UNIVERSITY OF CALIFORNIA. Class THE RESISTANCE AND PROPULSION OF SHIPS. BY WILLIAM F. DURAND, Professor of Mechanical Engineering, Leland Stanford* Junior University. SECOND EDITION, THOROUGHLY REVISED. FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. LONDON : CHAPMAN & HALL, LIMITED. 1909. GENERAL Copyright, 1898, 1909, I BY WILLIAM F DURAND. frbr g-rintttfir Sohrrl Srimunnnii anD Company 3f n fork PREFACE TO SECOND EDITION. DURING the ten years since the first edition of this book was prepared, the chief contributions to the material available for the discussion of the resistance and propulsion of ships consist, of the results of various experimental researches, notably in regard to the operation of the screw propeller. In the prepa- ration of the present revision no attempt has been made to change the general character of the theoretical and descriptive treatment of the subject. References have been given here and there, however, to some of the more important recent contri- butions to general theory. Regarding the results of experience, an effort has been made to summarize some of the more important researches in this field, and especially in relation 'to the problem of design. Some matter of relatively minor importance, and certain discussions which could well be superseded by new matter based on more recent experience have been omitted, thus permitting the inser- tion of a considerable amount of new matter without material change in the size of the volume. It is hoped that the present revision will serve as a reasonably comprehensive introduction ;o the principal problems of ship resistance and propulsion, and that in reference to the various problems of design, the data given and methods suggested will be found adequately represen- tative of present day practice. STANFORD UNIVERSITY, CALIFORNIA, January, 1909. iii 188508 PREFACE TO FIRST EDITION. DURING the last twenty or thirty years the literature relating to the Resistance and Propulsion of ships has received many valuable and important additions. Of the few books in English published in this period, those of the highest value have been restricted either in scope or in mode of treatment. For the most part, however, the important addi- tions to the subject have been published only in the transac- tions of engineering and scientific societies, or in the technical press. Such papers and special articles are far from provid- ing a connected account of the trend of modern thought and practice, and the present work has been undertaken in the hope that there might be a field of usefulness for a connected and fairly comprehensive exposition of the subject from the modern scientific and engineering standpoint. Such a work must depend in large measure on the extant literature of the subject, and the author would here express his general acknowledgments to those who have preceded him in this field. In many places special obligations are due, and it has been the intention to give special references at such points to the original papers or sources. With the material drawn from the general literature of the subject there has been combined a considerable amount of original VI PREFA CE. matter which it is hoped will contribute somewhat to what- ever of interest or value the work as a whole may possess. A free use has been made of calculus and mechanics in the development of the subject, the nature of the treatment requiring the use of these powerful auxiliaries. At the same time most of the important results and considerations bearing on them are discussed in general terms and from the descrip- tive standpoint, and all operations involved in the actual solution of problems are reduced to simple expression in terms of elementary mathematical processes. It will thus be possible, by the omission of the parts involving higher mathematics, to still obtain a fairly connected idea of most of the subject from the descriptive standpoint, and to apply all methods developed for purposes of design. In the development of such methods the purpose has been to supply plain paths along which the student may pro- ceed step by step from the initial conditions to the desired results, and to arrange the method in such a way as shall con- duce to the most intelligent application of engineering judg- ment and experience. In the design of screw-propellers especially, where the number of controlling conditions is necessarily large, the purpose has been to present a mode of solution in which the most important controlling conditions are represented in the formulae, and in which the determina- tion of the numerical values of these representatives by auxiliary computation, estimate, or assumption is forced upon the attention of the student in such a way as to call at each step for a definite act of engineering judgment. In so far as the method may differ from others, it is intended to present the series of operations in such way as to favor the recognition of a considerable number of relatively PREFA CE. vii simple steps, and the full and free application at each step of such judgment and precedent as may be at hand. Such methods are perhaps better adapted to the discipline of the student than to the uses of the trained designer, and it is obviously for the former rather than for the latter that such a work should be prepared. At the same time it seems not unlikely that even for the trained designer the splitting up of concrete judgments into their separate factors is often an operation of value, and one which will the more readily enable him to adapt his methods to rapidly changing prece- dents and conditions of design. The natural limitations of size and the need of homo- geneity have rendered the work in many respects less com- plete than the Author might have wished, and many important developments especially in pure theory have received but scanty notice. As presented, the work repre- sents substantially the lectures on Resistance and Propulsion given by the Author to students of Cornell University in the School of Marine Construction, and many features both in subject-matter and mode of treatment have been introduced as a result of the experience thus obtained in dealing with these subjects. CORNELL UNIVERSITY, ITHACA, N. Y., February 4, 1898. CONTENTS. CHAPTER I. RESISTANCE. SECTION PAE 1. General Ideas i 2. Stream-lines 7 3. Geometry of Stream-lines 16 4. Resume of General Considerations 28 5. Resistance of Deeply Immersed Planes Moving Normal to Themselves 32 6. Resistance of -Deeply Immersed Planes Moving Obliquely to their Normal 37 7. Tangential or Skin Resistance 42 8. Skin-resistance of Ship-formed Bodies 48 9. Actual Values of the Quantities/ and n for Skin-resistance 50 10. Waves 54 11. Wave-formation Due to the Motion of a Ship-formed Body through the Water 75 12. The Propagation of a Train of Waves 87 13. Formation of the Echoes in the Transverse and Divergent Systems of Waves 89 14. Wave-making Resistance 93 15. Relation of Resistance in General to the Density of the Liquid 104 16. Influence of Form and Proportion on Resistance 104 17. Modification of Resistance Due to Irregular Movement 107 18. Variation of Resistance Due to Rough Water 108 19. Increase of Resistance Due to Shallow Water or to the Influence of Banks and Shoals 109 20. Increase of Resistance Due to Slope of Currents .- 123 1 21. Influence on Resistance Due to Changes of Trim 124 22. Influence of Bilge-keels on Resistance 127 23. Air- resistance 129 24. Influence of Foul Bottom on Resistance 131 fa CONTENTS. 25. Speed at which Resistance Begins to Rapidly Increase 131 26. The Law of Comparison or of Kinematic Similitude 133 27. Applications of the Law of Comparison 147 28. General Remarks on the Theories of Resistance 150 29. Actual Formulae for Resistance 151 30. Experimental Methods of Determining the Resistance of Ship-formed Bodies 154 31. Oblique Resistance 157 CHAPTER II. PRO PULSION. 32. General Statement of the Problem 162 33. Action of a Propulsive Element 164 34. Definitions Relating to Screw Propellers 174 35. Propulsive Action of the Element of a Screw Propeller 177 36. Propulsive Action of the Entire Propeller 182 37. Action of a Screw Propeller Viewed from the Standpoint of the Water Acted on and the Acceleration Imparted 193 38. The Paddle-wheel Treated from the Standpoint of 37., 196 39. Hydraulic Propulsion 201 40. Screw Turbines or Screw Propellers with Guide-blades 203 CHAPTER III. REACTION BETWEEN SHIP AND PROPELLER. 41. The Constitution of the Wake 206 42. Definition of Different Kinds of Slip, of Mean Slip, and of Mean Pitch 209 43. Influences of Obliquity of Steam and of Shaft on the Action of a Screw Propeller 218 44. Effect of the Wake and its Variability on the Equations of 36 222 45. Augmentation of Resistance Due to Action of Propeller 227 46. Analysis of Power Necessary for Propulsion 228 47. Indicated Thrust 236 48. Negative Apparent Slip 237 CHAPTER IV. PROPELLER DESIGN. 49. Connection of Model Experiments with Actual Propellers 240 50. Problems of Propeller Design 252 51. General Suggestions Relating to the Choice of Values in the Equations for Propeller Design, and to the Question of Number and Location of Propellers 281 CONTENTS. xi SECTION PAGE 52. Special Conditions Affecting the Operation of Screw Propellers 288 53. The Direct Application of the Law of Comparison to Propeller Design 296 54. The Strength of Propeller-blades 299 55. Geometry of the Screw Propeller 306 CHAPTER V. PO\VERLNTG SHIPS. 56. Introductory 322 57. The Computation of W p or the Work Absorbed by the Propeller 323 58. Engine Friction 324 59. Power Required for Auxiliaries 332 60. Illustrative Example . 334 61. Powering with Turbine Machinery 336 62. Powering by the Law of Comparison 337 63. The Admiralty Displacement Coefficient 340 64. The Admiralty Midship-section Coefficient 344 65. Formulae Involving Wetted Surface 344 66. Other Special Constants 346 67. The Law of Comparison when the Ships are not Exactly Similar 347 68. English's Mode of Comparison 352 69. The Various Sections of a Displacement Speed Power Surface 354 70. Application of the Law of Comparison to Show the General Relation between Size and Carrying Capacity for a Given Speed., , , , 356 CHAPTER VI. TRIAL TRIPS. 71. Introductory 362 72. Detailed Consideration of the Observations to be Made 364 73. Elimination of Tidal Influence 368 74. Speed Trials for the Purpose of Obtaining a Continuous Relation be- tween Speed, Revolutions, and Power 372 75. Relation between Speed Power and Revolutions for a Long-distance Trial 378 76. Long-course Trial with Standardized Screw 380 77. The Influence of Acceleration and Retardation on Trial-trip Data 383 78. The Time and Distance Required to Effect a Change of Speed 384 79. The Geometrical Analysis of Trial Data 394 80. Application of Logarithmic Cross-section Paper 404 81. Steamship and Propeller Data 411 INTRODUCTORY NOTE RELATING TO UNITS OF MEASUREMENT. THE units of weight, and of force in general, used in naval architecture are the pound and the ton, the latter usually of 2240 Ibs. It will always be so understood in the present volume. The units of velocity are the foot per second, the/ Integrating, we have for the total resultant along OC Q = ^ sin 0,. This total force outward from O toward C may be equili- brated by a pair of tangential tensions at A and B. Let each of these be denoted by /. Then we have 2p, . - . . E Jp. FIG. 5. of value o-av* -r- g. The effect due to the straight part FE must be o, and the two tensions at F and E will balance each other. Hence for the entire resultant of the forces in the dotted part of the contour we shall have the equal tangential forces p, = AM and /> 3 = DN at A and D. Now since the system of forces for the entire contour is balanced, it follows that the forces due to the liquid in ABCD must be repre- sented by two forces equal and opposite to AM and DN. Hence AR and DS must represent in direction, point of application, and amount the resultant of the forces due to the water in ABCD. As a special case let ABCDE, Fig. 6, be a curved pipe of any varying sectional area, but with its ends A and E equal, 14 RESISTANCE AND PROPULSION OF SHIPS. opposite in direction, and in the same line. Applying the above principle, it follows that the resultant will be that of FIG. 6. the two forces AR and ES, equal and opposite in direction and in the same line. Hence in such case the resultant is o and there is no tendency to move the pipe in any direction, and in particular no force tending to move it in the direction AE. This serves to establish the initial proposition relative to the forces in such a tube of flow. In the most general case the straight portions at the two ends, while parallel, are not in the same straight line. The preceding treatment still holds, however, and we shall have for such case the resultant represented by two equal and opposite forces not in the same line and therefore forming a couple. Let now AB, Fig. 7, be a body about which the liquid flows without tangential forces in stream-line paths. Let FIG. 7. this body be immersed at an indefinite depth so that surface effects are insensible. We wish to show that the total force RESISTANCE. 1 5 communicated to the body by the stream-lines is o. This may be seen by considering PQRS a tube of flow entirely surrounding ACBD, with its straight ends equal in area, opposite in direction, and in the same line, and of variable sectional area, ACBD forming an internal boundary. The general proposition above then applies, and the resultant of all forces acting on the internal boundary ACBD will be o. Otherwise the tube PQRS may be considered as made up of an indefinite number of small tubes for each of which the y resultant may be a couple. It is easily seen, however, that the forces constituting these couples will, for the whole tube, constitute a system uniformly distributed over the ends PR and QS, and hence the entire resultant will be o. This important conclusion may also be reached by con- sidering that if a resultant force were communicated from the liquid to AB, we might obtain work by allowing AB to move and thus overcome some resistance. By the supposition of the absence of all tangential force between AB and the liquid, this would take place without affecting the velocity of the latter. In fact without tangential force the velocity of the liquid can in no wise be affected. Hence we should obtain work without the expenditure of energy, and the sup- positions leading to this result are therefore inadmissible. We may at this point note a further result of the relation- ship expressed by the general equation of hydrodynamics above. Suppose z to remain substantially constant and v to increase continually. The pressure will correspondingly decrease, and at the limit for a certain velocity we shall have / = o. An indefinitely small increase of velocity would lead to a tendency to set up a negative pressure or tension in the UNIVERSITY. OF 1 6 RESISTANCE AND PROPULSION OF SHIPS. liquid. This could not be actually realized, and the result would be, instead, a breaking up of the stream into minute turbulent whirls and eddies. The existence of such turbu- lence may therefore imply a previous condition in which there existed a tendency toward an indefinite decrease in the pressure, and such may be considered as its hydrodynamical significance. In the actual case the stream becomes unsteady and breaks up before the pressure becomes actually reduced to zero, although it is always much reduced as compared with the pressure for steady flow. A further result of the great reduction in pressure is usually found in the giving up by the* water of the air which it holds in solution. The air thus liberated appears in the form of small bubbles mingled with the water, causing the foamy or yeasty appearance which usually accompanies this phenomenon. . 3. GEOMETRY OF STREAM -LINES. Among the first to note the relation of stream-lines to the problem of ship resistance was Prof. Rankine. His investi- gations covered the examination of their general properties and the derivation of forms for ships which, under the special conditions assumed, would give for the relative motion of ship and water smooth stream-line paths. We will now show methods of constructing stream-lines for various special cases as developed by Rankine and other investigators since his time. The stream-lines which surround a body are in general space-curves. The properties of such lines of double curva- ture are so complex that their investigation is a work of much difficulty and labor. Many helpful and instructive sugges- RESISTANCE. I/ tions, however, may be derived from a study of plane water- lines to which the present notice will be restricted. In order to imagine a physical condition which would give such lines, suppose a body partly immersed in a liquid on the surface of which is a rigid frictionless plane. Let a similar parallel plane touch the under side of the body. Then let the liquid between the planes move past the body, or vice versa let the body move relative to the liquid. The relative motion of liquid and body may then be considered as taking place in water-lines contained in horizontal planes. At the limit we may suppose the planes approached very near, the body being then simply that portion contained between them. The liquid will then move in a thin horizontal sheet in plane water-lines. It is shown in hydromechanics that for such water-lines there exists a function fy such that i . u = velocity along x = and v = velocity along y = dx That is, that the velocity in any direction is the rate of change of this function in a direction perpendicular to the given one. It is also shown that this function is constant for any one water-line, and that it may therefore be considered as a characteristic or equation to such line. It is likewise shown that its value for any one line is proportional to the total amount of flow between such line and some one line taken as a standard or line of reference. 18 RESISTANCE AND PROPULSION OF SHIPS. For the simplest case let us assume two planes as above, indefinite in extent and very near. At a given point let liquid be introduced at a uniform rate. Such a point of introduction is called a source. In such case the liquid will move in straight lines radiating from the source. Similarly we might have assumed liquid abstracted at a uniform rate. Such a point of abstraction is called a sink. In such case likewise, the liquid will move in straight lines converging toward the sink. In Fig. 8 let O be the point and OA the line of reference from which the total flow is measured. Let a be the amount introduced per second or other unit of time, and V the angle FIG. 8. between OA and any line OP. Then the flow between OA and OP is that corresponding to the angle A OP. Hence we have The quantity a -r- 2n is called the strength of the source or sink. It is also shown in hydromechanics that the function ^ for a complex system of sinks and sources is simply the algebraic sum of the separate functions, considering i/> for a source as -|- and for a sink as . Hence for such a system in general we have RESISTANCE. For two sources of equal strength we have aO l aO, ib. = - and iL' = . ri 27T ^ a 27T Hence ^ + For a sink and source of equal strength we have (i , In all such cases of simple combination the actual stream- lines are most readily found by a construction due to Maxwell and illustrated in Fig. 9. From O l and <9 2 let radiating lines be drawn at angular intervals proportional to z~ -=- a. Then the number of lines will be proportional to a -f- 2ic. Take any point P. Then it is readily seen that by going across the quadrilateral PQRS lo R we find another point for which Oi is increased by 2O RESISTANCE AND PROPULSION VF SHIPS. RESISTANCE. 21 /\~~~~~--- \ """lliJW; 22 RESISTANCE AND PROPULSION OF SHIPS. the angle RO^S and ^ decreased by the equal angle QO^S. Hence at R we shall have (6 l + # 3 ) of the same value as at P. Hence P and R will be points on one of the stream-lines belonging to the two sources O l and 6> a . It follows that the series of stream-lines for these sources will be obtained by drawing continuous curved lines diagonally across the series of quadrilaterals formed by the intersections of the radiating lines from O l and O t . Similarly it may be seen that for Q and 5 the values of (0, 0,) are the same, and hence that these two points are on one of the stream-lines for a source 0, and a sink a . In like manner, then, the series of stream- lines for such a doublet, as it is termed, will be found by drawing continuous curvilinear diagonals in the other direc- tion across the same set of quadrilaterals. These construc- tions are illustrated in Figs. 10 and I i, and it is readily seen in the latter case that the curves are arcs of circles. Maxwell showed in general that for all cases of combination of two systems of sinks and sources or their equivalents, the resulting system of stream-lines could be found geometrically by laying down those for the two component systems, and then taking curved lines continuously diagonal to the series of quadrilaterals thus formed. In this way the result of a combination of any number of sinks and sources may be found. In practice the labor involved is very great when they exceed three or four in number. In case the sinks and sources are not all of the same strength, exactly the same method holds. We have now to show as a special case the result of the combination of a source and a continuous stream with straight stream-lines. The latter may be considered as a part of the system due to a source of infinite strength situated at KESIS TA NCE. 2$ an infinite distance away. The geometrical method for the determination of the resulting stream-line system is exactly similar to the preceding, and comes under the general rule of procedure as stated above. It is illustrated in Fig. 12. The straight lines radiating from O are replaced each by a curved line as shown. Outside of this system of curved lines spring- ing from O are the deflected lines of the uniformly flowing stream. It is readily seen that the line ABC separates one of these systems from the other; also that there is no flow whatever across this line, and hence that it separates the liquid introduced by the source O from that originally in the stream. It is also evident that if, instead of the source O and its system of stream-lines, we should substitute a friction- less surface ABC extending away indefinitely, the lines of the uniform stream would have exactly the same form which they now have as a result of the combination of the source and stream. Taking similarly a uniform stream in combination with a source and sink, we have the result of Fig. 13. In this case it is seen that the liquid introduced at (9, and withdrawn at O^ does not spread away indefinitely, but is confined to the portion bounded by the line ABC, and that, as before, this line separates the liquid thus introduced and withdrawn, from that originally in the stream. Hence also if, instead of the doublet 6\(? a , we should introduce a thin flat solid with fric- tionless contour ABC . . . , the resulting system of stream- lines in the uniform stream would be the same as that here resulting from the stream and the doublet. It is thus seen that we may in this way arrive at the system of stream-lines due to a frictionless solid of certain form plunged in a uniformly flowing stream. 24 RESISl^ANCE AND PROPULSION OF SHIPS. RESISTANCE. 26 RESISTANCE AND PROPULSION OF SHIPS. The form ABC ... is not, however, suitable for the water-line of a ship, and it is only by an extension of the method that stream-lines for ship-shaped forms may be determined. This extension involves the appropriate choice and location of sources and sinks such that the resulting line of separation shall as closely as may be desired resemble a ship's water-line. The most general method of effecting this extension is that due to Mr. D. W. Taylor,* which we may briefly summarize as follows: Instead of a system of separate sources and sinks, Mr. Taylor imagines a source-and-sink line or narrow slot through a part of which liquid is introduced, and through the remainder of which it is withdrawn. The variation of strength along this line corresponds to a difference in width of slot. The distribution of strength may be represented graphically by a line ABC, Fig. 14, where OB represents the source portion and BP the sink portion, the strength at any point being proportional to the ordinate of ABC. FIG. 14. The areas ABO and BPC must be equal, else the total amounts introduced and withdrawn would not be the same. The author then shows by graphical representation and inte- * Transactions Institute of Naval Architects, vol. xxxv. p. 385. RESISTANCE. gration how to find the current function or equation for the stream-lines corresponding to such a sink-and-source line, and how by varying the character of ABC, Fig. 14, to vary the character of the resulting contour ABC . . . , Fig. 13, so as to bring it to any desired degree of similarity to a given water-line. The distribution of stream-lines being known, it is a simple matter to find, by the help of the equation for steady motion, 2 (i), the corresponding distribution of pressure. This is illustrated in Fig. 15. ABC represents one quarter of the hori- zontal section of a body producing D o a system of stream-lines as shown. A Then at points along a line OY l f the excess in pressure over the normal may be represented by the ordinate to the curve DE relative c to O Y, as axis. Similarly the defect in speed at the same points may be represented by a curve such as FG. In like manner at points along the FlG - I 5- line BY^ the excess of speed is represented by the curve HI and the defect of pressure by JK. At a considerable distance from the body these values all approach indefinitely near to their normal values and the variations therefrom become indefinitely small, as indicated by the diagram. The experimental study of plane stream-line motion has received special attention at the hands of Hele-Shaw, who has devised means of showing and of photographing the distribution for ship-shaped and other forms of various kinds.