternoon D = 21° 30' sm.v = tara.43° 19' X tan.21° 22' = 0.3689 = 21° 20', on 91° 90' t = .,- = 4' 34' 40" o'clock in the morn. 15 sin in.v = tanAS° 19' x tati.21° 30' = 0.3714 = 21° 48' . 90 + 21° 48' 7.h97M9" '1 1 • .1. t = Y5 = ' ^' ^^ ° clock m the evemng. Example 3 . Formula 5 . — To find the apparent tvme. In April 17, 1853, the correct altitude of the sun was observed in the afternoon to be a = 31° 31' in the latitude ?= 38° 47' N; the sun's declination, at the time of observation, was D = 10° 37' N. Require the apparent time of observation? sm.v = tan.3S° 47' x tan.lO° 37' = 0.1506, ^ = 90 _ 38° 47' + 10° 37' = 61° 50' ^j^»in.31° 31- (1 + 0.1506) _o.i6o6=0.6813 5m.61° 50' = 57° 54', 57° 54' t = — -- — =3'' 51' 36", the apparent time. 15 When it is a forenoon observation, t will be , 90 + i: ^ = ~15~ When the difference in longitude is to be found by the appaa-ent time, and the time shown by the chrono- meter, it will be necessary to notice the diflference between the apparent and mean solar time. 230 APPENDIX. The difference between ajpparent time and mean time is attributable to the irregularity of the earth's rota- tion around its axis, which causes a number of days, "say 100," in one part of the year, to be half an hour longer than 100 days in another part of the year, when a day is the time between the sun's passage over one meridian; this time is called the apparent time. Mean time is the time shown by a chronometer, and is always uniform, so that 100 days in one part of the year are always equal to 100 days in any other part of the year. The difference between the apparent and mean time is called the equation of time, and al- ways found in the nautical almanac, where it is noted if it is to be added to or subtracted from the ap- parent time. Example 4. — In connection with the preceding ex- ample, suppose the chronometer shows the mean time at Greenwich 7'' 27' 55" at the time of observation. Eequire the longitude from Greenwich? Apparent time 3'' 51' 36" * Equation of time, subtract 30" Mean time subtract from 3" 51' 6" 7" 27' 55" Multiply by West longitude 3" 36' 49" 15 54° 12' 15" When the longitude is to be found by an altitude a of the moon, the equation of time need not be noticed, because it is contained in the time when the moon passes the meridian at Greenwich. APPENDIX. 231 Example 5. — On the 25th day of September, 1852, in north latitude 22° 35', and west longitude about 53° 9', at T*" 15' o'clock by watch, was taken an altitude Of the moon's lower limb . . .33° 42' Correction semd. parlx. refn. add . 45' D correct centre altitude . . a = 34° 27' D declination corrected . . i)= 13° 52' 43" S Passes the meridian at Greenwich . lO** 21'. 6 Correction add . a; = 58 x 0.14 = 0" 7'.4 3 passes the meridian . . . lO"* 29' Kequire the mean time and longitude from Greenwich? dn.v=:ztan.22° 35' x taw.l3° 52' 43" = 0.10275, ^ = 90 — 13° 52' 43" — 22° 35' = 53° 32' 17" COS.L = ^•^•34°27' (1-0.10275) ^0 -^Q^^S^O ^3385 sm.53° 32' 17" = 42° 47', 42° 47' V = — -— — = 2*" 51' 8" apparent moon time. 15 Divided by 0.965 will be « = 2" 57' 20". 3) passes the meridian at . . . 10'' 29' Subtract «= 2" 57' 20" ifein itme of observation . . . 7'' 31' 40" Watch too slow 16' 40" Time of observation by chronometer . IV" 13' 15" Subtract mean time of obs. ... 7'' 31' 40" 3H1'35" Multiply by 15 West longitude 55° 23' 45" 232 APPENDIX. The formula (6) is for calculating the latitude from, two altitudes of the sun or any other heavenly body, when the times of observation are known. If this formula is to be used for the moon, it is necessary to notice the difference in the declination at the times of observation; then the formulae will appear as , 7 dn.a' COS.L — sin.a L' + tan.D dn.a+^tan.iy svn.a' in which D and D' are declinations at the altitudes a and a'. These calculations are computed very rapidly by the Calculator, owing to the trigonometrical func- tions being instantly at hand, and no tables but the nautical almanac being required. THE END. ADVERTISEMENTS. John "W. Ntstrom, of Philadelphia, Engineer, furnishes Draw- ings of Screw Propellers and Propeller Engines, with all calculations connected with the same. Designs Engines and Propellers suitable for any desired description of vessels. Furnishes drawings of vessels, with their whole internal arrangements distinctly shown in sections and details of any desired scale. Designs Cabin and State-room arrangements, location of Machinery, Boiler, Coal-boxes, and Fire- Room, capacity for Cargo, &c., with all computations in reference to the entire capability of