* As pointed * Transactions Institute of Naval Architects, 1897-1898. 28 RESISTANCE AND PROPULSION OF SHIPS. out by him, the study of such systems may be of value in con- nection with the investigation of the distribution of pressure over a part of a submerged body such as a rudder. Since, however, under the special conditions assumed for the mathematical treatment of stream-lines, the total resultant force between the body and the liquid is o, the resistance to the motion of the solid through the liquid will also be o. The investigation of stream-lines is not therefore sufficient in itself to determine the actual resistance in any given case and must be considered rather as a useful auxiliary in the investigation of problems con- nected with the motion of a body through a liquid contained be- tween rigid confining boundaries. In the actual case the liquid is quite free instead of being thus confined. This introduces into the form of the stream- lines, especially near the surface, profound changes, so that the indications received from plane stream-lines must be used with caution, and only so far as comparison with the results of experi- mental investigation may seem to justify. See further 14, on wave-resistance. 4. RESUME OF GENERAL CONSIDERATIONS. In considering the application of the general principles developed in the present chapter to the resistance of ships, it must be borne in mind that for the present we take no account of any influence due to the presence of a propelling agent. The resistance which we here consider is the actual or tow-rope resistance the resistance which the ship would oppose to being moved as by towing through the water at the given speed. The modifications due to the presence of a propeller or paddle-wheel will be considered in 45. We have before us, therefore, the general problem of the RESISTANCE. 2 g resistance of a partially immersed body in an actual liquid, as water. For convenience we shall examine the subject under the following heads: (1) Stream-line Resistance. The liquid being no longer without viscosity or internal shearing stresses, the stream-line conditions of a perfect liquid will not be quite realized, and the maintenance of these lines in the actual liquid will involve a slight drain of energy from the body. This involves the transmission of a resultant force between the liquid and the body opposed to the relative motion of the two, or, in the present case, to the motion of the latter. This gives the first item of the total resistance as above. (2) Eddy Resistance. We come next to the water involved in the liquid prow and stern. As already noted, this does not follow open stream-line paths. It is thrown into irregular involved motion, the energy of which in the actual liquid is continually degrading into heat. The water involved in this motion does not, moreover, remain the same, but gradually changes by the drawing in of new particles and the expulsion of others. The maintenance of this liquid prow and stern will give rise, therefore, to a constant drain of energy from the moving body, the effect of which is manifested as a resistance to the movement. This gives the second item as above. (3) Surface, Skin, or Frictional Resistance. We take next the belt of eddying water produced by the action of the tangential forces between the surface and the liquid. These eddies stand in the same relation to the motion of the body as those giving rise to the eddy resistance so called. Their maintenance constantly drains energy from the body, and this gives rise to a resistance to the movement. This gives the third item as above. 30 RESISTANCE AND PROPULSION OF SHIPS. (4) Wave Resistance. Finally we have the wave systems which always accompany the motion of a body near the sur- face of the water. As remarked in I and as we shall later show, the energy involved in these systems is not retained intact, but a portion is constantly being propagated away and lost. The maintenance of the wave systems requires, there- fore, a constant supply of energy from the moving body, and this, as in the previous cases, giyes rise to a resistance to the motion. This gives the fourth item as above. It may be noted that in the most general view (i) and (4) are not independent. Waves may be considered as the result of modified stream-lines near the surface, and hence in its most general significance (i) might be considered as including (i) and (4). As here classified, however, we mean by (i) the resistance due to the maintenance of the system of stream- lines which would be formed were the body moved with the given speed at an indefinite depth below the surface. The modification due to wave-formation is then taken care of by (4). In any case (i) is very small, since, with the veloci- ties occurring in practice, water behaves nearly as a non- viscous liquid. Similarly (2) and (3) might with propriety be termed the eddy-resistance, since both items arise from the maintenance of systems of eddies. Since their immediate causes are distinct, however, it is convenient to consider them separately. The sum of (i) and (2) is frequently termed the head- resistance, though it is sometimes considered as of two parts, called head and tail resistances, the former being due to the excess of pressure in the liquid prow, and the latter to the defect in the eddying liquid stern. , Inasmuch as they cannot be separated, however, we shall use the one term for both. RESISTANCE. 31 With ships of ordinary form this is so small as to be relatively negligible. In other cases, as in the motion of the paddle-wheel, propeller-blade, bilge-keel in oscillation, etc., this is one of the chief items of the resistance to the motion of the given body. In ships of usual form we are therefore concerned chiefly with surface and wave resistance. The sum of (i) and (4) or of (i), (2), and (4) is often known as the residual resistance. This term has reference to the usual view of ship-resistance which recognizes but two chief subdivisions: (a) Surface or skin resistance. (b) Residual resistance, including (4), above, and all other parts not properly classified under (a). It must not be forgotten that this classification of resist- ance according to the various effects produced on the water is somewhat arbitrary. A more logical mode of subdivision might be made by starting from the body and considering that each element of immersed surface is subjected to a cer- tain force. Such force may be resolved into two com- ponents; one tangential, the other normal. This is entirely general and includes all effects no matter from what cause arising. Comparing with the other method of subdivision it is evident that the longitudinal components of the tangential forces will give by their summation the surface or frictional resistance. Hence the longitudinal components of the normal forces must give by their summation the remaining or resid- ual resistance, corresponding to (i), (2), and (4), or with ship- shaped forms almost entirely to (4). This subdivision is -entirely independent of any supposition with regard to the effects on the water, and is the most direct or immediate as- 32 RESISTANCE AND PROPULSION OF SHIPS. pect of resistance itself. The two parts may be termed tangential and normal resistances. No matter what view we may take of the subdivision of resistance, we are obliged to go to experimental investigation for all satisfactory information as to its amount. We pro- ceed, therefore, to the description of the various experimental investigations, and to the discussion of the results which may be derived therefrom. 5. RESISTANCE OF DEEPLY IMMERSED PLANES MOVING NORMAL TO THEMSELVES. If the body in Fig. I is supposed to be reduced to a plane normal to the line of motion, the result will be a system of stream-lines and eddies somewhat as in Fig. 16, though the FIG. 16. line of demarcation between the two is not fixed nor so defi- nite as must be suggested by a rough diagrammatic represen- tation such as the figure gives. The tangential resistance will be o because all tangential force is at right angles to the direction of motion. This does not mean that the actual resistance is independent of the character of the surface, but simply that the actual-tangential forces have no longitudinal component. The nature 4 ' of : the surface will presumably RESISTANCE. 33 slightly affect the stream-line formation and the amount and nature of the liquid prow and stern, so that ultimately it will have its effect on the resistance. Viewed immediately, however, the resistance is due wholly to the eddies and stream-lines in an imperfect liquid. Experiment shows that the resistance in such case may be approximately expressed by an equation of the form (I) where R = resistance in pounds; f = a coefficient; cr = density of liquid = 62.5 for fresh and 64 for salt water; g= acceleration due to gravity = 32.2 ft. ; A = area in square feet; if = velocity in feet per second. For salt water cr -f- 2g is very commonly put equal to I. In such case the equation becomes (2)' The values of the coefficient f which have been deter- mined experimentally have varied widely. This is doubtless due to the fact that such experiments have been carried on by different observers under different conditions, notably as to amount of surface, depth of immersion, and actual veloci- ties employed. The subject has not been examined with sufficient completeness to show satisfactorily the general rela- tion of the resistance to the three conditions, area, immer- sion > and velocity. In fact there is considerable doubt as to whether R should increase strictly as the area, we are not sure 34 RESISTANCE AND PROPULSION OF SHIPS. that the index of v should for all velocities be 2, and the relation of /to varying immersion is not satisfactorily known. The attempt, therefore, to express the results of experiments under widely varying conditions by the simple formula above, throwing on / the consequences of all differences between the true and the assumed law, may with justice lead to widely varying values for this coefficient. Among the earliest experiments were those by Col. Beaufoy made at the Greenland Docks in London between 1793 and 1/98. Taking A in square feet, v in feet per second, and R in pounds, the mean value of /deduced from these experiments is about i.i. Mr. Wm. Froude's experiments on planes about 3 ft. wide gave with similar units a mean value of about 1.7. Dubuat at a single speed of about 3.28 ft. per second found for a plane maintained stationary in a moving stream a value of /= 1.86. For a stationary liquid and moving plane, the speed being the same, he found /= 1.43. If in the first case the liquid moved steadily and without eddies or internal irregularities, and in the second case the liquid was strictly at rest, the relative speed being the same in each case, these two results should be the same. The discrepancy was doubtless due to the difficulty of making satisfactory observations on the mean velocity of a flowing stream, and also to the now well known difference in effect between a stream moving without and with turbulence. More recently Joessel's experiments in the river Loire, made on a plane of sheet iron .98 ft. high by 1.31 ft. long and immersed so as to have .66 ft. of water over the upper edge, gave a mean value of /= 1.6. The maximum velocity was not above about 4.25 ft. per second. RESISTANCE. 35 Still more recently Mr. R. E. Froude has determined this coefficient under conditions which insure the highest degree of experimental accuracy. The resulting value agrees very closely with the average of Beaufoy's values mentioned above, and was about i.i. An extreme divergence from all the values above given is indicated by the results of experiments on bilge-keels. An attempt to fit this formula of resistance to these results leads to a coefficient varying from 5 or 6 to 15 or 1 6. Its excessive varia- bility, as well as its great divergence from the other values, indi- cates that we have here involved certain elements or conditions not represented by the formula, and whose importance has been hitherto unsuspected. This problem has received special theoretical investigation by Bryan,* who shows that in large measure this extra effect may be accounted for by the following features found in the motion of a rolling ship fitted with bilge-keels: (1) The movement of the ship to and fro tends to set up currents in the water which lag in phase behind the roll of the ship, and thus run counter in some degree to the motion of the bilge-keel. These counter currents may be capable of producing a marked increase in relative velocity between keel and water, and hence a a marked increase in the resistance opposed to the motion of the keel. (2) It appears from a study of the stream-line motion about a ship rolling with bilge-keels that such motion will produce a marked change in the distribution of pressure over the surface of the ship, and that this change of pressure will be of such character * Transactions Institute of Naval Architects, 190x3. 36 RESISTANCE AND PROPULSION OF SHIPS. as to resist rolling and thus increase the apparent resistance met by the bilge-keel. (3) We may first note generally that if the immersion is suffi- ciently great to reduce the surface disturbance to a negligible amount, the resistance will be practically independent of any further increase of depth. This is readily seen from the fact that the resistance arises, not from the pressure, which of course increases with the depth, but rather from a difference in the distribution of such pressure, and that this is practi- cally independent of depth so long as surface disturbance is absent. Still otherwise we may view the resistance as due to the energy absorbed by the stream-line and eddy systems; and since below the minimum depth above mentioned the configuration of these will be constant, the energy absorbed and the resulting resistance will likewise remain the same. These conclusions are borne out by Beaufoy's experiments. It will be observed that, according to definition, head- resistance is the resistance due to the eddy and stream-line formation supposing the immersion of the body so great that surface effects are negligible. It is not the resistance due to the actual stream-line and eddy formation if near the surface, for this would include the wave-resistance as well. In the present case, therefore, tangential resistance being absent, we must consider the change in the total resistance as the plane approaches the surface as due to the resistance arising from the wave-formation. In any experiment, therefore, if the plane is near the surface, the resistance actually measured will consist of two parts: head-resistance proper and wave-making resistance. As is known furthermore, a slight wave may involve con- siderable energy, especially if its length is great. Hence appar- RESISTANCE. 37 ently slight surface effects may count for much in the actual total resistance. These wave systems will be generated primarily by the periodic or alternate discontinuous movement of the ship due to which large bodies of water will be impressed with like dis- continuous motion, and a portion of the energy of which will become dissipated in the form of surface disturbance as noted, and a part in the form of eddy and vortex motion. The summation of these various influences, combined possibly with others, may presumably become sufficient to account for the extreme difference as noted, between the values of the coeffi- cient resulting from the application of the formula to experiments on planes in continuous motion and for which it is intended, and resulting from its application to experiments on ships carrying bilge-keels in alternate or discontinuous motion, and to the circumstances of which it is not properly applicable. 6. RESISTANCE OF DEEPLY IMMERSED PLANES MOVING OBLIQUELY TO THEIR NORMAL. If the plane, instead of standing at right angles to the direction of motion, is inclined at an angle 6, we have a result somewhat as indicated in Fig, 17. In such case the force in FIG. 17. 38 RESISTANCE AND PROPULSION OF SHIPS. the line of the normal EB as determined by Joessel's experi ments is given by the formula . a . 39+ -61 sm 6 2g . The longitudinal component, or the resistance in the line of motion, is therefore R 9 = 1.622 - p 2 - : TJ Av*. (2) .39 + -61 sin 6 2g P 9 sin 9 ^ = .39 + .61 sin ....... <3) sin' ^ = .39 + .61 si^ ........ According to Beaufoy's results we may put approxi- mately ^ = sin ^8 a7O 1. 04 .314. .260 I Q1 271 207 Tinfoil 2.16 30 .295 1. 09 .278 .263 I.qo .262 .244 1.83 .246 .232? Calico 1.93 87 .725 I.Q2 .626 1 .504 1.89 531 447 1.8 7 474 .423 Fine sand. .. 2.00 .81 .600 2.OO .583 450 2.00 .480 384 2.06 .405 337 Medium sand 2.00 .90 730 2.OO .625 .488 2.OO 534 465 2.00 .488 4S6 Coarse sand . 2 .00 1. 10 .880 2.OO .714 -520 2.OO .588 .490 Column A gives the value of n for the particular length and character of surface. Column B gives for the whole surface the mean resistance in pounds per square foot at a speed of 600 feet per minute or 10 feet per second. Column C gives for the same speed the resistance per square foot at a distance from the forward end equal to that stated in the heading. It is thus seen that the resistance per unit area decreases as we go from forward aft. Thus for varnish the resistance per square foot at distances of 2, 8, 20, and 50 feet from the bow are respectively .39, .264, .24, .226. It necessarily results, as is shown by the table, that the mean resistance per square foot decreases with increased length. The explana- tion of this as given by Mr. Froude is as follows: 44 RESISTANCE AND PROPULSION OF SHIPS. The board or surface under test moves surrounded by a skin of eddying water, the maintenance of the energy of which gives rise to the resistance under consideration. The forward part of the board is constantly entering water at rest, while the after portions come in contact with water already acted on by the forward end, and possessing, therefore, to a greater or less extent, this eddying motion. In these eddies the movement of the water-particles next the board is forward, more or less of the water being involved, and with a greater or less velocity according to the length of time they have been acted on, and to the other circumstances of the sur- face and motion. It follows that the eddying motion will be more and more pronounced as we go from the bow aft. Hence a part of the board near the forward end advancing into still water will naturally meet a much greater resistance than the same area located near the after end, and entering water which already possesses in large measure the eddying motion. We will now consider in detail the character of the equa- tion for tangential resistance as given above, especially as to the relation between the .quantities involved and the circum- stances of the experiment. The quantity A is taken as the area without modification or change. It is seen as above that the length factor of area is the one of importance, so that we may consider, so far as its effect on the quantities of the formula is concerned, A to be represented by length. Suppose now that we have given a series of values of R for a range of values of quality, length, and speed. Each such value will give a single equation involving the two unknowns /"and n. These may be both variable from point to point, and not knowing on what they RESISTANCE. 45 may depend we cannot consider any two equations as simul- taneous, and thus use them for the determination of f and #. In other words, without other conditions the relations of/ and n to the circumstances of the experiment are indetermi- nate. Some arbitrary assumption must therefore be made in regard to one, and the other determined in accordance there- with. Taking any set of values of R for fixed quality, and length at varying speeds, it seems most natural to assume that /will be constant for all such values of R, and to then determine n in accordance with this assumption. This is equivalent to assuming that f is dependent on quality and length, and independent of speed. We may also consider this coefficient as providing for effects due to changes of tem- perature and density of liquid. It may also be well to note that this coefficient, and in fact the entire equation, is con- sidered as independent of the depth of immersion. While thus taking /as independent of speed variation, it is not independent of the amount taken as the unit of speed. This appears as follows: In the formula, give v a value I. Then R=fA, or f= R-i- A. Hence /= the mean value of R per square foot at the unit speed. The value of /being thus known, it maybe substituted in the various equations representing the values of R at vary- ing speeds, and the values of n thus determined. It is seen that the values of n may vary with the speed, and they will likewise depend on the particular value of /used. It thus appears that /will depend on length, quality of surface, density, and temperature, while n will depend on speed and / and hence on speed and the other various con- ditions above. Applying these principles to the results of Froude's 46 RESISTANCE AND PROPULSION OF SHIPS. experiments * it is found in general for a given quality, that / decreases with increasing length at a decreasing rate: also that / varies with quality in the way we should naturally expect. The relation of n to length, quality, and speed is much more obscure. Actual values vary from about 1.8 to 2.0, with some evidence of a slight decrease for increase in length, at first rapid and then slow. There is also evidence of a generally higher value for rough surfaces than for smooth, the value for sanded surfaces being about 2 for all lengths. Early experiments on tangential resistance were limited in speed to 6 or 8 knots, while the results obtained have been applied to the estimation of such resistance for speeds of 30 knots and more. This gap has been in large measure filled in by experi- ments at Washington in the U. S. Naval Tank, the results of which have generally confirmed those drawn from earlier experi- ments, at least for smooth surfaces, and have indicated for n an average value of about 1.854 and for smooth surfaces a value of /=.oo 9 7. With regard to the use of such values, it may be observed that for any individual case but little significance can attach to the third decimal place in the value of n, and the second will like- wise involve some uncertainty. On the other hand, we may remark that a difference of i in the hundredths of n will cause for usual speeds a difference of only about i per cent in the value of v n ^ so that such an amount of uncertainty is not greater than that to which we are accus- tomed in most engineering work. In this connection the ideas of M. Risbec as given in a * See complete report in British Association Reports for 1874, of which the table on page 43 is a partial resume. RESISTANCE. 47 recent paper * will be of interest. This author considers that it is more rational to assume that fundamentally, skin-resist- ance, involving as it does the energy of eddying water, will be proportional to the square of the speed, and to write the formula The coefficient f is then to be determined in accordance with the experimental data, and is considered as including a certain function of "perturbation," the value of which offsets the varying values of n in the more usual mode of consideration. The possibility of such a treatment of the original data will be obvious from the preceding discussion of the relation of the quantities in the equation to the conditions of the experiment. We have simply to find for the various values of quality, length, and speed the quotient M. Risbec then represents the quotient/" as a function of four parameters for each quality of surface, of the speed ^, and of the length L. The complexity of the resulting equa- tion and the uncertainty of the exact nature of the param- eters, especially as affected by their extension to lengths and speeds far beyond those contained in the fundamental data, make it questionable whether any advantage is to be derived from this mode of consideration. It is, however, of interest as showing a possible mode of considering the rela- tion of the formula to the characteristics of the experiment. * Bulletin de 1' Association Technique Maritime, vol. V. p. 45. 48 RESISTANCE AND PROPULSION OF SHIPS. 8. SKIN-RESISTANCE OF SHIP-FORMED BODIES. While in the preceding section we have referred in general to the application of the data therein discussed to the resist- ance of ship-formed bodies, it must be remembered that fundamentally the data relate simply to thin boards, sensibly equivalent to planes. Now at any point of a ship-formed body let the horizontal component of the tangential force be denoted by/. Let ds be an element of area, and the inclination to the longi- tudinal of the water-lines at this point. Then / cos Ods is the longitudinal component of the tangential force or the tangen- tial resistance for this element. For the whole ship we shall have, therefore, for the skin-resistance R^ C p cos Bds\ or if/ denote a mean value of/, we have R = / / ds cos 6. The value of the integral is known as the reduced wetted surface,* and it thus appears that strictly speaking it is with this rather than with the actual surface that we are concerned in dealing with skin-resistance. The difference between the two, however, is usually less than I per cent, and the uncer- tainties surrounding the whole question as already discussed * As shown in the theory of statical naval architecture the value of the reduced wetted surface is more simply expressed by Reduced surface =fds cos 6 = fGdx, where G is the variable girth or length of section, and dx is the element of length. UNIVERSITY Of .CALIFI RESISTANCE. 49 show that no significant error will be made by using either the one or th other. The experiments of Wm. Froude on ,i the jjreyhound led him to use for A the actual wetted surface, and the coefficients derived from his results and given in Table II, p. 51, are intended to correspond to this value of the area. In continental Europe the reduced surface is frequently used in this connection. The value of the wetted surface may be derived by approximate integration as shown in the theory of statical naval architecture, or by the use of one of the following empirical and approximate formulae: Let 5 = wetted surface in square feet; D = displacement in tons- L = length in feet; H = draft in feet; B = beam in feet ; b = block coefficient of fineness. Then we may have S=L[i.>jH For the reduced surface we have, from the foot-note on p. 48, S r * Transactions Society of Naval Arhcitects and Marine Engineers, rol. I. p. 226. t Transactions Institute of Naval Architects, vol., xxxvi. p, 72. 50 RESISTANCE AND PROPULSION OF SHIPS. With the same notation as before we have the following empirical relation between S and S r : With a correct value of S,, the error of this formula is usually less than one-tenth of one per cent. 9. ACTUAL VALUES OF THE QUANTITIES / AND n FOR SKIN-RESISTANCE. We will now give a resume of the practical values which have been proposed by various authorities for the coefficient /"and the exponent ;/. The general formula is R = where R = resistance in pounds; f = coefficient; A = area of wetted surface in square feet; v speed in knots; n = index. * Transactions Society of Naval Architects and Marine Engineers, vol. in. p. 138- RESISTANCE. TABLE I, SHOWING FROUDE'S EXPERIMENTAL VALUES. Nature of Surface. Length of Surface. 2 feet. 8 feet. 20 feet. 50 feet. n / / n f n / 2.00 1-95 2.l6 1-93 2.00 2.00 2.00 ,OII7 .Olig .0064 .0281 .0231 .0257 .0314 I.8 5 1.94 1.99 1. 92 2.00 2.00 2.OO .OI2I .0100 .008l .0206 .0166 .0178 .O2O4 1.8 5 1-93 1.90 1.89 2.OO 2.OO 2.OO .OIO4 .0088 .0089 .0184 .0137 .0152 .0168 1.8 3 1.83 1.8 7 2.06 2.00 .0097 .0095 .0170 .0104 .0139 Paraffin TABLE II. VALUES FOR SHIPS BASED ON FROUDE'S EXPERIMENTS. (SALT WATER.) Index for the Index for the Coefficient of Variation of Coefficient of Variation of Length in Feet. Skin-resistance. Resistance with Speed. Length in Feet. Skin-resistance. Resistance with Speed. f n f n 8 .01197 1.825 80 .00933 1.825 9 .01177 " 90 .00928 IO .OIl6l " 100 .00923 12 .01131 " 120 .00916 14 .01106 140 .00911 16 .OIO86 ' 160 .00907 18 .01069 " 180 .00904 20 .01055 200 .00902 25 .OIO29 " 250 .00897 30 .01010 " 300 .00892 35 .00993 < 350 .00889 40 .00981 400 .00886 45 .00971 ii 450 .00883 50 .00963 41 500 .00880 60 .00950 " 550 .00877 c 70 .00940 " 600 .00874 < RESISTANCE AND PROPULSION Of SHIPS. TABLE III. VALUES FOR SHIPS BASED ON TIDMAN'S EXPERIMENTS. (SALT WATER.) Iron Bottom Clean and Copper or Zinc Sheathing. Well Painted. Length in Feet. Smooth. Rough. ''f n f f 10 .OII24 853 .OIOO 9175 .01400 .870 2O .01075 .849 .00990 .900 .01350 .861 30 .01018 .844 .00903 .865 .01310 .853 40 .00998 8397 .00978 .840 .01275 .847 50 .00991 .8357 .00976 .830 .01250 .843 100 .00970 .829 .00966 .827 .01200 .843 150 .00957 829 .00953 .827 .01183 .843 2OO .00944 .829 .00943 .827 .01170 .843 250 .00933 .829 .00936 .827 .OIl6o .843 300 .00923 .829 .00930 .827 .01152 .843 350 .00916 .829 .00927 .827 .01145 .843 400 .00910 .829 .00926 .827 .01140 .843 450 .00906 ,829 .00926 .827 .01137 843 500 .00904 1.829 . 00926 .827 .01136 843 TABLE IV. VALUES FOR PARAFFIN MODELS BASED ON TIDMAN'S EXPERIMENTS. (FRESH WATER.) Index for the Index for the Coefficient of Variation of Coefficient of Variation of Length in Feet. Skin resistance. Resistance with Speed. Length in Feet. Skin-resistance. Resistance with Speed. f n f 2 .01176 1.94 12 .00908 1.^94 3 .01123 12.5 .00901 4 .01083 13 .00895 5 .01050 13-5 .00889 6 .01022 14 .00883 7 .00997 14.5 .00878 8 .00973 15 .00873 9 .00953 16 .00864 10 .00937 17 .00655 10.5 .00928 18 .00847 n .00920 19 . 00840 ii. 5 .00914 20 .00834 RESISTANCE. 53 The various values for / given in the above tables are based on the experiments of Froude and Tidman. As other special values we may mention the following: As noted in 7, the values commonly employed at the U. S. Naval Tank are /=.oo97, w = 1.854. Colthurst * gives the value /= .0167 for rough-sawn wood and /= .01 for smooth-planed wood. The same authority states that for fine " grass" or marine growth / may rise to a value from .048 to .062. For clean copper the value .0073 is given, and for fresh-painted iron the value .01. This last value is also that used by Prof. Rankine. These values all relate to ships of usual length, or at least over 50 feet. In this connection the results given in a recent paper by Chief Naval Constructor Hichborn f are of interest. From the data given in this paper the following tabular presentation is derived : Column a shows the percentage increase of total resistance with foul bottom over that with clean, reckoned on the latter as base. Ship. Speed. a. Charleston 8.8 70 Yorktown 9 72 Philadelphia II 52 San Francisco 9 33 Bennington 7 200 Baltimore 1 1 20 It thus appears, as we should expect, that the term " foul " is entirely indefinite, and may mean almost any con- dition in which the resistance is sensibly increased. In the * Cited by Pollard and Dudebout, Theorie du Navire, vol. m. p. 376. t Transactions Society of Naval Architects and Marine Engineers, vol. II. p. 159. 54 RESISTANCE AND PROPULSION OF SHIPS. extreme case of the table the total resistance of the Benning- ton when foul is three times that when clean. At the low speeds at which these runs were made, most of the resistance will be due to the skin, so that while the above figures refer to total resistance, the ratios for the skin-resistance will not be far different, though slightly greater. These possibilities impress in the strongest manner the extreme importance of a clean bottom especially for economi- cal cruising at moderate speeds, and for the possible attain- ment of the highest speeds. 10. WAVES. As introductory to the subject of wave-resistance we shall give in the present section a brief account of the commonly accepted theories of wave-motion in liquids, with a statement of the more important results, but without the details of the mathematical development.* In order to examine in a satisfactory manner the influence of waves upon a ship, some working theory as to their dynamical constitution is required. It must be understood that such a theory is not to be viewed as final truth in itself, but rather as a convenient method of correlating as nearly as possible the known facts, and of deducing by appropriate * Among others, the following works may be consulted by those in- terested: Rankine et al.: "Shipbuilding, Theoretical and Practical." London, 1866. J. Scott Russell: " System of Modern Naval Architecture," vol. i. Lon- don, 1865. Encyclopedia Britannica, article Wave. Pollard et Dudebout: Theorie du Navire, vol. HI. Paris, 1892. Gatewood : Journal U. S. Naval Institute 1883, p. 223. Annapolis. Stokes: Cambridge Transactions 1847. Encyclopedia Metropolitana, article Tides and Waves. RESISTANCE. 55 mathematical means the probable nature and extent of the influences which are to be examined. The most simple of the theories proposed to this end is that known as the trochoidal. The theory receives its name from the trochoid, which is the form of the contour of waves resulting from this mode of viewing their constitution. The trochoid may be defined as the locus of a point P, Fig. 20, which revolves uniformly FIG. 20. about a center O, which center itself moves uniformly along a straight line OO 6 . In this case if O moves uniformly along OO K , and OP revolves uniformly counter-clockwise, the curve Pf^ . . . P t will result. For comparison there is shown in dotted lines a sinusoid of equal length and height, and the difference between the two is indicated by the shaded area. A trochoid may also be defined as the locus of a point in the radius of a circle, which circle rolls uniformly along a straight line. Thus in Fig. 21 let a circle of radius OQ roll along on the under side of AB. Let Pbe the tracing-point. Then the locus of P will be a curve CPDE, and the identity in character between the two curves in Figs. 20 and 21 is readily seen. If the tracing-point P is on the circumference of the circle the locus becomes the common cycloid as a special case of the trochoid. If the point passes beyond Q, as to R, the locus is still called a trochoid. The specific 56 RESISTANCE AND PROPULSION OF SHIPS. names prolate-trochoid and curtate-trochoid are given to these two classes of curves. It is only with the former as shown in the figures that we are here concerned. Waves whose contours are very nearly trochoidal are but rarely met with, though a moderately heavy swell in calm weather at a considerable time after the subsidence of the storm to which it owes its existence, quite closely conforms FIG. 21. to this type. The trochoidal wave is, moreover, the type most readily examined by mathematical means, and is there- fore naturally taken for purposes of theoretical investigation. The ideal constitution of a trochoidal wave involves the following suppositions: (1) That for any given wave all particles revolve in fixed circular orbits, in vertical planes parallel to the direction of propagation and normal to the lines of crests and hollows, with the same constant angular velocity, and in direction with the propagation when in a crest. (2) That for any layer of water originally horizontal the radii are equal and the centers are on the same horizontal line. RESISTANCE. 57 (3) That the radii decrease with increase of depth accord- ing to a law to be stated later. (4) That all particles originally in the same vertical are always in the same phase. It follows that the surface layer of particles and all suc- cessive layers originally horizontal will take the form of trochoidal surfaces of continually decreasing altitudes accord- ing to the law of decreasing radii. It further results that all corresponding crests and hollows for successive layers will lie in the same vertical line, and hence that all wave-lengths for the successive trochoidal subsurfaces will be equal, and that the velocity of propagation for all must be the same. This ideal constitution of a wave may be illustrated in Fig. 22. The diagram represents a section of a certain por- FlG. 22. tion of still water divided by horizontal and vertical planes into rectangular blocks as shown in dotted lines. The full lines represent the same blocks of water when thrown into wave-motion. The horizontal planes o, i, 2, etc., for still water are, in the wave, thrown into the trochoidal surfaces #, b, c, etc. The corresponding circular orbits are shown on 58 RESISTANCE AND PROPULSION OF SHIPS. the left, and their centers are seen to be elevated above the corresponding still-water planes by successively decreasing amounts as we go from the surface downward. To avoid confusion in the diagram the circular orbits are elsewhere omitted. The motion involved in the figure may be viewed from several standpoints. Thus we may consider the trochoidal layers and note, as the wave progresses, the change in thickness and form at any one point. Or again we may consider the vertical columns of water between the distorted verticals af,gh, kl, etc., and note that as the wave progresses these columns shorten and broaden and then lengthen and narrow, at the same time waving to and fro with the period of the wave. Again, each rectangular block of still water is represented in the wave by a distorted block, and the successive forms into which one of these blocks is thrown may be seen by fol- lowing along the series of distorted quadrilaterals between the pair of corresponding trochoidal surfaces for the wave. In order that such a wave may actually exist in a liquid it must fulfil certain hydrodynamic conditions. These are: (i) That at the upper surface the pressure must be constant and normal to the surface. (2) That the liquid cannot undergo expansion or compression during the process of wave-propagation. These conditions may be made to furnish a relation btween the length and the velocity of propagation, and a law for the decreasing radii of the circular orbits. These are as follows: Let L = length in feet; V = linear velocity in feet per second; R = radius of rolling circle: QO =. angular velocity; RESISTANCE. 59 T = periodic time ; g = value of gravity = 32.2. Then V-- or F= 2.265 ^L ft- per sec., and F= 1.341 VL knots, also (2) Let r a = radius of orbit at surface; r = radius of orbit at depth of center z below center of surface orbits; z = any given depth of orbit center below center of surface orbits; e = base of Naperian logarithms. Then or z _ 2irz r= r.e~~R r.e~~^ (3) If these various conditions are fulfilled, therefore, it appears that the geometrical motion here supposed, if once inaugurated, is hydrodynamically possible. This does not prove that in any actual wave the particles move as herein assumed. Observation shows, however, that these supposi- tions are near the truth, and the actual motion and character- istics of waves, so far as admitting measurement, are for the most part in fairly satisfactory agreement with those resulting from the type of wave-motion assumed. It seems, therefore, reasonable to accept the theory as a working basis for the 60 RESISTANCE AND PROPULSION OF SHIPS. investigation of such characteristics of wave-motion as we are here concerned with. We may now state a further series of results following upon the conditions assumed. The point of maximum slope of any trochoid corresponds to a value of 3 given by cos 6 = . (4) or If is this maximum slope, then r nh , N and sin = -=--, ...... (5) where h = height of wave = 2r. The normal at any point intersects the vertical through the center of the rolling circle at the constant distance R above it, and therefore always at the point of contact of the rolling circle with the line on which it rolls. The pressure on any trochoidal subsurface is constant and equal to that due to the depth of the corresponding plane surface in still water. Let z = depth of the line of centers; r = radius of surface orbit ; r = ' orbit with center at depth z\ R= " " rolling circle. RESISTANCE. Then the pressure in the crest of a wave is less than that for an equal depth in still water in the ratio z 2R (6) Similarly the pressure in the hollow of a wave is greater than that for an equal depth in still water in the ratio 2R 2 r, -)- r (7) At any point the resultant force due to gravity and to centrifugal force acts along the normal and hence through the instantaneous center of motion of the rolling circle as specified above. In Fig. 23 let OA = R. Then if the force due to gravity is represented by OA or R, that due to centrifugal force will be represented by the radius OP, and the resultant by the line AP. Let DI = mass of particle; g = value of gravity = 32.2 ; co angular velocitv ' L FIG. 23. /= resultant force represented by AP. Then we have #. ... (8) The line of orbit centers for any trochoidal surface or 62 RESISTANCE AND PROPULSION OF SHIPS. sheet is raised above the original still-water level of the corresponding plane by the distance L " 2R 2R (9) Let H denote the still-water depth of a layer correspond- ing to the trochoidal sheet with line of centers at a depth z below the center line of surface orbits. Then ' (10) 2R The total energy of a wave is always half kinetic and half potential. Let E denote total energy in foot-pounds; & " density; L " length in feet; r " radius of orbit at surface; h '* height of wave = 2r ; R " radius of rolling circle. Then *= vLk'l lh\\ 01 R -- I 1 -- 4.935$;. If we put & = 64, we have E =%Uf(i- 4-935 The total power involved in a series of waves per foot of breadth is Power = .0329 VLtf (i 4.935 UJ J. . . (12) RESISTANCE. 63 Combinations of Wave Systems.' Given two trochoidal systems in general with parallel crests, moving in the same direction. The result of such a combination geometrically, as is well known, is to produce a series of groups of waves, each group reaching a maximum of height where two crests correspond, and a minimum where crest and hollow combine. Each group therefore culminates in a wave of maximum alti- tude with comparatively small disturbance betweeen the suc- cessive groups. It may be shown that such geometrical combination does not fulfil the necessary hydrodynamical conditions, and therefore cannot exist permanently as a com- bination of actual trochoidal systems. The actual result of the combination of two wave systems is to produce a more or less mixed resultant system, with characteristics relative to which those given by the geometrical combination of trochoidal sys- tems are intended simply as approximations. We do find, however, actual combinations of wave systems agreeing in all their general characteristics with those given by the geometri- cal combination referred to, and it therefore seems fair to use the method as a working basis for the examination of such points as we are interested in. As a special case let us assume that the two series are of Components Resultant FIG. 24. equal altitude, h, but of different lengths, Z, and. Z,. Then the geometrical combination will give a result as indicated in Fig. 24. At A and C the phases are opposite, and the values 64 RESISTANCE AND PROPULSION OF SHIPS. of h being equal the resultant disturbance is o, and we have for a little space practically undisturbed water. At a point B midway between, the phases are the same and the two effects are added, giving a wave of height 2//, the maximum possible in the entire series. The group from A to C is then indefinitely repeated to the right and left as far as the given conditions hold. For such a group the length AC is given by the equation Let v = velocity of propagation of the resultant group as a whole; v l = velocity of propagation of the component of length Z,; v % = velocity of propagation of the component of length Z a . Then v = i i i or - = - + - *7 *7f ' *7f (H) As L l approaches Z 2 in value the length of the group AC will increase and the velocity will approach more and more nearly to one half that of the components. At the limit where Z, = Z, the length would become indefinite and the velocity would become one half that of the components. Since any given group is the expression of a certain distribu- tion of energy, it is considered that the velocity of propaga- tion of the energy is the same as that of the group. RESISTANCE. We will now take the special case of two trochoidal systems of different altitudes, but of equal lengths and there- fore of equal velocities. In Fig. 25 let O denote the orbit center for a given sur- face particle as affected by either component. Strictly speaking the two centers are not c at the same level, but the differ- ence is very small and is usually neglected. At any given instant of time let DO A denote the phase- angle B 1 of the first component, and DOB the phase-angle 6^ of the second component. Then AOB will represent the constant differ- ence of phase between the two components. Also let OA = r } and OB = r^. Then the combination of the two components will result in a trochoidal system of wave-length equal to that of its components, and of surface orbit radius r = OC, and with phase relationships with its components as given by the angles CO A and COB. In other words, if the parallelogram OACB is made to revolve uni- formly about O while the latter moves uniformly along x, the two speeds being appropriate to the wave-length taken, then the point A will trace the contour of the first com- ponent, B that of the second, and C that of the resultant. Let a denote the linear difference between similar points on the two components, <*> the angular velocity, J^the linear velocity, and R the radius of the rolling circle. Then the angle denoting the phase difference AOB is given by FIG. 25. 66 RESISTANCE AND PROPULSION OF SHIPS. coa a 2ita . A OB = = -75- = :, in angular measure, 3600 . or A OB = j in degrees. _z> From the triangle OAC we derive for the altitude of the resultant system = *; + *. + 2* A cos. . . . (15) As to the degree of fulfilment which the ideal theory finds in actual waves it may be said in general, and without entering into details, that, so far as observations can be made, most of the actual characteristics are in fair agreement with the results derived from theory. The agreement is by no means complete, and in fact differs in certain particulars in such way as to lead to the belief that a truly trochoidal wave is but rarely if ever met with. On the other hand the agree- ment is sufficiently close to seemingly warrant the use of the theory for^ all purposes of the naval architect, at least in so far as he is concerned with waves in deep water. In regard to the limiting sizes of waves the following results may be given: From reliable observation it may be fairly concluded that waves longer than 1200 to 1500 feet are very rarely met with, though there seems ground for accepting reports of excep- tional waves of a length approaching 3000 feet. The corre- sponding periods would be 15 to 17 seconds for the first named, and about 24 seconds for the last. With regard to height the evidence is much more conflict- ing. It appears, however, that from 40 to 45 feet is about RESISTANCE. 6 7 the maximum value which can be accepted as reliable, and values approaching these figures are met with but rarely. From 25 to 30 feet may be taken as the ordinary limit. It should be noted, however, that these heights relate to a series of regular waves as contemplated by the theory, and not to exceptional results arising from the interference and combination of different systems. With regard to ratio of height to length, it is found usually to decrease with increase of length. The highest values of this ratio appear to be about I : 6, such being found only with comparatively short waves. More commonly the ratio is from I : 12 to i 125. The corresponding values of the maximum inclination are respectively about 32, 15, and 7. Shallow-water Waves. When we come to waves in shallow water we find, as nearly as can be determined, the paths of the particles to be elliptical with the longer axis horizontal. In order to meet this departure from the tro- choidal system, the so-called ellip- tical trochoidal or shallow-water wave system is used. This ideal system may be generated as fol- lows: Given an ellipse, Fig. 26, with its major axis horizontal, moving uniformly along X without angular change. Draw any line OB at an angle 6 with the verti- FlG - 26 - cal. Through B draw a vertical and through A a hori- zontal. They will intersect on the ellipse at P. Let OB revolve uniformly to the left. Then the locus of the point P 68 RESISTANCE AND PROPULSION OF SHIPS. thus determined by the movement of the ellipse along OX and of OB about O will be an elliptical trochoid. Comparing this contour with that resulting from the circle as a base, as in Fig. 27, the former is seen to lie slightly within the latter, thus making a flatter hollow and steeper crest, these differ- ences becoming more pronounced as the difference between the two axes of the ellipse is greater Full line, Circular Trochoid Dotted line, Elliptical Trochoid FIG, 27. In the present connection the feature of especial interest is the velocity of propagation. For this the following ex- pression may be derived : Let V = velocity in feet per second; r = semi-major axis at surface; = " minor " *' " L = wave length ; g = value of gravity = 32.2. Then F=-v/t=\/? (16) This equation shows by comparison with (i), that the velocity of a shallow-water wave is less than that of a deep- water wave of equal length, a conclusion borne out by experience. Let d = depth of the water. Then when 2nd -^ L is small it may be shown that (17) RESISTANCE. 