the vessel. Several years of experience in this profession enable him to furnish complete and correct drawings on a very short notice. It is very important to have full and complete drawings hefove the keel of the vessel is laid, so as to insure unity of action between the Ship and Engine Builders, and enable them to obtain in time proper materials; and furthermore to afford an opportunity for a clear understanding in reference to contracts. The drawings wiU contain all new and useful improvements, suit- able for any desired purpose. Letters directed to J. W. Nysteom, Philadelphia, will be promptly attended to. NYSTROM'S CALCULATOR, This instrument ought to be in the hands of Engineers, Shipbuild- ers, and all whose business requires frequent and extensive calcula- tions. The price of the Calculator is from fifteen to twenty dollars; and it can be obtained by addressing J. W. Nystkom, Philadelphia. It is a complete instrument, made of brass, and will not get out of order. These Calculators received the FIRST PREMIUM At the Exhibition of the Franklin Institute, Philadelphia, Pa. 16 PUBLICATIONS HENET OAEET BAIED, SUCCESSOR TO E. L. CARET, No. 7 Mart's Building, Sixth Street, above Chestnut, Philadelphia. SCIENTIFIC AND PRACTICAL THE PRACTICAL MODEL CALCULATOR, For the Engineer, Machinist, Manufacturer of Engine Work^ Naval ArcMtect, Miner, and Millwriglit. By Oliver Byrne, Compiler and Editor of the Dictionary of Machines, Mechanics, Engine Work and Engineering, and Author of various Mathematical and Mechanical Works. Illustrated by numerous Engravings. Now complete. One large Volume, Octavo, of nearly six hundred pages $3.50 It will contain such calculations as are met with and required in the Mechanical Arts, and eatablish models or standards to guide practical men. The TableB that are introduced, many of which are new, will greatly economize labour, and render the every-day calculations of the practical man comprehensive and easy. I'rom every single calculation given in this work numerous other calculations are readily modelled, so that each may be considered the head of a numerous family of practical results. The examples selected will be found appropriate, and in all cases taken from the actual practice of the present time. Every rule has been tested by the unerring results of mathematical research, and confirmed by experiment, when such was necessary. The Practical Model Calculator will be found to fill a vacancy in the library of the practical working-man long considered a recLuirement. It will be found to excel all other works of a similar nature, from the great extent of its range, the exemplary nature of its well-selected examples, and from the easy, simple, and sys- tematic manner in which the model calculations are established. NORRIS'S HAND-BOOK FOR LOCOMOTIVE ENGINEERS AND MACHINISTS : Comprising the Calculations for Constructing Locomotives. Manner of setting Valves, &o. &c. By Septimus Nobkis, Civil and Mechanical Engineer. In One Volume, 12nio, with illustrations $1.50 "With pleasure do we meet with such a work as Messrs. Norris and Baird have given us. — Artisan, In tliis work, he has given what are called the " secrets of the business," in the rules to construct locomo- tives, in order that the million should be learned in all things. — Sckntijlc American. A TREATISE ON THE AMERICAN STEAM-ENGINE. Illustrated by numerous Wood Cuts and other Engravings. By Olivee Btenb. In one Volume. (In press.) PUBLICATIONS OF HENBT CAEEY BAIKB. THE PEACTICAL COTTON-SPINNER AND MANUFACTTJREE ; OR, THE MANAGER'S AND OVERLOOKER'S COMPANION. This work contains a Comprehensive System of Calculations for Mill Gearing and Machinery, from the first moving power through the different processes of Carding, Drawing, Slabbing, Eoving, Spinning, and Weaving, adapted to American Machinery, Practice, and Usages. Compendious Tables of Yams and Eeeds are added. Illustrated by large Working-Drawings of the most approved American Cotton Machinery. Complete in One Volume, octavo $3.50 This edition of Scott's Cotton-Spinner, by Oliver Byrne, is designed for the American Operative. It will be found intensely practical, and will be of the greatest possible value to the Manager, Overseer, and Workman. 