6 9 It should be noted that the ideal wave system thus derived, does not, and seemingly cannot, be made to fulfil the hydrodynamic conditions necessary to its actual existence as assumed. This does not prevent, however, such form of motion from being, possibly, a close approximation to what actually takes place, nor does it seriously interfere with its usefulness as an approximate working theory of shallow-water waves. Waves of Translation. The waves thus far considered occur naturally in indefinite series, and represent the result of a widely distributed disturbance. The typical wave of translation is individual in character, and represents the result of a local disturbance. In Fig. 28 let AB be a trough or canal cc, DD, FIG. 28. of a length AB, large relative to its breadth and depth. Let CD be a false end or partition movable within the canal along the direction AB. If now CD be moved rapidly from CD to C^D^ the water will become at first banked up in front of it; but as CD is retarded and finally brought to rest at CJ)^ the banked-up water will leave C^D^ and travel on as a crest or hump elevated entirely above the general level of the tank, somewhat as indicated in the diagram. Such a wave is called positive. Let us again suppose that the partition was at C l D l and the water at rest. Let it then be moved from C,D^ to CD. 70 RESISTANCE AND PROPULSION OF SHIPS. ' As a result a hollow or depression will be formed existing wholly below the general level of the surface. This hollow will then be propagated on in a manner similar to the crest previously described. Such a wave is called negative. A positive wave may also be formed by plunging into the water at one end of the canal a block which will occupy the volume CDDfv or by the sudden introduction of a certain additional amount of liquid into the end of the trough. Similarly a negative wave may be formed by withdrawing a block, or by withdrawing, as by a lifting-pump, a certain amount of water. Positive waves if properly formed are quite permanent, only slowly losing their energy and form by external friction and internal viscous forces. If not properly formed the wave will soon break up into an irregular series, the members of which are propagated with different velocities, so that the energy is soon dissipated and the form disappears. The negative wave is much less permanent than the positive, and can be made to preserve its integrity for a short time only. The horizontal movement of any sheet of particles sit- uated in a transverse plane seems to be the same. The vertical motion varies according to its distance from the bottom of the canal, and seems to be very nearly proportional to such distance. The path of any particle seems to be very nearly a semi-ellipse, as shown at AGO in Fig. 29. It there- fore appears that, due to the passage of the wave, a particle which was formerly at A has been transported and left at O; furthermore, that all particles in the transverse plane of A will undergo the same longitudinal translation, and will be left in a transverse plane containing O. Care must be here taken to distinguish between the motion of the particle and the RESISTANCE. 7 1 motion of propagation. While the particle moves through its path as A CO, the wave-form will move forward its wave- length OF or LM, Fig. 28. FIG. 29. The approximate contour of a wave of translation may be determined by the following construction: Let AGO be the ellipse representing the path of a surface particle, and let AH be the depth of the canal. Then lay off AF = 27r/i and construct the sinusoid AEF. Next lay off to the right from points on this curve, as A, c, g, etc., distances AO, ab, ef, etc., or the intercepts between the ellipse and the vertical OP. The points O, d, /, etc., thus determined will give the desired contour of the ideal wave of translation corresponding to the conditions assumed. Let v velocity of propagation ; h = depth of water when at rest ; k = height of wave above still-water level ; L = wave-length = OF, Fig. 29; a = range of translation = AO. Fig. 295 V volume of water constituting wave; 72 RESISTANCE AND PROPULSION OF SHIPS. b = breadth of canal; A area of profile of wave. Then v = Vjffi+t) (18) When k is relatively small we have v = V'gh (19) It is found that k cannot exceed //. On reaching this value, or before, the wave breaks at the crest. Hence as a limiting value of v we have v = V2gh (20) For the wave-length L = 27th a, (21) or where OL is relatively small L 2Tth (22) For the value of A we have A -ah (23) As with oscillatory waves, the energy is one-half kinetic and one-half potential, the entire amount being Energy = 2", or as a summation of such terms, according as the law is assumed to be constituted. Hence we may consider that in reference to speed the total wave-making resistance will probably vary with some index between 4 and 6, or as a sum of terms with these or intermediate exponents. The prob- ably decreasing values of P and Q with increasing speed would, if they were taken as constant, decrease the exponents of u from the values they would otherwise have, and thus perhaps partly compensate for the increase in exponent due to the variation of the term B. The actual relation, however, of the total wave-making resistance to the speed and to the dimensions of the ship is not known, nor can it be satisfactorily determined from the data available. There seems to be ground, however, for the belief that in many cases it shows a higher rate of variation than that given by the exponent 4. See also 26. For the present, therefore, we are thrown back on more empirical formulae, intended simply to express as nearly as may be the results of experience. Such formulae will be found in 29. With the accumulation of data from model experiments we may perhaps hope ultimately for a satisfactory determination of the constants in (i) and (2) or in other like formulae shown by the data to be more suitable to the purpose in view. 104 RESISTANCE AND PROPULSION OF SHIPS. 15. RELATION OF RESISTANCE IN GENERAL TO THE DENSITY OF THE LIQUID. For the head-resistance and skin-resistance per unit area we have seen that the density of the liquid enters as a factor. Similarly here the density must be a factor of the energy of a wave, and thus all parts of the resistance, and hence resist- ance as a whole, will vary with the density. The variation in the resistance of a given ship with change of density of the liquid is, however, in large measure offset by the oppositely varying amount of immersed body of ship. As between fresh and salt water the difference is slight, and is usually neglected except in the case of model experiments or very careful estimates. The densities of fresh and salt water are usually taken in the ratio of i.o: 1.032, or less accurately in the ratio 35 : 36. 16. INFLUENCE OF FORM AND PROPORTION ON RESISTANCE. In 1876 Wm. Froude published a paper* showing the results of experiments on the comparative resistance of models of four ships of the same displacement but of certain differences of form. These models were called A, B, C, D; A being that of the Merkara. The dimensions correspond- ing to the four models were as follows: Displace- ment. Length of Fore Body. Length of Middle Body. Length of After Body. Total Length. Beam. Draft. Wetted Surface. A i r 144 72 144 3 6o 37-2 16.25 18660 B l i 179-5 179-5 359 45-88 18 19130 C i i 154-5 154-5 309 49.4 19.32 17810 D j 1 95 95 95 285 45.56 17.89 16950 * Transactions Institute of Naval Architects, vol. xvn. p. 181. RESISTANCE. 10$ The forms of the fore and after bodies of B, C y and D were derived from those of A by expansion longitudinally, transversely, and vertically as required. The fore and after bodies themselves have therefore the same ratios of fullness and other characteristics as those of A, and the resulting differences in resistance may therefore be considered as due to the effect of the presence or absence of the parallel middle body. The results may be summarized by saying that at all speeds B and C had less resistance than A or D. As between B and C, the latter having the less wetted surface had the less resistance. As between A and D, the latter, though having less wetted surface, had at the speeds tried (from 9 knots upward) greater resistance, though at the lowest speeds the directions of the two resistance-curves were such as to indicate an intersection, and lower values for A for very low speeds. At high speeds B and C changed places relatively, the smaller wave-resistance due to B's greater ratio of L to B being more influential at such speeds than its greater wetted surface. At all speeds A had less resistance than D. For the former the resistance began to increase rapidly at about 17 knots and for the latter at about 14 knots. For C this relatively rapid increase in resistance began to appear at about 19 knots, while for no speed up to 2O knots was such tendency noticeable with B. It appears therefore in general that resistance will be decreased if, instead of a parallel middle body, the fore and after bodies are expanded so as to obtain a ship of the same displacement as with parallel middle body, even if the ratio B to L is thereby somewhat increased. 106 RESISTANCE AND PROPULSION OF SHIPS. Later experiments, notably those by Rota,* have thrown additional light on the influence of a slight change in proportions with the same characteristics of form. In such case secondary forms are derived from a primary by changing all dimensions in one direction (as transversely) in a constant ratio. As the result of a large series of experiments Rota reaches the following con- clusions : (a) For the same change in displacement obtained by, such "expansion" the power required for a given speed is affected in larger degree by a change in breadth than in depth. (&) The increase in longitudinal dimensions for the same displacement results in a notable decrease in the residual resist- ance and on the other hand in a marked increase in the frictional resistance the latter effect however, less than the former at usual speeds. (c) Every form derived by change in the longitudinal dimen- sions has a particular speed for which the increase or decrease of displacement resulting from such change is accompanied by a small or negligible change in the power required. The length and speed at which such relation obtains are connected approxi- mately by the formula, v = VL, where v = speed in knots and L= length in feet. (d) Change in the displacement resulting from change in the transverse dimensions, so long as the percentage of change is moderate, will result in a change of resistance at the same speed of approximately the same percentage value; that is, 10 per cent increase in displacement will require about 10 per cent increase in power for the same speeds. Changes in the displacement resulting from change in the * Bulletin de L' Association Technique Maritime, 1900, p. 49. RESISTANCE. 107 vertical dimensions, so long as the percentage of change is moderate, will result in a change of resistance at the same speed of approximately .6 the same percentage value; that is, 10 per cent increase in displacement will require about 6 per cent in- crease in power for the same speed. With the displacement slightly increased by longitudinal change and the speed in question greater than \/L, there will be a decrease in power for the same speed; and with the dis- placement slightly decreased, there will be an increase in power for the same speed. With the speed in question less than VZ the above relations are reversed. 17. MODIFICATION OF RESISTANCE DUE TO IRREGULAR MOVEMENT. It is well known experimentally in all cases of bodies moving in a liquid, that a certain amount of water is momentarily bound to the body and must be considered as practically taking part in its motion, especially in any rapid changes involving accelerations and retardations. This amount, it also appears, may be from 15 to 20 per cent of the displacement of the ship. It follows that the resistance to an acceleration will not be that due simply to the mass of the ship, but rather to the combined mass of ship and water. At the same time it must be remembered that during retar- dation a corresponding effort is given out. It is, however, a general principle that more energy is required to maintain a system in irregular movement, the velocity oscillating about a mean value u, than at the same uniform value of the velocity. This is because in general R varies with u accord- 108 RESISTANCE AND PROPULSION OF SHIPS. ing to a higher index than I, and because in such case the mean of a series of such powers of u is greater than the mean value of u raised to such power. This is readily verified for an index 2 or 3. The mean resistance of a ship in irregular motion but at a mean speed u will therefore in general be greater than if the velocity could be made uniform at the same amount. The actual modes of propulsion employed usually involve slight irregularities in motion, and these may be more or less increased by irregularities of wind, current, tide, depth of water, etc. Under usually favorable conditions it is not likely that the increase of R due to this cause is important, although its existence and explanation are of interest. 18. VARIATION OF RESISTANCE DUE TO ROUGH WATER. The resistance of a ship in a seaway is known actually to be very considerably greater than that in smooth water for the same average speed. The causes of this may be given under three heads: (1) The direct action of the waves and the rolling and pitching tend to increase the irregularities of motion, thereby affecting the resistance as described in the last section. (2) The wave-disturbance of the water tends to confuse and disturb the regular stream-line motion, thereby entailing a greater drain of energy from the ship than in smooth water at equal speed. (3) The pitching and rolling, notably the former, place the ship continually in less favorable positions relative to RESISTANCE. propulsion, so that on the whole the mean resistance is in- creased. These causes also react unfavorably on the propelling apparatus, lowering its efficiency and increasing the irregulari- ties of motion. While this last result is secondary,- the final effect on the propulsion is the same as though the resistance of the ship were virtually increased. No rules can, of course, be given for the estimate of such irregular modifications. It is a matter of experience, how- ever, that the following qualities are favorable to uninter- rupted speed in rough water: (1) Considerable length, good free-board, and steadiness, especially absence from pitching. (2) Large size and consequently great weight. The reasons for the good effects of the former are ap- parent. The good effects of the latter are due to the increase f inertia and the consequent decrease in the irregularities of movement. 19. INCREASE OF RESISTANCE DUE TO SHALLOW WATER OR TO THE INFLUENCE OF BANKS AND SHOALS. It is a well-known fact that the resistance of a ship or boat at any given speed is very much increased by shallow water or by the proximity of banks, as in the navigation of canals and narrow rivers. The primary cause of this excess of resistance is readily found in the disturbed stream-line motion. As a result of the decreased cross-section within which the stream-lines must be contained, there arises an entire change in their form and in the wave configuration about the boat. The latter in general is very much exagger- ated relative to its condition in open deep water, and the net HO RESISTANCE AND PROPULSION OF SHIPS. consequence is a very considerable increase in the amount of energy necessary to the maintenance of the configuration of the water, and hence a corresponding increase in the resistance. We may next inquire as to the limits within which these results become of notable significance. Taking first the question of shallowness of water, the other dimensions being unlimited, we have a large number of instances of trial-trips in which the increase of resistance due to shallow water is more or less clearly shown. White* gives the following instance: The Blenheim in water averaging 9 fathoms made 20 knots with 15750 I.H.P. The wave phenomena were most striking and unusual. In water from 22 to 36 fathoms with the same power the speed was 21.5 knots. In the trial of the U. S. S. New York both ways over a 4O-mile course it was found that at the same point each way, where the depth of water was charted as 37 fathoms, the revo- lutions fell off slightly while the steam-pressure became increased. The water over the remainder of the course was from 15 to 20 fathoms deeper than at this point. The mean speed was 21 knots. Experimental investigation of these relations reported on by Marriner and H. Yarrow | has brought out the following facts : It is a matter of common observation that as a boat runs into shallow water the wave length of the transverse system increases in length. The relation between speed and length of wave for deep water, as noted in 10, is * Manual of Naval Architecture, 1894, p. 467. f Transactions Institute of Naval Architects, 1905, p. 344. RESISTANCE. Ill A graphical representation of such law is shown by the heavy curve of Fig. 42. Now when a ship is run in shallow water the relation between speed and wave lengths follows this line for low and moderate speeds, and then as the speed becomes higher the curve rises more quickly, as shown by the light lines, implying a more rapid increase in length with increase in speed than for deep water. Finally, at some limiting speed the curve approaches the vertical, implying an indefinite increase in length for such speed. If then the speed is increased beyond this value there is no corresponding wave or wave system. The speed at which this condition is approached is called the critical speed for that depth, and the depth at which the length of wave becomes in- definite for a given speed is called the critical depth for such 112 RESISTANCE AND PROPULSION OF SHIPS. speed. The relation between speed and depth for this critical combination is approximately where v = speed in knots and d = depth in feet. In going from deep to shallow water the increase in length might be expected to follow the law of shallow water waves, as noted in 10. Actually, however, the increase is less rapid than the formula would indicate. In explanation of this it has been suggested by Marriner that if the reaction between the ship and the wave system results in the latter being dragged along, as it were, and thus prevented from lengthening as rapidly as it other- wise would, then such reaction would result in an added drag on the ship manifested as an increase in resistance. This con- sideration may aid in explaining the great increase in resistance in approaching the critical speed for a given depth. In accordance with these various observations on speed and wave system, it is found furthermore that the maximum wave disturbance and the point of maximum resistance both occur at or near the critical combination of speed and depth. Certain experiments, moreover, seem to indicate that the hump in the resistance curve is independent of displacement. This point, however, needs wider assurance through further investigation. Certain other experiments by Rasmussen on the torpedo-boat Makrelen, and reported by Laubeuf,* bear out the same general conclusions. In these experiments a sharp rise in the speed horse-power curve was noted at points corresponding closely to * Bulletin de L' Association Technique Maritime, 1898, p. 207. RESISTANCE. 113 the relation 1^ = 11.3^, as noted above. The conclusions drawn by Laubeuf are as follows: (1) The various curves present a hump more and more marked as the depth is reduced. At depths greater than 50 feet there was no noticeable hump within the range of speed em- ployed (to 20 knots). (2) The speed corresponding to the hump was, for a given depth, nearly independent of the two displacements at which the boat was tested (105 and 118 tons). The power for such speed, however, increased in marked degree with the displace- ment. The important question, however, relates rather to the con- ditions under which the resistance will be sensibly that due to water of indefinite depth, or, in other words, for what depth will the influence of the bottom be negligible? Regarding this prob- lem various view-points may be taken. Thus it has been sug- gested that so long as the depth is sufficient to allow the wave corresponding to the speed of the ship to assume sensibly its trochoidal form and constitution no sensible increase in resist- ance will result. The propriety of considering this as more than a roughly approximate relationship may be questioned, since we do not know that the -natural wave formed by the ship is more than roughly trochoidal in form. By -reference to 10, Table V, it appears that if the depth equals one half the wave-length the difference between the two kinds of waves should be quite negligible. This leads by the aid of 10 (i) to the following table : 114 RESISTANCE AND PROPULSION OF SHIPS. Minimum Depth, Speed, Knots. Feet 10 28 12 40 14 55 16 71 18 '. 90 20 in 22 135 24 160 26 188 28 218 3 2 5 32 284 34 322 36 3 6 38 400 40 444 From considerations drawn from a study of the distribu- tion of stream-lines under certain ideal circumstances, D. W. Taylor* concludes that a depth of water equal to ten times the draft is an outside limit, and that no sensible increase in resistance is likely to be met with so long as the depth is not less than six times the draft. The result in the case of the New York, if reliable, indi- cates that the influence extends to a considerably greater depth than six times the draft, or than the amount given by the table above. On the other hand the experiments reported by Marriner and Yarrow showed that for torpedo-boats at * Transactions Institute of Naval Architects, vol. xxxvi. p. 234. RESISTANCE. 115 speeds of 25 to 30 knots, a depth of water of 100 feet was insuf- ficient and would not permit of realizing contract conditions, while an increase to 120 feet proved adequate for the purpose and permitted substantially the same performance as the Skelmorlie course with 240 feet depth. The depth for critical speed is seen to be "3 while, according to Marriner and Yarrow, the worst combination of conditions was reached at about 10 It therefore seems most rational to assume that the safe depth may be stated as some multiple of this depth, modified suitably by the size and form characteristics of the ship. It is readily seen that the above table representing half wave-lengths gives simply values of v 2 + $.6. While perhaps none too deep for large ships of moderate or full form, the values of this table indicate a depth somewhat greater presumably than is needed for torpedo- boats and other like relatively small boats of fine form. Further investigation is needed on these points, and especially hi relation to the influence of form and dimension on the limiting depth. Turning now to the conditions existing in canals, we have not only the restriction due to the bottom, but also that due to the sides as well. Proceeding at once to a representative case, we may note briefly the phenomena attending propul- sion in a canal of which the cross-sectional area is only some four or five times that of the greatest section of the ship. In such cases at moderate speeds the bow wave is quite marked, rising more or less transversely on each side or a Il6 RESISTANCE AND PROPULSION OF SHIPS. little ahead of the bow, and followed amidships or astern by an equally notable hollow. Following astern is a train of waves, each member of which moves with the velocity of the boat, while the train as a whole moves at a lesser velocity, as explained in 12. The slower the boat moves the more are the phenomena like those noted in connection with motion on the open sea. As the speed increases the more definitely does the boat place herself on the rear edge of a wave as in Fig. 43, and the more pronounced does the series of follow- ing waves become. As the speed of the boat approaches that corresponding to the propagation of a permanent wave of translation, u = Vgh, the waves begin to fake on more and more the characteristics of waves of translation. They transmit a greater and greater proportion of energy and the train becomes shorter and shorter. The resistance, however, continually increases due to the rapidly growing size of the waves and their consequently increased demand for energy. As the boat reaches the critical velocity or passes slightly beyond it, the train of waves contracts to a single member, which takes its place with crest amidships or with the boat slightly on its forward slope. The resistance then decreases, and the boat may be towed along at this speed with a much less expenditure of power than just before. The cause of this sudden change in the law of resistance is found in the relation of the boat to the wave. The necessary energy to RESISTANCE 117 form the permanent solitary wave having been expended, and the boat having been placed in the most favorable posi- tion relative to such wave, the maintenance of the wave and the velocity of the boat may be obtained at a comparatively small further expenditure. The resistance for the most part is that due to the maintenance of the wave in an imperfect liquid, with the increased tendencies toward degradation due to irregularities in the sides and bottom. Following is a table showing results of experiments made by J. Scott Russell, the pioneer of experimental investigation in this direction. Weight of Boat in Pounds. 10239 Speed in Miles per Hour. 12579 Resistance in Pounds. 112 201 275 250 268.5 250 500 400 280 Speeds at which these conditions obtain cannot, however, be practically maintained in canals, due chiefly to the wear and tear on the banks from the wash of the wave, and the comparatively high speeds necessary for depths of water exceeding 10 or 12 feet. We are therefore rather concerned as to the effect of the banks and bottom on the resistance at moderate speeds, and Il8 RESISTANCE AND PROPULSION OF SHIPS. as to the necessary ratio between the section of the canal and that of the ship in order that the increase in resistance may- be negligible. Satisfactory data for a general discussion of these points are not available. We have, however, certain partial results as follows: Elnathan Sweet * gives the following formula as repre- senting the results of a series of experiments made by him on the Erie Canal: .10303^ 1 r- .$97 ' R = resistance in pounds; s = wetted surface ; v = speed in feet per second; r = ratio of section of canal to that of boat. The speed varied from 2 to 3 feet per second, or from 1.27 to 2.09 miles per hour, s varied from 2870 to 3090; r from 4.28 to 5; depth of water from 7 to 8 feet; draft of boat from 6 to 7 feet; displacement from 260 to 310 tons. The formula is a special case of the general formula pro- posed by J. Scott Russell and Du Buat for resistance in a restricted channel, viz., ID where A and B are constants to be determined by experi- ment. The ranges of values of r, v y and s are manifestly too * Transactions American Society of Civil Engineers, vol. IX. p. 99. RESISTANCE. small to warrant the application of these results to conditions widely different from those covered by the experiments. Conder* states that in one instance where r =. 3 a speed of 7 miles in the open was reduced with the same power to 5 ; while in the discussion of the same paper Gordon states that in the Rangoon, where r 10, a speed of ten miles was similarly reduced to 7. M. Camere gives the following results cited by Pollard and Dudebout:f Let resistance be represented by a formula R = fsv\ where the letters have the same significance as above and /is a coefficient. r Values of /at Speeds per Second of 3'-28 4 '. 9 2 6'. 56 I6. 4 9.0 3-45 .00464 .00836 .0280 .00513 .00891 .00564 .00946 More recently M. de Mas has conducted extensive experiments in France, the results of which have been made public by Derome.J Among the many interesting features of this investigation the following may be men- tioned : * Minutes of Proceedings of Institution of Civil Engineers, London, vol. LXXVI, p. 660. f Theorie du Navire, vol. in. p. 458. J Proceedings Sixth International Congress on Inland Navigation, 1894. Quoted also by Leslie Robinson, Proceedings Institute Mechanical Engineers, i8 9/ . 120 RESISTANCE AND PROPULSION OF SHIPS. Three boats were tested by towing on the Seine. The ratio of the section of the stream to the immersed section of boat was about 55, for which the results should be, presum- ably, not far from those for a ratio indefinitely large. The results are given in Table I, from which it appears that the TABLE I. Displace- Resista nee in Pou nds at Spe eds per M nute of Boat. Length. ment. Q 8'. 4 ig6'.8 295'.2 393'-6 492' 124.6 286 liq JC7 783 1464 2467 Rene QQ A. 112 qe-i 78-} 1466 24.60 Adrian 67 4 148 112 -ie-7 78-3 1466 2460 resistance was independent of the length within the limits of speed and length employed. The same three boats were afterward tested in the Bourgogne Canal at speeds increasing by increments of 49.2 feet per minute, from 49.2 to 246 feet per minute, with the same result as regards the substantial independence of resistance on length. The general truth of this proposition must by no means be assumed from these experiments, though there seems reason for believing that, due to the peculiar form of canal-boats, a change of length has relatively small influence on the resistance. A number of boats with particulars as in Table II were tested in both river and canal, with results as given in Table III. RESISTANCE. 121 TABLE II. PARTICULARS OF BOATS. Name. Length. Average Breadth at Midship Section. Block Coefficient of Fineness. TO; . *J 16 4 Flute 122.8 16 e, yy 118.4 i6.5 Q7 III .Q 16 i Q-J TABLE III. rt c Q. u o Speed Speed 1 *! 984 Feet per Min. 196 8 Feet per Min. Cross- s ~ = /. section IS 5 TT O. Draft. Boat. of <- y ~ - Resistance. Resistance. Canal. w t?"x R R A rt'7) 5 ^ is Canal River T Canal River r OS'S R r R T ( Peniche 3I7 ; 9 86.66 3-67 379 225 1.69 1896 664 2.86 5'.25- Flute 86.44 3.68 247 119 2.07 1060 357 2.97 1 Toue 86.44 3-68 240 97 2. .18 IO2T 278 3-67 ( Flute 70.30 4-52 154 97 1-59 626 315 1.99 4'. 27 - Prussian 68 68 4 62 119 49 2-45 474 176 2.69 / Margotat 69.98 4-54 H7 46 2.52 434 148 2.94 3'.28 Flute 54-04 5-88 106 86 1.23 421 284 1.48 li 1 An interesting test was also made in various waterways of the boat Jeanne, 99 feet long, 16.4 feet wide, and of the Flute class. The results are given in Table IV. 122 RESISTANCE AND PROPULSION OF SHIPS. TABLE IV. TEST OF JEANNE IN DIFFERENT WATERWAYS. C-* 3 49'. 2 per Min. 98'. 4 per Min. i47'.6perMin- i96'.8perMin. 246' per Min. Draft. Boat. .2 $ a R R R R R R r R r R r R r R ! r y r r ^ Seine 116. 26.5 26.5 .00 28.7 28.7 .00 146 146 .00 245 245 .00 384 384 1. 00 A 8.31 26.5 .00 28.7 .00 150 3 271 .11 459 i .20 3'. 28 B 5.89 35-3 " 33 103.6 .42 227 56 4 o8 ' .67 692 ' 1.83 C 4.64 37-5 i .41 108.2 .48 238 .63 452 85 79 6 1 2.07 D 3.82 39-7 ' .5 I2 3-5 .70 284 95 558 .28 1019 ' 2.66 Seine 89-3 26.5 ' .00 83.8 83 8 .00 172 172 .00 295 295 .00 474 474 1. 00 A 6 -39 3-9 ' .17 105.8 .26 238 38 44 1 .49 756 i. 60 4'.27 -{ B 4-54 48-5 ' 74 154.3 .84 342 .98 622 .1011074 ' 2.26 1 C 3-57 5-7 4 .02 178.6 13 410 .38 814 75 T 5 01 3- T 7 I D 2.94 1 .67 2 55-7 5 657 3.82 1389 " .70 2646 ' 5-58 r Seine 72.5 2^7 28.7 .00 90.490.4 .00 187 187 i .00 320 320 .00 5 11 5" I. 00 A 5- J 9 39-7 42 141.1 .56 326 1.74 620 '94 1124 2. 2O 5 . 25 | B C 3-68 2.90 70.5 72.8 " .46 .56 244.7 264.6 ' 7 1 93 562 688 3.00 3-67 1047 " 3.28 4-79 1839 2965 4 3-59 5.80 While the information relating to many points of this problem is still quite meager, it is sufficiently well established that where the ratio of section of canal to section of ship is not more than six or eight, a very considerable increase in the resistance will result, even at moderate speeds. Again, mere sectional area of canal is not sufficient, for we might have a very wide and shallow canal in which the ratio of sec- tions would be very great, but in which we should have the shallow-water resistance as already discussed. In order to reduce this increase of resistance to a small or negligible quantity, it seems likely that the transverse dimensions of the channel should be everywhere from six to ten times the corresponding dimensions of the greatest section of the boat. In this connection it is interesting to note that the influ- KESIS TA NCR. 123 ence of fine extremities is less and less useful to reduce resist- ance as the ratio of section of canal to boat is less. It is readily seen that a point may be reached where the resistance may be less for a somewhat narrow and full form than for an equal displacement and equal length, but with fine ends and a larger midship section, and hence with a greater constric- tion of available cross-section for stream-line motion. 20. INCREASE OF RESISTANCE DUE TO SLOPE OF CURRENTS. In ascending rivers where the current is noticeable there must be a corresponding up grade and a resulting vertical component to the movement. The total energy expended in any given time will therefore include, in addition to that necessary to overcome the water-resistance, an amount Db y where D equals displacement and b is the total change in elevation effected during the interval in question. The corre- sponding resistance is naturally D sin a Da, where a is the angle of grade and always small. Naturally the amount of this part of the total resistance in such a case is usually small, though it may reach such an amount as to prevent the ascent of a river where the speed of the current is consider- ably less than the possible speed of the ship in smooth water. In special cases ex. may reach a value of from .001 to .0015. At moderate speeds the smooth-water resistance may not be more than .oiD or even less, so that it is readily seen that this part of the resistance may reach an amount considerable in comparison with that for such speeds under usual conditions. 124 RESISTANCE AND PROPULSION OF SHIPS. 21. INFLUENCE ON RESISTANCE DUE TO CHANGES OF TRIM. In treating of resistance we have thus far considered only the horizontal component. It is quite evident, however, if the ship were to be maintained in her statical trim and draft and towed through the water at any given speed, that the resultant of all the changes of pressure would in general have a vertical as well as a horizontal component. In many cases, especially at high speeds, it becomes important to take cognizance of the existence of this vertical force. Considered as a single vertical resultant, it usually acts through a point forward of the statical center of buoyancy. In the actual case, the ship being free to yield to this force, she will change trim, the bow rising somewhat more than the stern sinks, thus decreasing the displacement by a small amount. At certain high speeds these modifications, especially the change of trim, become very noticeable. At still higher speeds, especially with the cut-up form of after body mentioned in 11, it would appear that the trim may begin to go back toward its normal value, though the displacement will prob- ably continue to decrease with the increase of speed. From another point of view we may consider that the motion of the boat, especially at high speeds, gives rise to a wave system, and that the boat naturally accommodates itself in trim to the characteristics of this system. In general at high speeds such system will involve a hollow under or near the stern and a crest near the bow, thus placing the boat on the back slope of the wave and naturally giving rise to the change of trim. An instance is given by White* of measure- * Manual of Naval Architecture, p. 470. RESISTANCE. 125 ments taken by Yarrow on a torpedo-boat about 80 feet long at a speed of 18.5 knots. The change of trim was about one- half inch per foot of length or forty inches between bow and stern. Relative to the water surface forward the bow raised rather more than one foot, while relative to the surface aft it settled less than six inches. The remainder of the change in trim is accounted for by the change in the level of the water itself due to the wave formation. Like the other phenomena attending the motion of a solid in a liquid, this change of trim and of displacement cannot at present be treated by abstract mathematics. The investiga- tion of the performance of models and such comparative methods are alone of use in aiding to form an estimate of the amount of such change in any given case. The influence of the change of trim on resistance is prob- ably slight. Directly it is a result and not a cause of resistance. Indirectly it may be a cause of a modification of resistance on account of the difference which it may cause in the form of the immersed bodv. With usual forms the influ- ence of such slight changes of this character as usually occur is not likely to be important. It is probable that the prin- cipal loss arising from a change of trim is due to decreased efficiency in the propulsive apparatus rather than to actual increase in the resistance. In connection with these considerations it may be pointed out that there is reason to believe that for extremely high speeds relative to length, such as are attained by some torpedo-boats and fast launches, and notably by recent torpedo-boats and torpedo-boat destroyers, the coefficients in the formulae of resistance which hold for larger ships and for more moderate speeds undergo considerable change. A 126 RESISTANCE AND PROPULSION OF SHIPS. direct extension of the results for lower speeds to such extreme values is not therefore allowable, and the various constants and characteristics of empirical formulae for resist- ance and power must be determined by direct experiment under such conditions, or under others comparable with them. See 26. In the boats mentioned speeds of 30 to 35 knots and upward have been attained on lengths of from 180 to 200 feet. The resistance at such speeds is shown to vary with a lower index than for more moderate speeds, as, e.g., 18 to 24 knots, and the performance as a whole seems to indicate a general modification in the relation of the resistance to the attendant conditions. As causes of this change in relationship we may look to the change in displacement, and consequently in wetted sur- face, and to the changed location of the wave system relative to the boat. ^ Photographs of boats at very high speeds show that under such conditions the decrease in wetted surface may be quite considerable, though data are lacking to furnish a satisfactory basis for estimate in any given case. The change in the resistance of a boat in a canal when it approaches a speed such that it can be forced over on to the top or forward slope of the primary wave has already been mentioned. * It is not to be expected that these conditions can be paralleled in open water; but we know that as the speed increases the location of the bow primary wave falls farther and farther aft, and photographs taken under these conditions as well as the decreased change of trim seem to show that at a speed sufficiently high the boat is forced at least partly KESIS TA NCE. I2J over the primary bow wave, so that such wave, once formed, might require a less expenditure for the maintenance of its energy and the speed of the boat than would be indicated by an extension of the law for lower speeds to this extreme point. It is not likely that this favorable region in the law of resistance can at present be taken advantage of by other than boats of the types mentioned. For larger ships, e.g., of L = 400 feet, such speeds would exceed 40 knots, and the power required would be so great that with present engineer- ing resources its development on the weight available and its economical application would be impracticable. 22. INFLUENCE OF BILGE-KEELS ON RESISTANCE. Bilge-keels are usually located near the turn of the bilge, and stand normal to the surface rather than vertical. They usually run for one half or two thirds the length of the ship, and are supposed to follow as nearly as may be the natural stream line path in order to interpose the minimum resistance. Experiments by Wm. Froude and others have shown that the additional resistance due to bilge-keels is slight. Some of the model experiments by Mr. Froude indicated that the additional resistance due to the bilge-keels was less than that due to their surface friction as usually computed ( 8). It has been suggested that this may be due to the fact that the keels are in a skin of water moving forward more or less with the ship, and that in consequence the resistance per unit area is less than that normally due to the relative velocity of ship and water as a whole. A more just view is perhaps to consider that with the bilge-keels the 128 RESISTANCE AND PROPULSION OF SHIPS. average skin-resistance coefficient for the ship as a whole is somewhat less than for the ship without the keels; so that while with the keels the total area is greater, the coefficient is less, and the product is but slightly increased. The reason for the decrease in the average coefficient may be readily seen by comparing two bodies of cylindrical form moving axially through the water. Let one be smooth and the other deeply corrugated or channelled longitudinally, both of the same outside diameter. Both will tend to set in motion a cylinder of water of about the same diameter, and with the channelled cylinder this tendency will be so effective that at and near the bottom of the channels the relative velocity of the sur- face and the water will be very slight, and hence the resist- ance small. The average coefficient of surface resistance for the channelled cylinder will therefore be less than for the plain, and while its area is much greater, the total surface resistance will not be increased in a proportional degree. Recent Italian experiments on the model of the Sardegna* showed for speeds of the ship from 16 to 21 knots an increase of resistance from I to 5 per cent increasing with speed, or with given power a decrease in speed from practically nothing to a little over I per cent. These experiments seemed to indicate a somewhat rapid increase in the resistance due to the bilge-keels from speeds of about 18 knots upward. It may be suggested that this was due in large measure to a change of trim at these speeds and to a disturbance of the stream-line flow, in virtue of which the keels became placed oblique to the line of flow and thus offered a certain amount of head-resistance. * Rivista Marittima, Oct. 1895. RESIS TA NCR. 1 2 9 23. AIR-RESISTANCE. All above-water portions of a ship in motion are, of course, exposed to air-resistance. This from the irregularity of the motion of the air and the great irregularity of the sur- face exposed is very difficult of computation. Except in rigged ships with a high velocity of the wind, however, this effect is not considered of great importance, and usually no attempt is made to estimate its amount. From data obtained from planes moving in air" it appears that the resistance may be expressed by the formula R=fAv\ In this equation A is the projected area of the surface exposed, the projection being made on a plane perpendicular to the direction of motion. For an unrigged ship this is readily found, being simply the area of an end view of the part of the ship above water. Where the ship is rigged the amount to be taken is very indefinite, and no estimate can be made except by comparative methods, for which, unfortu- nately, the amount of data available is insufficient. Wm. Froude was led to believe that in the case of the Greyhound the influence of the rigging was about equal to that of the hull. This would depend very much, however, on the nature and amount of rigging, and doubtless on the speed of the wind. The velocity v is to be taken as the longitudinal component of the relative velocity of the wind and ship. Thus in calm v will be the speed of the ship, in a following- wind it will be less, and in a head wind more. The value of the coefficient / rests on experimental determination. The experiments of Wm. Froude and others 130 RESISTANCE AND PROPULSION OF SHIPS. indicate that with pounds, feet, and seconds as units its value may be taken as about .0017, and with pounds, feet, and knots as about .0048. Later experimenters, among others Dines and Langley, find values somewhat less and ranging for pounds, feet, and knots about .004. The experimental values are derived from planes of small or moderate size, and there is considerable uncertainty as to the values suitable for use on large areas. In the case of the Greyhound unrigged, with a relative speed v = 15 knots the effect produced was measured by 330 pounds. For v = 10 knots this would correspond to about i^o pounds. This is about 1.5 per cent of the water-resist- ance at the same speed. With rigging it was assumed that this might be doubled, so that in a calm the ship would experience an air-resistance of perhaps 3 per cent of her water-resistance. For moderate speeds, or so long as the water-resistance varies sensibly as the square of the speed, we might expect the relation between the air and water resistance to remain sensibly constant. For higher speeds the water-resistance will increase more rapidly than the square of the speed, so that the ratio of air to water resist- ance will decrease at these speeds. If instead of calm air we suppose a speed of 10 knots through the water against a head wind of velocity say 30 knots, we should have v = 40, and the air-resistance would be some 16 times as much as before, or unrigged about 25 per cent of the water-resistance and rigged about 50 per cent. While these figures may not be exact, they are sufficiently so for illustrative purposes, and show plainly the importance which wind-resistance may assume in extreme cases. On one hand the constant decrease of sails and rigging on all types of steam-vessels tends to decrease the importance of air-resistance. On the other, the constant effort to RESISTANCE. 13! increase speed on ocean-liners, war-ships, torpedo-boats, etc., calls increased attention to all causes which in any manner may affect the resistance. Additional experiments in this direction are greatly needed. Such experiments might be carried out by allowing a ship to drift in a fairly smooth sea under the influence of the wind alone, and measuring the velocity of the ship relative to the water and that of the wind relative to the ship. Then from model experiments or other known data relative to the ship her resistance at the speed attained may be known, and this must equal that due to the air moving past the ship with the relative velocity observed. 'Otherwise a ship might be moored and subjected to the wind, the strain being measured by suitable dynamo- metric apparatus. 24. INFLUENCE OF FOUL BOTTOM ON RESISTANCE. There seems no reason for believing that the condition of the bottom of a ship will essentially affect anything but the skin-resistance. The effect of foul bottom on skin-resistance has been dis- cussed in 9, and it is again mentioned here simply to make complete the special causes which may affect the total resist- ance. 25. SPEED AT WHICH RESISTANCE BEGINS TO RAPIDLY INCREASE. The analysis of curves of resistance usually shows that for very low speeds the total resistance varies nearly as the square of the speed, such relation remaining nearly constant until a speed is approached at which the resistance begins 132 RESISTANCE AND PROPULSION OF SHIPS. rather suddenly to increase more rapidly, the index becoming 3 or even 4 for a time, while at still higher speeds the index may fall again to 2 or sometimes apparently to even less. Such characteristics in the curve of residual resistance are shown in Fig. 40. The speed at which the first rapid in- crease is located depends chiefly, no doubt, on the wave- making features of the ship; and while it is not expressible with any great accuracy by any simple formula, yet it is sometimes considered that for general purposes it may be taken as proportional to the speed of a wave of length equal to that of the ship. Hence we should have /&. U ~ A / - - = V 27T bVL, where b is some constant. With knots and feet as units, b is frequently taken as approximately I, in which case we have u = VI. This must be considered as giving simply an indication of the locality of a region where the resistance will begin to rapidly increase, rather than the a'ctual location of a definite speed. It must also be remembered that generally the loca- tion of this region will be higher as the ratio of length to beam is greater, and as the prismatic coefficient is less. Normand has developed formulae for what he terms the maximum normal speed as follows: (i.oim-b)L "* - 1 1.39(1.01^ b)L or ^ RESISTANCE. where 5 = beam of ship in feet; H = draft in feet; L = length in feet; b = block coefficient; m = midship section coefficient. These formulae are intended to take account of form charac- teristics, and to indicate the maximum speed which, under normal or ordinary conditions, such ship may be given without marked departure from the usual resistance and power relations. 