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A TREATISE ON SCREW-PROPELLERS AND THEIR STEAM- ENGINES, With Practical Rules and Examples by which to Calculate and Construct the same for any description of Vessels. By J. W. Ntstbom. Illus- trated by over thirty large working Drawings. In one Volume, octavo $3.50 THE ANALYTICAL CHEMIST'S ASSISTANT : A Manual of Chemical Analysis, both Quahtative and Quan- titative, of Natural and Artificial Inorganic Compounds ; to which are appended the Rules for Detecting Arsenic in a Case of Poisoning. By Fkedekik Wcehlee, Professor of Chemistry in the University of Gottingen. Translated from the Ger- man, with an Introduction, Illustrations, and copious Additions, by Osoak M. LiEBER, Author of " The Assayer's Guide." In one Volume, 12mo $1.25 THE FRUIT, FLOWER, AND KITCHEN GARDEN. By Patrick Neill, L. L. D., F. R. S. E., Secretary to the Royal Caledonian Horticultural Society. Adapted to the United States, from the Fourth Edition, revised and improved by the Author. Illustrated by fifty Wood Engravings of Hothouses, &c. &c. In One Volume, 12mo |l-25 This volume supplies a desideratum much felt, and gives within a moderate compass all the horticultural information necessary for praoti.^ ^'fT-S^-iZ^ ™2w«^, PnM^t A valuable addition to the horticultorist's library.-BoBimore Fotrmt. PUBLICATIONS OF HENBY CAEEY BAIKD. THE ENCYCLOPEDIA OF CHEMISTRY, PRACTICAL AND THEORETICAL : Embracing its Application to the Arts, Metallurgy, Mineralogy, Geology, Medicine, and Pharmacy. By James C. Booth, Melter and Kefiner in the United States Mint, Professor of Applied Chemistry in the Franklin Institute, &c. ; assisted by Campbell Morfit, Author of " Chemical Manipulations," &c. Complete in One Volume, royal octavo, 978 pages, with numerous Woodcuts and other Illustrations. Second Edition. Full bound $5 It covers the whole field of Chemiatry as applied to Arts and ScieDces. * * * As no library is complete without a common dictionary, it is also our opinion that none can be without this Encyclopedia of Chemis- try. — Sdenbijic American. A work of time and labour, and a treasury of chemical iuformation.-s-iVoriA American. By far the best manual of the kind which has been presented to the American public. — Bostofp Courier. An invaluable work for the dissemination of sound practical knowledge. — Ledger. A treasury of chemical information, including all the latest and most important discoveries. — Baltimore American. ELEMENTARY PRINCIPLES OF CARPENTRY. By Thomas Tredgold. In One Volume, quarto, with nume- rous Illustrations $2.50 RURAL CHEMISTRY: An Elementary Introduction to the Study of the Science, in its relation to Agriculture and the Arts of Life. By Edward Sollet, Professor of Chemistry in the Horticultural Society of London. 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In One Volume, 8vo...37J eta PTTBLICATIONS OF HENRY CABEY BAIBB. HOUSEHOLD SURGERY ; OR, HINTS ON EMERGENCIES. By J. F. South, one of the Surgeons of St. Thomas's Hospi- tal. In One Volume, 12mo. Illustrated by nearly fifty Engravings $ 1 . 2r- THE COMPLETE PRACTICAL BREWER; Or, Plain, Concise, and Accurate Instructions in the Art of Brewing Beer, Ale, Porter, &o. &c., and the Process of Making all the Small Beers. By M. Lafayette Bykn, M. D. With Illustrations. 12mo $1.00 THE COMPLETE PRACTICAL DISTILLER; By M. Lafayette Bten, M. D. With Illustrations. 12nio $1.00 THE PYROTECHNIST'S COMPANION; Or, A Familiar System of Recreative Fire- Works. By G. W. MoKTiMEB. Illustrated by numerous Engravings. 12mo 75 ota. ELECTROTYPE MANIPULATION: Being the Theory and Plain Instructions in the Art of Work- ing in Metals, by Precipitating them from their Solutions, through the agency of Galvanic or Voltaic Electricity. By Chaeles V. Walker, Hon. Secretary to the London Electrical Society, &o. 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Thoroughly Revised, with the Addition of New Receipts. In One Volume, 12mo, half bound, or in sheep $1 In preparing a new and carefully revised edition of this my first work on cookery, I have introduced improvements, corrected errors, and added new receipts, that I trust will on trial he found satisfactory. The success of the book (proved hy its immense and increasing circulation) affords conclusive evidence that it has obtained the approbation of a large number of my countrywomen ; many of whom have informed me that it has made practical housewives of young ladies who have entered into married life with no other ac- quirements than a few showy accomplishments. Gentlemen, also, have told me of great improvements in the family table, after presenting their wives with this manual of domestic cookery, and that, after a morn- ing devoted to the fatigues of business, they no longer find themselves sutgected to the annoyance of an ill-dressed dinner. — Pr^a/x. 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