26. THE LAW OF COMPARISON OR OF KINEMATIC SIMILITUDE. It is the purpose of the law of comparison to furnish for two ships of similar geometrical form but of different size, and moving at different speeds, a relation between the know, principally the wave-making or modified stream-line resistance. This being the purpose, care should be had in noting the fundamental assumptions on which the statement of the law depends. Consider the stream-lines in an indefinite liquid moving past a body as in I. Let the mo'Jon throughout these stream-lines be steady. That is, let there be no discon- tinuity as in the breaking of waves at the surface or the formation of eddies within the body o'f the liquid. The usual equation for steady motion will then apply, and de- noting the total head by k we shall have for any stream- line, z + + = .. (i) z =*. (3) In this equation let us take z as measured from a base- line OX. The pressure/ will be simply the statical pressure due to the height FG, and p -f- a will in feet be equal to this FIG. 44. height. Hence 2 -\- p +- v is here equal to XG, the total depth of the reservoir to the base-line OX. From (3) this is the value of k. Hence throughout the entire course of the stream-line represented by (i), the constant k represents this depth GX. Now the second system is a magnification of the first in the linear ratio A. There will be therefore a similar point f similarly situated in a similar indefinite reservoir whose depth to a similar base-line will be A/, and this will be the value of /,, the constant for the stream-line in the second system, whose equation is given in (2). I$6 RESISTANCE AND PROPULSION OF SHIPS. We have thus far simply considered the velocity-ratio n such that the stream-lines in the two systems will be geometrically similar. We must now determine what the value of this ratio will be in terms of the linear ratio A. Since the stream-lines are similar throughout, and since the flow is steady and the condition of continuity is fulfilled, we must have the velocities along similar lines exactly pro- portional at similar points. That is, if v ot v lt and F , V l denote velocities in the first and second systems at pairs of similar points, we must have V,_v. F, V, - '-'- This shows that n is a constant, and that if we can determine the velocity-ratio for any pair of similar points the constant value of n will thus be known. The velocity-ratio at a pair of corresponding points is most readily found by aid of Fig. 44, by considering A an open end in each system. We shall then have / = o, and for the first system .and for the second =U-\z= A( - A Hence, dividing, we have n 1 = A and n = 4/A. It follows that in order to fulfil the conditions of simi- RESISTANCE 137 larity throughout the two systems we must have a constant velocity-ratio equal to t x A. Putting this for n in (2) we have (4) whence, comparing (i) and (4), /, = \p. The same relation will hold for every other stream-line, and it follows that throughout the entire second system the pressure will be A times that at corresponding points in the first system. In other words, the magnification in pressure will exactly correspond to that in linear dimension. This relation will therefore hold between the pressures at corre- sponding points on the surfaces of the two bodies, because by supposition the surfaces throughout are in contact with the stream-lines. In the stream-lines of Fig. I the body is supposed to be indefinitely immersed, so that no surface changes take place. All of the preceding equations and con- clusions hold, however, for any stream-line in a perfect liquid, and therefore for a stream-line system about a body partially immersed, so long as the conditions of continuity of flow are fulfilled. Now it is due to the distribution of pressure throughout such a stream-line system that the stream-line resistance arises. Hence this resistance will be the value of the integral of the longitudinal component of the pressure acting at the inner surfaces of the stream-line sys- tem. Denote an element of the surface by ds and the angle between its normal and the longitudinal by 6. Then we have for our first system = Cp cos ds. 138 RESISTANCE AND PROPULSION OF SHIPS. In the second system the entire distribution of pressure is multiplied by A, and the linear dimensions are increased in the same ratio. Hence the ratio between corresponding elements of surface will be A a , and in the second system we shall have R' = A cos e Kds = A 3 cos 6 ds = That is, with two systems under the conditions we have assumed, the ratio between the resistances due to stream-line pressure is A 3 . We may also note that this is likewise the relation between the volumes of the two immersed bodies, and hence between their displacements. The law of comparison thus derived involves three con- siderations, to which careful attention should be given: (1) The supposition of geometrical similarity; (2) The definition of corresponding speeds- (3) The statement of the law. Geometrical similarity for the two systems being assumed, corresponding speeds are described as those which will pro- duce similar stream-line or similar wave configurations, and are defined as speeds in the ratio of the square root of the linear-dimension ratio A. , Based on these conditions, we then have the following statement of the law: At corresponding speeds the resistances of similar ships are in the ratio of the cubes of like linear dimensions ; or as the cube of the linear -dimension ratio ; or as the volume -ratio. The truth of the law thus derived, it must be remem- bered, rests on the application of the formula for steady motion to stream-line flow. It is therefore only applicable RESISTANCE. 139 to that part of the total resistance which is due to the stream-line motion, and hence, so far as the above derivation is concerned, is not applicable to skin or eddy resistance. It also presupposes the following as necessary conditions for its exact truth: (1) A liquid without viscosity; (2) The absence of discontinuity of flow such as would be caused by breaking waves or eddies. So far as the first condition is concerned, the viscosity of water relative to the velocities with which we are concerned is so slight that this departure from the exact conditions will make no essential difference in the application of the law. With regard to the second condition we must distinguish between the effect of the eddies, and that due to breaking waves. Eddies will be formed by possible irregularities in the surface, and as an expression of the general tangential action or skin-resistance. Such eddies may, however, be considered as virtually a part of the ship so far as the trans- mission of pressure between it and the stream-line system is concerned. The actual stream-line system involved is there- fore that which envelops the eddies. If the two ships are geometrically similar, we may evidently assume with safety that the existence of these eddies will not essentially disturb the geometrical similarity of the enveloping systems of stream- lines. Hence while we are not here authorized to apply the law of comparison to eddy and skin resistance, it appears that the existence of the eddies as the expression of these two forms of resistance should not interfere essentially with the application of the law to the systems of stream-lines actually formed. The breaking of a wave involves a discon- 140 RESISTANCE AND PROPULSION OF SHIPS. tinuity of stream-line flow, and in such case we are not strictly entitled to extend the law to the stream line or wave resistance. Within the limits prescribed by this condition, however, we may consider the law as satisfactorily estab- lished. We may also consider it as highly probable that even with breaking waves' the general configurations in the two systems will, at corresponding speeds, still be essentially similar, and that the resistances involved will still essentially follow the same law of comparison. It must be remembered, however, that while such may be the case, its verification will depend on experiment rather than on mathematical reasoning. It is possible to view the question of the law of compari- son from an entirely different standpoint, and thus to extend the scope of its possible application. It will be noted that the derivation above given establishes the law under certain limited conditions. It does not follow, however, that the same law may not be applicable under wider conditions. To this question we now turn our attention. Assume a series of ships all of similar geometrical form, but diverse in dimension. Assume that the resistances of such ships as a family at varying speeds may be represented by a general equation consisting of a series of terms all of the form Afi-^v". We shall then have R = 2At( 3 ~~ 2 \ n (5) In this equation / is some typical dimension, necessarily the same throughout the series, v is the speed, and A is one RESISTANCE. I 4 I of a series of constants depending on form characteristics, and hence constant throughout the series. Such a general equation might therefore consist of a series of terms, as for example R = Al^v + Bl*tf + Cl^v* + Dlv*, . . (6) in which n was given successively the values I, 2, 3, 4. It is by no means necessary that n should be integral, so that terms of the form /'-V- 2 , -F/'-V- 8 , etc., may occur. In fact, so far as we are at present concerned, n may have any value whole or fractional, and the number of terms may be indefi- nite. In regard to the limits for the values of these exponents we note that if n = o, if n = 6, The first would give a term independent of speed and dependent only on size, and varying directly as volume or weight. This seems hardly likely, and we may consider n = o as the lower limit of values of n. The upper limit n = 6 gives a term independent of dimension and depending wholly on speed. As indicated elsewhere, it has been thought that such a term is probable, or at least possible. In any event, however, it is evidently the upper limit for n unless we admit the possibility of a term involving a decrease of resistance with increase of size, which seems quite unlikely. !42 RESISTANCE AND PROPULSION OF SHIPS. Hence n may presumably have any value between o and 6 r whole or fractional. Taking therefore (6) simply as an illustration of the kind of terms to be found in such an equation, we note again that A, , C, etc., are form constants, and remain unchanged throughout the particular family of ships. We assume then that it may be possible to express the resistance of any ship of this family at any speed by substituting in the appropriate equation using the form constants A, J3, C, etc., and the given values of / and v. For any other type of form there will naturally be another series of form constants with their series of values of exponents for / and v, not necessarily the same as in the equation for the first type. We see also that for the same ship at varying speeds this equation assumes the possibility of representing the resist- ance as a summation of terms involving the variable v with various exponents, whole or fractional; while for different ships at the same speed we should have the resistance expressed as a sum of terms invoking the variable / with another set of exponents, whole or fractional. A little thought will show that these assumptions are more elastic than those involved in the derivation of the law from the equations of hydrodynamics. They admit a wide range of variation, and an indefinite number of terms in the expression for R, binding the exponents of / and v only to the one condition that the exponent of / plus one half that of v shall equal 3. Taking (5) therefore as the symbolic equation let us compare by its means the resistances of two ships similar in form with a linear-dimension ratio A, and at RESISTANCE. 143 speeds in the ratio |/A. We have then, using subscripts I and 2, or Hence for any resistance which follows the law as ex- pressed in (5) the law of comparison as above stated will hold. Furthermore, the elasticity of equation (5) makes it quite possible or even likely that it may properly represent a larger portion of the total resistance than that due directly to the stream-line or wave-formation, and hence that the law of comparison may be applicable to a somewhat greater part of the total resistance than that to which the first mode of derivation properly makes it. Eddy-resistance, for example, is usually considered as varying with some area, and as the square of the speed, 5, or as represented by a term Bl*v*. This is seen to fall into place as one of the general series of terms in (5). Next as to skin-resistance. This is represented by a term of the form fAv n , 7, where f is a coefficient which may decrease with length, A is area, and n is an exponent usually less than 2. This term is seen to violate the condi- tions for members of (5), and the amount by which n is less than 2 and the amount of decrease of /with increase of /and ,ience of A, furnish together a general index of the amount by which skin-resistance is not amenable to the law of com- parison. With smooth bottom and the values derived from 144 , RESISTANCE AND PROPULSION OF SHIPS, Froude's experiments, the difference between the values of the resistance as computed and as derived by comparison may be considerable. To take an illustrative case, suppose / 3 -T- /, = 4, and hence v 9 -r- v l 2. Also, let the values of /be .01 and .0093, and the exponent n = 1.85. Then the ratio between the resistances by comparison will be 64, while by computation it will be A J = i6x .93X2^ = 53.64. The ratio of these is .84, showing that by formula the value is 16 per cent less than by comparison. If the skin- resistance were about one half the total, this would involve a difference of about 8 per cent of the total resistance. In many cases, however, especially with rough bottom, the value of the exponent is nearly or quite 2, and the decrease in f is very small. In such cases this term would likewise fall practically under the general law with the others. The question of the form of the terms expressing wave- resistance has been discussed in 14. If we take the two- terms A/v* and Bv*, we find that each comes into place as one of the terms of (5), and hence both fulfil the law of com- parison. We will now give a series of equivalent expressions for the velocity and resistance ratios for two similar ships at corresponding speeds. We have _ R, ~ \ij - \A ~ - A > f^ 1 _ f^ 1 &Y _ (A y f^.) 1 _ (.} ~z,v "vy V w V W' ' ' ' RESISTANCE. 145 Any and all of these, and others which may be written similarly, are equivalent forms of the ratios involved in the law of comparison for resistance. The actual confidence felt in the application of this law, either to the total resistance or to the residual resistance, must be considered as resting rather on the result of experi- ment than on any abstract proof necessarily involving as- sumptions not perfectly fulfilled in the actual case. This agreement may also be considered as indicating the degree of closeness with which we may consider either the residual or total resistance as capable of expression by the sum of a series of terms as symbolized in (5). As direct experimental investigation relating to the law of comparison, we may mention the experiments referred to in greater detail in 30, in which in two separate cases the resistances of a model and of its corresponding ship when compared according to this law were found to be in very satisf acton' agreement, the greatest errors being within 3^ per cent. In this connection we may also note the experimental determinations made by Wm. and R. E. Froude on the wave-, configurations made by similar forms moving at correspond- ing speeds. The wave systems due to a madel of the Greyhound and to the boat herself at corresponding speeds were compared by Wm. Froude. Their similarity was found striking even to details of the bow wave system. It was also found for the model that up to a speed of about 1.6 feet per second the resistance-curve followed almost exactly the curve of skin- resistance as computed by formula. It was a matter of observation that up to this speed the model moved without 146 RESISTANCE AND PROPULSION OF SHIPS. iaising sensible waves. At higher speeds the total resistance increased somewhat rapidly over that due to skin-resistance alone, and seemed to follow the plainly visible augmentation in the size of the wave system. Mr. R. E. Froude also mapped with care the wave-configuration about two models, one four times the size of the other, the larger travelling at twice the speed of the smaller. The smaller represented a launch 83 feet long at 9 knots per hour, and the larger a ship 333 feet long at 18 knots per hour. The configurations thus resulting have been already referred to and are shown in Figs. 31 and 32, and their great similarity in position and relation to the ship is strikingly seen. In Fig. 32 is also shown the system for the smaller model at the same speed as the larger, so that we have here the comparison of two wave systems made by different similar ships at the same speed. The general similarity of distribution relative to the water is quite apparent. The first comparison indicates therefore that similar ships at corresponding speeds will produce similar wave-con- figurations relative each to the ship; while the second indicates that similar ships at the same speed will produce approximately similar wave-configurations relative to the water. This latter consideration may lead to the suggestion that the wave pattern for a given type of ship is in large measure independent of the size of the ship, and simply dependent on character of form and on speed. Hence the energy involved in the waves would be likewise dependent only on the same functions, and likewise the resistance. It is these considera- tions pushed to their full limit which gives Bv* as the form of the term for wave-resistance in 14. In one important RESISTANCE. 147 feature, however, the wave pattern cannot be independent of the size of the ship. This feature is the composite stern system which in general depends on the relative positions of the components due to the primary bow and stern waves, and this in turn depends on the relation between the wave- making length and the speed. Hence in part, at least, wave-making resistance must depend on the dimensions, and its proper expression may perhaps involve terms in both v* and v' t as already suggested in 14. 27. APPLICATION OF THE LAW OF COMPARISON. CASE I. Denote two similar ships by S l and S t . If the law of comparison is assumed to apply to the entire resist- ance, we have to form simple proportions as expressed by any of the various forms in 26 (7) and (8). The given in- formation must necessarily involve three terms: (1) A given ship 5, at a given speed; (2) The resistance R t at this speed ; (3) A similar ship 5 2 at a corresponding speed. The fourth term, R 9 , is then readily found from the pro- portions in 26 (8). Let Fig. 45 be a graphical representation of the resist- ance of a ship at varying speeds. We now wish to show that the same curve may also be considered as a diagram of resist- ance for all similar ships, the speed and resistance scales being suitably modified. To this end we see that any ordinate as AB gives the resistance R, for the speed v = OA. Suppose now we have a ship A times as large in linear dimension. Then at a speed \/\v the resistance will be KR^. Hence if the scales are so 148 RESISTANCE AND PROPULSION OF SHIPS. changed that OA to the new scale denotes ^/Az>, and AB denotes A 3 ^, the one will properly correspond to the other; and the same relation holding for other points, we shall have to the new scales a diagram of the resistance of the new ship o A v FIG. 45. at varying speeds. We note that to effect this change the linear amount representing unit speed must be divided by A*, and the linear amount representing unit of resistance must be divided by A 8 . A curve of this character, therefore, by the proper treatment of the scales will represent graphically for this type or family, the resistance of any ship at any speed, within the limits determined by the original data and by the modification of the scales. CASE II. Let OP lt Fig. 46, denote the curve of total resistance for a given ship S 1 at varying speeds. Let the skin-resistance be computed according to the methods of 8, and set off from OP 1 downward at the various speeds as CB at A. This will give a curve OQ as the graphical representa- tion of the residual resistance, while the intercept between OQ and OP, will show the value of the skin-resistance. The RESISTANCE 149 residual resistance as given by OQ is then treated directly by the law of comparison, either by proportion, or by a change of scales, as just explained. On the latter supposition OQ may also be considered as representing to appropriate scales FIG. 46. the residual resistance of the second ship S, at varying speeds. The residual resistance of 5, being thus found, the frictional resistance is computed as in 8 by an appropriate selection of the constants. The two being added the entire resistance is thus determined. For graphically representing the entire resistance of 5, we may proceed as follows: The values of the skin-resistance of 5 2 having been computed, find ordinates for its representation to a scale /I 3 times as large as that used for S^ Lay these off from OQ, using a speed scale j/A times as large as that for 5,. The result will be a curve OP t , representing to these changed scales the total resistance of 5,. If instead of the ship 5, we have a model, the principles and operations are exactly similar, and we may thus consider the above as a general explanation of the 150 RESISTANCE AND PROPULSION OF SHIPS. method of connecting the results of a model experiment with the ship which it represents. 28. GENERAL REMARKS ON THE THEORIES OF RESISTANCE. From the preceding sections treating on the resistance of ships, it is quite evident that pure theory is quite unable to furnish the indications necessary to the solution of any given problem. The true function of theory as at present existing must be considered as that of providing a means to a sys- tematic study of the phenomena attending the motion of a ship-formed body through the water, and of establishing the basis for an intelligent comparison of the results of one vessel with those of another, in order that experimental data may be made available for purposes of prediction and design. It may be here remarked that the so-called " form of least resistance " in its abstract sense has for the designer only an indirect interest. It is of course true for any one condition of displacement, character of surface, state of the liquid, and speed, that there must be some form of least resistance, or at least some form than which none can offer less resistance. Such form, however, will doubtless change with every change in the four fundamental characteristics above, and with the question of resistance must always be associated those of safety, strength, and carrying capacity, and often adaptation to special conditions. The purpose is not therefore in any given case to produce a form of least resistance as such, but rather a form which shall the most economically combine the several qualifications in the par- ticular proportion called for by the circumstances of the problem. RESISTANCE. !-! 29. ACTUAL FORMULA FOR RESISTANCE. Of the many formulae which have been proposed for the direct computation of resistance we shall mention but few. In practical application it is usually power rather than resist- ance which is computed, though with the necessary assump- tions, of course, the one may be derived from the other. In Chapter V the question of power will be discussed, and various additional formulae and methods will be given for its determination, from which, by inverse processes, the resist- ance might be determined if desired. We have first where D is displacement in tons; v is speed in knots: R is resistance in tons; a and k are two variable coefficients the coefficient of propulsion and the admiralty coefficient. These coefficients are explained in 46 and 62, and, as is readily seen, the formula is simply derived from the so-called admiralty formula for power, and except where used with the law of comparison, as explained in 62, its results are only to be considered as roughly approximate. Middendorf* has deduced from numerous experiments on screw-steamers the following formula, of which the first term expresses the residual and the second term the skin resist- ance: R =* Where the surface is quite smooth the exponent of v in the second term may be made 1.85 instead of 2. * Biisley, Die Schiffsmaschine, 1886, vol. II. p. 579. 152 RESISTANCE AND PROPULSION OF SHIPS. The units are pounds, feet, and knots. B is beam; Z, length ; M, area of midship section ; S, wetted surface ; while e is a coefficient to be taken from the following table, using the prismatic coefficient as argument: f .70 and under. . e 2 P .81 e 1.50 71 I.QQ .82 1.42 72 1. 08 .83 . 1.32 71 1. 06 .84., 1.18 7A I.Q3 .85., i. 06 7S 1. 80 .86 .QO 76 ' Z7 i.8<; .87 . 74 77 1.81 .88 .. . , 55 78 1.7^ .80.. .31 7Q I 60 .QO. . 02 .80.. . wy 1.62 Biisley considers that this formula may be expected to give good results except for very long, fine ships, in which case the values are somewhat too large. Taylor* suggests for speeds up to that for which v* in knots divided by the wave-making length is not more than about 1.2, the following: R (3) where the first term gives the skin resistance and the second the residual or wave-making. The units are pounds for R, tons for Z), and feet and knots for the other quantities. D is displace- ment; L, length; s, surface; /, the coefficient of skin-resistance as discussed in 8, 9; and b the block coefficient. * Transactions Society of Naval Architects and Marine Engineers, vol. n. p. 143. RESISTANCE. !^ Regarding the application of this formula, Johns states* that where b varies from .60 to .65, R is generally correct for a speed such that v 2 + L is about .85. Below this the results are generally about 7.5 per cent larger than experimental results, while above this they are somewhat smaller. When b varies from .50 to .55, R is generally correct where v 2 + L = i. Above this the calculated values are too small, and below they are gener- ally some 10 per cent too large. Reference may also be made at this point to diagrammatic representation of resistance relations by both Johns f and Taylor { by means of which the results of experimental investigation may be more accurately and satisfactorily shown than by formula. In this connection we may also mention Calvert's formula for the resistance of sections of propeller-blades or bodies of similar form moving at a slight obliquity to the face. Let v = speed in feet per second; B = breadth in feet; D = depth in feet; 8 = angle of inclination; m = an exponent depending on 6, as indicated by the following table : 6 m 5 20 io 35 20. 50 30 6o 40 69 m 5o u 77 60 84 70 90 80 96 90 i. oo We then have R=6V l * s B m Dsme (4) * Transactions Institute of Naval Architects, 1907, p. 181. f Ibid. % Marine Engineering, July, 1905. Transactions Institute of Naval Architects, vol. xxvui. p. 303. 154 RESISTANCE AND PROPULSION OF SHIPS. 30. EXPERIMENTAL METHODS OF DETERMINING THE RESIST- ANCE OF SHIP- FORMED BODIES. Experiments on resistance have been made in a few cases on actual vessels. For the most part, however, such experi- ments have been limited to models usually from 10 to 20 feet in length. Where the actual vessels have been used, they have been towed either from a boom or off the quarter of the towing ship, in order to bring them clear of the wave system formed by the latter. The tow-rope strain was then measured by a suitable dynamometer and the speed by specially constructed and carefully rated propeller logs. In this way tests were made by Wm. Froude * on the Greyhound at a time when tests of models were generally considered of little value. The comparison of the data thus derived with that given by a model of the same ship reduced in the ratio I : 16 showed a very satisfactory agreement throughout, and did much to establish confidence in model experiments, the value of which has come to be accepted by the leading maritime nations of the world. A later comparison of the resistance of an actual vessel with that of the model was made by Yarrow in 1883 f between a torpedo-boat and its model. The range of speeds covered by the comparison was up to 15 knots. Here also there was virtual agreement, the actual values being some 3 per cent greater than those determined by means of the model. * Naval Science, vol. in. p. 240. f Transactions Institute of Naval Architects, vol. 24, p. in. RESISTANCE. ^5 These and other results, as well as the general agreement (when special disturbing causes are eliminated) found between predictions based on model experiments and actual results, have all led to a high degree of confidence in the value and reliability of model experiments for the determination of the resistance to be expected in full-sized ships. The models may be made either of paraffine or wood. In the former case they are shaped on a special machine in which the block cast roughly to shape moves under rapidly-revolving cutters which are moved by a pantagraphic connection with a tracing-point, the latter being carried by the operator along the water-lines of the vessel whose form is to be reproduced. In this way successive water-lines are traced out at the proper suc- cessive heights on the surface of the future model, the cross- sections at the end of the operation resembling series of steps, the inner corners of which are on the surface desired. The outer corners are then worked off smoothly by hand-tools until the inner angle of the path cut by the tool barely remains. The surface is then smoothed and polished, and is ready for the trials. At the United States experimental tank at Washington the models are made of wood. A so-called " former " model is first made by sawing out sections from the body plan, these being clamped in place, covered by thin strips, and then smoothed down. A suitable block of wood, made by gluing up lifts of pine, is then placed, together with the " former " model, on a special cutting machine. This machine is provided with saws, cutters, sanders, etc., so connected by pantagraphic means with a guide which is moved over the surface of the " former " model that the form of the latter is reproduced in the block of wood. After completion as far as possible on the machine, the ends are 156 RESISTANCE AND PROPULSION OF SHIPS. hand finished and the model, after painting and varnishing, is ready for use. The tank at Washington consists of a canal or basin some 470 feet long, 42 feet wide, and 14 feet deep. A truck running on rails spans this canal and carries the necessary apparatus for holding the model and measuring the pull at the various speeds attained. The latter are measured by electrical contacts at known distances on the track, and also through the revolutions of the truck-wheels. All the elements of the entire determination, including seconds of time, dynamometer pulls, changes of trim and draft, are automatically registered on a drum driven by con- nection with the truck-wheels. The models are taken hold of in such way that they are free to assume whatever draft and trim may suit the conditions from moment to moment. If desired also, special measurements by photography and otherwise may be made of the wave systems produced. For investigating the effect of the propeller on resistance a special truck is provided carrying the propeller connected with driving-power in such way that it may be driven at any desired .number of revolutions per minute. The propeller thus driven and carried entirely independent of the model is then brought up into its proper position astern of the model, and the revolutions adjusted until the total thrust developed is equal to the modified resistance of the model. The con- ditions are then similar to those which would exist were the propeller by means of this thrust driving the model at the same speed. A comparison of the resistance thus found with its value without the propeller shows the increase due to the presence of the latter. With the same apparatus the performance of model propellers may be investigated, and thus most valuable infor- RESISTANCE. 157 rnation bearing on the design of full-sized propellers may be determined. To these matters we shall refer later in 49. 31. OBLIQUE RESISTANCE. Euler's Theory. Let the resistance for motion longi- tudinally ahead be denoted by R l A t v*, and that for motion transversely by R^ = AJJ*. Then if the actual motion of a vessel is with a velocity u in a direction making an angle 6 with the longitudinal, the components of the velocity lon- gitudinally and transversely will be u cos 6 and u sin 6. Euler then assumed that the corresponding components of the total resistance will be Arf cos a and Aj? sin 2 6, the resulting total resistance being cos 4 + ,4,'sm 4 0, . . . (i) and the tangent of its direction with the longitudinal A, sin' A, These equations can only be considered as giving a rough approximation to the actual values. They serve, however, to introduce the questions of principal interest which are the direction and point of application of this total force R. These have relation to stability of route when the vessel is moved obliquely as by a tow-line. In Fig. 47, for a certain direction OV and speed u, let TS be the direction of R, and hence ST the direction of the applied towing force. Let M be the location of the point of application of the resistance. The tow-line must be attached somewhere in the line ST. If the point of application is toward T from M, it is readily 158 RESISTANCE AND PROPULSION OF SHIPS. seen that the couple developed by any slight yaw or deviation from the given direction OF would give rise to a moment FIG. 47. tending to return the ship to this direction, and to the fulfil- ment of the conditions originally assumed. Under these RESISTANCE 159 circumstances, then, there will be stability of route. If, on the other hand, the tow-line were attached beyond M toward 5, the moment due to a yaw would tend to still further increase the divergence, and there would hence be instability of route, with a tendency on the part of the vessel to swing about and go with the port side forward. For each value of the direc- tion OV there will evidently be a point M, and the collection of these for all possible directions will give a locus, as shown in the figure. There is very little experimental data relating to the form and characteristics of this locus. Certain general conclusions, however, may be reached as follows: It is evident that the line ST must touch the locus at M. Also, since physically there will be but one such point for any given position of OV, it follows that ST can touch the locus but once, and hence will be tangent to it at M. Hence the locus will be the envelope of the series of lines ST. From the results of Joessel relative to the location of the centre of resistance for planes, 6, it seems reasonable to assume that the center for longitudinal motion will be at some point A, and that as the angle is varied it will somewhat rapidly fall aft on either side, thus giving a cusp at this point. Similarly there would be like cusps at B y the center for motion astern, and at C and D, the centers for motion nearly transversely, the exact angle depending on the amount of difference between the fore and after bodies of the ship. We then assume the remainder of the locus to be as indi- cated in the diagram. If therefore any point of attachment on the ship be taken for the tow-rope, the angle between the latter and the keel for stability of route will be determined by drawing a tangent from this point to the envelope. There will be two or four such tangents in general, according i6o RESISTANCE AND PROPULSION OF SHIPS. as the point is within or without the envelope. Observation will show, however, which one is applicable to any given case. This determines a, the angle between the keel and the tow-rope. In regard to 6, the inclination of the direction of motion, there seems physical reason to believe that the latter will always lie between the keel and the direction of the tow- rope. This does not agree with the indications of (2) for very small values of 6, but we may well doubt the exactness of VALUES OF ANGLE a FIG. 48. this equation at the limit where one of the component veloci- ties is very small. We may expect, therefore, that the rela- tion between a and 6 will be somewhat as indicated in Fig. 48, where values of (a 6) and a are plotted in rectan- gular coordinates. The value of (a 6) is (-(-) between O and A t or for a approximately between o and 90, and ( ) for a between A and B. For a =. o, 180 and some angle near 90, a and will coincide and (a 0) will be o, as indi- cated. If, therefore, the information expressed by these two RESISTANCE. !6i diagrams is at hand, any problem involving the relation between the point of attachment of a tow-rope, its direction for stability of route, and the direction of the motion, is readily solved. Unfortunately we have no such accurate knowledge for representative cases, and in lieu the problem must practically be solved by trial and- error. Guyou has determined by experiment for a launch that the point A lay about 1 1 per cent of the length forward of the center. The character of the two diagrams above will depend on the ship fundamentally, and in the second place on the fol- lowing conditions: (1) Trim. The greater the trim by the stern the greater the difference between the forward and after bodies, and the greater the difference between the branches AC and CB of Fig. 47. The farther aft as a whole, relative to the ship, also, will the envelope be located. (2) Speed. It is found by experiment that when v in- creases, a increases slightly. This indicates that the trans- verse component of resistance increases more rapidly than the longitudinal. The value of a is not therefore independent of speed, as in Ruler's equations. (3) Helm. The position of the rudder is capable of changing the whole character of the envelope of Fig. 47 by introducing an additional oblique force at the stern and by modifying the stream-line motion. It follows that within certain limits the relative and absolute values of 6 and a may be varied at pleasure by the appropriate use of the helm. The manoeuvring of ships when moored by single cable or in a bridle, or when towed by single line or bridle, are familiar illustrations of the practical application of these' principles. CHAPTER II. PROPULSION. 32. GENERAL STATEMENT OF THE PROBLEM. THE fundamental problem of propulsion is to find a thrust whereby we may overcome the resistance which the ship meets when moving through the water. Leaving aside pro- pulsion by sails, the water about the ship presents itself as the only medium by means of which this necessary thrust can be developed. Now water is a yielding medium, and the conditions are entirely different from those encountered in pushing a boat in shallow water by means of a pole, where a firm and unyield- ing hold is obtained on the bottom. With water or any such fluid medium the development of a thrust must depend fundamentally on the utilization of its inertia and viscid forces. We may view the production of a thrust from two standpoints: (a) We know that the production of a change of momen- tum requires the action of a force and the expenditure of energy, and that conversely the matter acted on will react on the agent producing the change of momentum, such reaction being in fact the resistance opposed to the change of momen- tum. If therefore we provide an agent attached to the ship which shall produce a change of momentum in matter of any iiind, such change of momentum being directed astern, or at 162 PROPULSION, 163 least having a sternward component, there will result on the agent a reaction having a forward component, such reaction being then available as a propulsive thrust. The fundamen- tal conditions necessary are, therefore, that a change of momentum must be produced in matter by an agent attached to the ship, and that such change must have a sternward component. In the usual case water is the matter acted on, and that in which the change of momentum is produced. A propulsive thrust would, however, be given by a gun firing projectiles over the stern or by a boy throwing apples or stones in the same direction, as well as by the means found more useful for actual propulsion. fc We now turn to the second method of viewing a thrust. () We have seen in Chapter I that no body can be moved through a liquid without experiencing a resistance. Let us then consider the pair of bodies made of the ship and a propeller of some kind, and let us attempt to produce rela- tive motion by so moving the latter relative to the former that its motion shall have a sternward component. Such motion will meet with a resistance which, as we know, is simply the resultant of the distributed system of pressures and tangential forces acting on the surface of the moving body. This resistance will react along the line of relative motion, and will therefore have a forward component, and may hence be made to yield a propulsive thrust. It results that the resistance of the ship and that of the propeller, both taken along the line of motion of the former, or what is the same thing, along the line of motion of the system as a whole, must be equal. We shall next examine geometrically the conditions attendant on this second mode of view. 164 RESISTANCE AND PROPULSION OF SHIPS. 33. ACTION OF A PROPULSIVE ELEMENT. Let 55, Fig. 49, represent the ship and C the element, the latter being simply any body so connected with the ship that its relative motion is along a line AB with a velocity v. The line AB is not necessarily horizontal, but may have any FIG. 49. direction in space, and is simply defined as making an angle B with the longitudinal HL. As a result of this motion of C let the ship move with a velocity u along LH. In the usual way we determine AF = w as the direction and amount of motion of C relative to the surrounding body of still water. This we lay off at CK, remembering that its direction in space will be determined by LH and AB. While the actual motion of C is thus along CJ , let its form be such that the resultant of the system of distributed forces acting on its surface is along some line in space CR, making an angle y with CJ. Denote the inclination of CR to AB by yw and to the longitudinal by A, and denote the value of this resultant by R. For simplicity, Fig. 49 is drawn as though all the lines were in the horizontal plane. As noted above, such is PROPULSION. 165 not necessarily the case, but with the angles A, /*, y, and defined as above, the statement of the configuration is entirely general. The resistance to the relative motion of C along AB will evidently be R cos #, and for the total work performed by the propelling power we shall have W = R cos fj. . v = Rv cos // ..... (i) For the available thrust we shall have the longitudinal component of R, T = R cos A = resistance of ship ; . (2) and for the work done on the ship, W, = R cos A . u = Ru cos A ..... (3) The resistance along the line of motion of C relative to still water is R cos y, and for the work thus expended we have IV 9 = R cos y . w = Rw cos y ..... (4) We must also have or v cos // n cos /\ -f- w cos y. ... (5) This last result may also be derived directly from the triangle AEF. Definition of Efficiency. In the remaining chapters of this work we shall have frequent occasion to use the term efficiency. In its general sense it is simply the ratio of returns to total investment; or of the useful result received, to the total expense necessary to its attainment. The fun- damental definition must be carefully kept in mind, for in i66 RESISTANCE AND PROPULSION OF SHIPS. some cases it is not easy to say what is the useful return, or what the total expense. In the present case there is no ambiguity, and we have for the efficiency u cos u cos W v cos /< u cos A -\- w cos y' (6) We may note that the three velocties v cos ju, u cos A, and w cos y are the projections of the velocities v, u, and w on the direction of total resistance CR, and that the efficiency is simply the ratio of the two projections of u and v along this line. As a particular case, let the foregoing equations relate to an element of the surface of C instead of the whole body, or otherwise we may consider the element as a small plane. This is represented in Fig. 50. In this diagram, as before, we FIG. 50. assume that the lines are not necessarily in one plane. The total force R may here be represented by its two components PROPULSION. 167 P and Q normal and tangential to the plane. Denote the inclination of P to the longitudinal by a and to the line AC by ft. Then the inclinations of Q to the same lines are go _j_ a and 90 ft, respectively. We have then for the thrust T = P cos a Q sin or, . . . . (7) and for the component of R along AC R cos ft = P cos ft + Q sin ft (8) Hence for the useful work W^ (P cos a Q sin a)u, .... (9) and for the total work W= (Pcos/3 + Q sin ft)v\ . . . . (10) (P cos a Q sin a)u whence e = 7-5 a , ^ . -^-. . . (u) (P cos ft -\- Q sin ft)v It Q = o, we have ?^ cos a u v cos ft ~ v cos y3 sec a' Suppose now the element supported on a smooth unyield- ing surface as indicated by UV. Then when it has moved relative to the ship the same distance AN as before, the ship will have moved to A, and the element to C^ In such case the only work expended is that on the ship. Hence e must equal I. Hence in (12) v t a, and ft remain the same, while // must equal v cos ft sec a. Or in other words, v cos ft sec ft is the speed which the ship would have if C were supported on the smooth unyielding surface. This is readily seen l68 RESISTANCE AND PROPULSION OF S H I PS. geometrically if the lines of Fig. 50 are considered in one plane ; otherwise an application of spherical trigonometry is necessary to a geometrical proof. The difference between the velocity v cos ft sec a and the actual velocity u is called the slip. This we denote by 5 and the ratio 5 H- v cos ft sec a. by s. Hence we have u = v cos ft sec a S, and =. - = i s (13) v cos ft sec a This is therefore the limiting value of the efficiency when the tangential forces Q are o. It will be noted that the results thus far developed are entirely independent of any supposition relating to P or Q or the laws according to which they vary; and further, that the limiting value of e in cq (TC) is purely a geometrical function. Application to Typical Cases. (a) Suppose the element a small plane normal to the direction of motion and moved in the direction of motion, and hence normal to itself. In this case all tangential forces on the element balance, and R is simply the difference of the two normal forces, one on the front and the other on the back. Referring to Fig. 49, we have, therefore, o as the value of the angles A, //, y, and 6. Hence T = R, and W Rv = TV; W l = Ru= Tu; w = v u = S' PROPULSION. 169 (b) Let the element be any solid symmetrical body, or at least of such form that the total resistance R is in the direc- tion of motion, and let the latter be longitudinal. Then in Fig. 49 we have o for the value of A, ju, y, and 0*. Hence the same equations apply as for (a). (c) Let the element be a plane revolving about a trans- verse axis attached to the boat as in a radial paddle-wheel. If we suppose the boat propelled by a wheel consisting of a large number of such elements, we know that in order to obtain the necessary reaction the elements must move back- ward somewhat relative to the water, and that if u be the speed of the boat and v the peripheral speed of the paddles, i' will be greater than u. If v were equal to u, it is readily seen that the element relative to the water would describe a cycloid, the motion being in fact exactly as though the boat were carried on wheels of a radius equal to that of the element, such wheels rolling without slip on a rigid plane horizontal foundation. In the actual case the motion will be as though the wheels were reduced in diameter in the ratio of u to v, and likewise rolled without slipping on such reduced diameter. As a result the element will now travel in a curtate trochoid, as illustrated in Fig. 51. In this figure the paddle is denoted by the heavy line, and the lower por- tion of the path of a point P is shown by the curve PQRS. The line of centers is at AB and the slip being taken at 20 per cent, the effective radius OO is 80 per cent of OR. The rolling circle will then touch the line CD, which therefore will contain the instantaneous centers of the path of the point P. The location of the center of the wheel for succes- sive angular positions of the element being as shown by the numbers along AB, the corresponding instantaneous centers I/O RESISTANCE AND PROPULSION OF SHIPS. PROPULSION. are located at points vertically underneath and similarly numbered on the line CD. The location of these centers serves to show in any position the direction of motion of the point P relative to still water, and hence approximately the like motion of the entire blade. Taking the blade as a whole, the various elements lying between the inner and outer edges will have varying values of r and hence of v, and hence will travel in different trochoidal curves, all deter- mined, however, by the conditions and in the manner above described. For this case in Fig. 50 we should have =*; Hence in (7) we have T=Pcos 6 Q sin ..... (15) The sign of sin 6 changes as the element passes through the vertical. Since, however, before reaching the vertical the element is inclined on one side, and after passing it the inclination is on the other side, it follows that Q itself also changes sign, and hence that the horizontal component of Q is always subtractive. Hence the value of T will always be correctly given by (15), considering 6 as always plus. We have also in (8) R cos /* = P. Hence W, = (P cos B - Q sin ff)u ; IV= Pv; _ (P cos e Q sin 0)u T 7 2 RESISTANCE AND PROPULSION OF SHIPS. (d) Let the element be a plane moving as on the end of an oar, but suppose the plane of the oar to remain always vertical. This case is readily seen to be similar to the radial paddle-wheel, and so far as the general equations go, those given above in (c) apply here as well. (e) Let the element be a plane always vertical and revolv- ing about an axis fixed to the boat, as in certain forms of feathering paddle-wheels ( 38). The path of the element relative to the water will be the same as before, but we shall have a = o; ft =6. Hence in equations (7)-(i i) T=P; R cos /* = Pcos 0+ Q sin 0; = (P cos (9+ Q sin 0)v\ ~ (Pcos 6>+ Qsin ff)v (/) Let the element be as in (c) and (e), but so connected that its plane shall always pass through the upper point of the circle which it traverses relative to the boat, as in certain forms of feathering paddle-wheel. (See Fig. 52.) In this case we have = 2' FIG. 52. PROPULSIOX. 173 Hence in equations (/)-(i i) T = P cos - Q sin -; e e R cos p = P cos - + Q sm - ; W,= 7, W-'= 7ez/ cos //; /) /) c/ " cos - Q sin - 2 2 and ^ = - ( P cos - + Q sin - ) v \ 2 ix/ 2 / Let us now compare the limiting efficiencies of these three styles 'of paddle-wheel, elements, assuming that u and v are the same in each case. The latter condition is necessary in order to make a comparison possible, and is perfectly allowable and natural, since it merely requires an adjustment of the amount of surface to the size or resistance of the ship. Remembering that our equations apply simply at a given instant, we have, for for (e), e = v cos e 9 for (/), The order of excellence, so far as efficiency goes, is seen to be (e), (/), (<:). For (/) the efficiency is constant or inde- pendent of 0, and therefore this is "the value for all parts of the revolution during which the element is acting, and there- 174 RESISTANCE AND PROPULSION OF SHIPS, fore the value for this element as a whole. For other like elements the value will vary inversely as v, and hence inversely as their radial distances from the axis of revolution. The resultant limiting efficiency is, of course, a mean of these elementary values. For (c) and (e), however, the case is different. The efficiency for an entire stroke will be the useful work divided by the gross work for the same period. Hence we should have, 2Pu cos for (<:), e = ~ SPu for (e), e = cos Now P being really a function of 6 and the velocities, it is evident that no simpler general expression can be obtained except by making special assumptions as to the form of P. It seems reasonable to believe, however, that under similar conditions the order of excellence with regard to both limit- ing and actual efficiency might be expected to remain the same as above. 34. DEFINITIONS RELATING TO SCREW PROPELLERS. If one line revolves about another perpendicular to it as an axis, and at the same time move its point of contact along this axis, then points in this revolving line will describe helical curves, and the line as a whole or any part of it will generate a helical surface. A screw propeller may for our present purpose be defined as consisting of one or more blades having on the rear or driving side an approximately helical surface, such blades PROPULSION. !75 being joined to a common boss or central portion through which they receive their motion of rotation in a transverse plane relative to the ship. Fig. 53 shows a four-bladed right-hand propeller: FIG. 53. A propeller is said to be right-hand or left-hand according as it turns with or against the hands of a watch when viewed from aft and driving the ship ahead. The face or driving-face of a blade is the rear face. It is that face which acts on the water, and which in return receives the excess of pressure which gives the driving thrust. The back of a blade is on the forward side. Care must be taken to avoid confusion in the use of these terms. The leading and following edges of a blade are respectively the forward and after edges. 176 RESISTANCE AND PROPULSION OF SHIPS. The diameter of a propeller is the diameter of the circle swept by the tips of the blades. The pitch is the axial distance between two successive convolutions of the helical surface. In an actual propeller the term should be understood as strictly referring to a small element of the driving-face only. With this understanding the/z'A:// may be defined as the longitudinal distance which the ship would be driven were such element to work on a smooth unyielding surface, as for example the corresponding helical surface of a fixed nut. Pitch is thus seen to be purely a geometrical function of the propeller. Its value may vary from point to point over the entire driving-face, or it may be constant. In the latter case the propeller is said to be of uniform pitch. If it increases as we go from the hub toward the outer circumference, the pitch is said to increase radially. If it is greater on the following than on the leading edge, the pitch is said to increase axially. The latter mode of variation is usually implied by the simple term increasing or expanding pitch. The pitch-ratio is the ratio of the pitch to the diameter, or -pitch-ratio = pitch -r- diameter. The area, developed area, or helicoidal area of a blade is the actual surface of the driving-face. For the propeller it is, of course, the sum of the areas of the blades. The projected area is, correspondingly, the area of the pro- jection on a transverse plane of one blade or of all the blades. The disk area is the area of the circle swept by the tips of the blades. The area-ratio is the ratio of the helicoidal or developed area to the disk area, or area-ratio = developed area -f- disk area. The boss or hub is the central body to which the blades are all united, and which in turn is attached to the shaft. PROPULSION. 177 35. PROPULSIVE ACTION OF THE ELEMENT OF A SCREW PROPELLER. It follows from the definition in the preceding section that a helicoidal path must lie in the surface of a cylinder, and that the pitch is equal to the distance between two successive convolutions of the helix measured along the same element of the cylinder. It is shown in geometry that if the surface of such a cylinder is developed, the helical path becomes a straight line, the diagonal of the rectangle representing the portion of the cylinder containing one convolution. Thus in Fig. 54, if the cylindrical surface be cut along the line AB, and the lower part rolled out and flattened, the helix AHB will become the diagonal AC of a rectangle ABCD, Fig. 55. In this rectangle AB denotes the pitch and AD the circum- ference of the circle of diameter A D, Fig. 54. The angle CAD, Fig. 55, is called the pitch-angle, and the ratio AB -f- AD, Fig. 54, is called the pitch-ratio. A radius AL moving so as to always rest on LL and AHB, Fig. 54, at the same time keeping perpendicular to FIG. 54- FI G 55- LL, will then describe a helical surface such as we are now concerned with. Let us consider a small element of this sur- 1 78 RESISTANCE AND PROPULSION OF SHIPS. face at the radius LA. If this element acted without slip, as already defined, then it is evident that for one revolution the ship would be driven ahead a distance AB and the element would travel in the helical path AHB. In the actual case due to the yielding of the water the longitudinal component of the motion will be AB^ instead of AB, while the circular component will remain the same. The element will hence actually traverse a helical path AH 1 B 1 instead of AHB. Let this be represented in the developed diagram, Fig. 55> by AE. Then DE-=AB^ the amount of longi- tudinal motion, and CE BB^ the amount of slip. The angle CAE is often termed the slip angle. Let diameter AD, Fig. 54, = d; radius AL " " = r; pitch A3 " " =/; pitch-ratio = c\ revolutions = N\ area of element = dA ; angle CAE, Fig. 55, = 0. Then, using the nomenclature of Fig. 50, we have 6 = 90 ; CAD = a\ PAD = ft. Hence a + ft = 90 ; and 'an "= p = 2 nr tan a ; P P c = ~j = = 7f tan a. d 2r PROPULSION.'^ jyg We will now apply the general equations of 33 to the element under consideration. We have for the thrust T= Pcos a Q sin a ........ (i) Also, R cos }JL = P sin a -\- Q cos at; ....... (2) W, = Tu = (P cos a Q sin d)u ; . . . . (3) W = Rv cos jJi = (P sin a -f- Q cos a)v ; . . (4) (P cos a Q sin a)u ~ (P sin a -f- Q cos a)v' **' For the limiting value when Q = o, as in previous cases, v sin a v tan a DC u cos a u DE . e = : -- = - = -7=^ ( I s). We have next to derive expressions for the forces P and Q considered as determined by the motion of the element. We may first note that these forces will vary directly with the density of the liquid. Inasmuch, however, as this factor enters equally into the resistance of the propeller and that of the ship, it is evident that so far as the relations of the two are concerned, the design of the propeller will be independent of the density. That is, the same propeller should drive the ship with equal efficiency in either fresh or salt water. In any event the density factor will be provided for by the con- stants used to express the values of these forces. The actual direction of motion of the element relative to still water is determined by the angles a and 0. Now let Fig. 55 represent to an appropriate scale the dis- tances moved per minute instead of per revolution. l8o RESISTANCE AND PROPULSION OF SHIPS. Then AD = 27rrN = v\ DC=pN = 27rrNtan or; DEpN(i s) 27rrNtan a (i s) = u\ AC= 2nrNVi + tan* a. Taking 5 (i) and 6 (5) as the general expression of P y we should have P = a dA AE sin 0. EK But sin = -r- and EK = CE cos a = spN cos or. Hence P=adAAEEK. Whence, substituting and reducing, we find P= 4*VWV sin a Vi + (i s)* tan 2 a a dA. Likewise from 7, taking the relative gliding, velocity of water and element as AC, and assuming a variation as the square of the speed, we have Q=f~AC' l dA = 4?rVW(i + tan 2 a)/^4. The coefficient f will depend on the length of the ele- ment, character of surface, and angle 0. Now let dT, dU, and dW denote for this element the thrust, the useful, and the total work, instead of T, W lt and W. PROPULSION Then from (i), (3), (4), we have dT= 4**r*N*dA(as sin a cos a Vi + (i -- s)* tan 2 a /sin a(i -j- tan 3 a)}; N\l s}dA(as sin' a Vi + (i - j) 1 tan 2 a f sin a tan <* (i -f- tan a a) ) sin' a Vi + (i - *)' tan a a + / cos a ( i + tan' at) ). Put sin 8 a Vi + (i sf tan 3 a = B\ sin a tan or (i -f- tan 3 a) = C\ cos (i + tan 3 a) = E. Whence C = E tan" a. Then the above equations become dT = n*fN*dA(asB cot a /C cot a) ; . . ' (6) :8^VA r3 (i - s)dA(asB - fC)\ ....(/) ^n^N^dA(asB-\-fE) (8) Whence ./?-/<: J sB ~ c e = ^ l -^^B+TE = (l - S} ^ (9) ~cff+* Now put 2r = jv/ where 7 is a variable denoting the loca- tion of the element in terms of the pitch. It is called the 182 RESISTANCE AND PROPULSION ?^ SHIPS. diameter ratio, and is evidently the reciprocal of the pitch- ratio c for the element in question. For the propeller as a whole the values of c and y will be, of course, those of the outer element. Where we wish to especially distinguish these limiting or outer values we shall denote them by c l and jv We then have dT = n*p*N*ffdA\^sB cot a - C cot a) ; . . (10) . . . . (11) (12) 36. PROPULSIVE ACTION OF THE ENTIRE PROPELLER. If the expressions in the preceding section for dT, dU, and dW could be integrated over the surface of the blade, we should have values for the entire T, U, and W. Assuming / constant,, these may be represented by T = f?p*N % y(asB cot a fC cot a)dA ; . . (i) U = n*p*N\i - s)fy*(asB-/C}dA ; .... (2) (3) Also, dividing (2) by (3), we have, similar to 35, (9), .. (4) PROPULSION. jg^ Of the variables involved in this integration we may remark that y, s, B, C, E, and dA are determined by the geometry of the propeller and by the slip. They are there- fore readily known. It is not the same, however, with a and f. While these are called constants, there is no reason for assuming that their values will be the same for different elements of the surface, or for different aggregations of such elements. In fact such indications as we have point toward a decided variation in value with different values of the total area, pitch-ratio, slip, thickness of blade, condition of sur- face, and location of element. In regard to the coefficient f it must be remembered that it here represents not merely a surface effect, but rather the resultant force in the direction of the plane of the driving- face. Due to the increase of thickness from the tips inward, we should expect a corresponding increase in the value of this coefficient. The coefficient for direct resistance is taken in the form a sin 0, and the supposition of constancy in the value of a throughout the integration virtually assumes that the normal resistance varies directly with the surface. This, however, can only be even approximately true up to a certain point. We have specified in 32 the two general methods of con- sidering the propulsive action of an element. Thus far we have made use of the second one only. In considering the variation in value of this coefficient a, we shall find it useful to here briefly consider the propulsive action of the element of the screw propeller viewed from the first standpoint. In 37 and 38 a more general discussion of the application of this method will be found. From this point of view the action of the element is ta 1 84 RESISTANCE AND PROPULSION OF SHIPS. accelerate in a sternward direction the water acted on. This, for the element, is a cylindrical shell of water of radius equal to that of the Ibcation of the element, and of thickness determined by its height. For the propeller it would be a cylindrical body of water of diameter sensibly equal to that of the propeller, and with a hollow core of diameter sensibly equal to that of the hub. As we shall show later, the maximum amount of acceleration depends on the slip and on the geometrical configuration of the element or of the propeller. If, now, we consider the element very narrow, as for example a part of a narrow lath-like blade, the amount of acceleration produced either by the element or by the blade as a whole will be very much less than this geometrical maximum above referred to. The thrust obtained will therefore be corre- spondingly less. As we increase the width of such a blade the factor a will at first be nearly constant for each element of increase, so that for a time the thrust will increase directly with the area. At length, however, as we approach a condi- tion where the maximum acceleration and maximum thrust are nearly attained, the increase in thrust for each additional element of area will be less and less, or in other words, the thrust will increase at a slower and slower rate relative to the increase of area. We shall therefore finally reach a width of blade such that no additional increase will add sensibly to the thrust. It is evident that as we thus increase the area from our very narrow initial blade the average value of a or the thrust received per unit area will steadily decrease, at first slowly and then more rapidly. It even seems possible that, under certain conditions, after having passed a certain point the increase of area might be attended by an actual decrease in total thrust due to the increasing confusion of the streams PROPULSION. jg 5 of water acted on. These points will be again referred to in 49. We wish now simply to indicate that this coefficient will probably vary with the total area in the general manner above described. Again, the actual values of due to the influence of the thickness of the leading edge may perhaps be quite different from the values determined geometrically, and the law of variation with the sine of the angle is, again, not quite exact. It is therefore evident that the coefficient a instead of being constant will probably vary for different locations and configurations of the element, for different values of the slip, and for different amounts of total area of blade. If therefore we wished to make direct use of these expressions for T, U, and JFas they stand, we should need a series of experimental values of these coefficients appropriate to the circumstances of the case in hand. There is, however, no such set of values available, so that, even if it were desirable, no direct use can at present be made of these expressions in the form given above. For most purposes, however, we are not so much concerned with the distribution of the values of a and /as with mean values, or rather with values of the integrals of (i), (2), (3), (4). These integrals depend on the values of y\ and s, and on the area and shape of the blade. It is obvious that an experimental observation on a pro- peller of given form, proportion, and dimensions, and operated under a given set of conditions, will give for such propeller and conditions the values of work, thrust, etc., and hence the values of the integral expressions above. If, furthermore, such data covers a series of propellers all of the same diameter and form of blade, but of different values of pitch-ratio and area-ratio, l86 RESISTANCE AND PROPULSION OF SHIPS. and operated at different values of the slip ratio, then for such series we may find the values of the integral expressions in (i), (2), (3), and (4), and by interpolation extend such data to include all intermediate values of the three variables involved. The propriety of extending the use of data drawn from ex- periments on comparatively small or model propellers to those of full size will be considered in Chapter IV. We shall in the meantime here refer to the typical form of the relations between the various quantities involved, as drawn from a series of experi- ments conducted by the author and elsewhere reported.* The propellers used in these experiments were of the follow- ing general proportions and dimensions: Diameter of ah 1 propellers 12 inches Diameter of hub 2.4 ' ' Form of blade. elliptical Number of blades 4 Axis of blade at right angles to axis of propellers. Pitch-ratios included: .9, i.i, 1.3, 1.5, 1.7, 1.9, 2.1. The values of the pitch were determined by measurements on the driving-face. Maximum widths of blade as fraction of radius: .2, .3, .4, .5, .6, .7, .8 Resulting area ratios: .18, .27, .36, .45, 54, 63, .72 The general character of hub and blade and the relation of the elliptical contour to the hub are shown in Fig. 56. The form of cross-section of blade at the root is that of a circular segment, as shown in Fig. 57. * Transactions Society of Naval Architects and Marine Engineers, 1906; Publications of Carnegie Institute, No. 79. PROPULSION. I8 7 The maximum thickness varies with blade width in accord- ance with the following table : Blade Width, Area Ratio. Maximum Thickness. ,2 .18 . 20 inch -3 .27 .25 " 4 .36 -30 " -5 45 -35 " .6 -54 -40 " 7 .63 .45 " .8 .72 50 " As the full number of blade areas were made for each value of the pitch-ratio, the total number of propellers included in the experiments was 49. In the reduction of the observations it was found desirable to relate all results to standard conditions represented either FIG. 56. FIG. 57. by a speed of advance of 100 feet per minute, or a rotative speed of 100 revolutions per minute. In Figs. 58-66 are shown typical forms of various relations as indicated, and the character of which may with advantage be carefully studied. As shown in Figs. 58 and 62 the general form of the curves showing the relation between thrust and slip, or work and slip, when reduced to a standard of 100 revolutions per minute and l88 RESISTANCE AND PROPULSION OF SHIPS. Constant Pitch, Area and .X Revolutions * Slip FIG. 58. Constant Pitch, Area and Speed of Advance Slip FIG. 59. Area Ratio FIG. 60. Constant Area and Slip Pit/>h Ratio FIG. 61. Constant Pitch, Area and Revolutions Slip FIG. 62. Constant Pitch, Area and Speed of Advance Slip FIG. 63. PROPULSION. I8 9 with area and pitch-ratio constant, shows a nearly straight line slightly concave to the axis of slip. A special investigation with certain selected propellers showed that as the slip is in- creased the curves bend over toward the horizontal in a more and more pronounced manner, and at 100 per cent slip reach values only some 40 or 50 per cent in excess of those for 40 per cent slip. The slope of the lines, such as those in Figs. 58 and 62, increases with area-ratio and with pitch-ratio, according to some complex law which will be a function of these two variables. Constant Pitch and Slip Area P.atio FlG. 64. At values of the slip = o, the thrust is seen to be positive. This is due to the influence of the rounded back, and simply implies that for a blade with a rounded back the distribution of the stream lines is such as to give the same result as the face of an ideal thin blade inclined at a slightly greater angle. This is equivalent to saying that the effect of the rounded back is to increase the effective or virtual pitch of the propeller. See also 42 and 52. As shown in Figs. 59 and 63 the general form of the curve showing the relation between these same variables when re- duced to a standard of 100 feet speed of advance is distinctly parabolic, increasing beyond limit as the slip is increased. The 190 RESISTANCE AND PROPULSION OF SHIPS. reason for this difference in the character of the two sets of curves will be readily apparent on a little reflection. As shown in Figs. 60 and 64 the form of the curve showing the relation between thrust and area or work and area, with slip and pitch constant, shows a line concave to the axis of area and in the case of thrust becoming nearly or quite parallel to such axis for large values of the area. This is in line with the previous discussion regarding the value of a and shows that after reaching area-ratios of .50 to .60 but little additional thrust can be obtained by further increase of area. Increase in the Constant Area and Slip Pitch Ratio FIG. 65. value of the work is more continuous, as we might expect, and these two variations taken conjointly indicate a decreasing effi- ciency for excessive values of the area-ratio. The curves of Figs. 61 and 65 show the nature of the relation between thrust and pitch-ratio for the former, and work and pitch-ratio for the latter, each for constant values of the area-ratio and slip-ratio. The former curve is concave to the axis of pitch, showing a value of the thrust continuously increas- ing within the limits covered by the observations, but with a continuously decreasing rate of increase. The second curve is nearly straight, usually with a slight convexity to the axis of PROPULSION. pitch, though in some cases this feature is obscure and indica- tions of a double curvature are sometimes found. The general slope of the lines increases with slip and with area according to some complex law which is obviously a function of these two characteristics. The curve of Fig. 66 shows the typical form of the effi- ciency curve plotted on variable slip and for constant values of the pitch and area. Such curves show invariably a low value of the efficiency for a low value of the slip rising to a maximum in the neighborhood of 20 or 25 per cent, and then falling off .10 .20 .30 Blip FIG. 66. .40 more gradually for increasing values of the slip. There are general indications that a propeller of very small area-ratio reaches its efficiency at a relatively high slip, usually 25 per cent or over, while that of large area reaches its best value at about 20 per cent, or slightly below. The highest values of the efficiency seem, furthermore, to be only slightly dependent on either area or pitch-ratio, most of the propellers under test reaching at some value of the slip an efficiency close to 70 per cent, with in general the best results for fairly high values of the pitch-ratio and moderate areas. In general propellers of high pitch-ratio reach their best 192 RESISTANCE AND PROPULSION OF SHIPS. efficiency at a somewhat higher slip than those of low pitch- ratio. For low values of the slip, such as 10 to 15 per cent and low pitch-ratios, increase of area is accompanied first by increase of efficiency and then by some falling off, while for higher values of the pitch-ratio and low slip there is a general improvement of the efficiency up to some value at which it remains nearly stationary. For high values of the slip there is, in general, a continuous, though slow, decrease in efficiency with area through the range under investigation. Intermediate between these extremes we find for moderate values of the slip but slight change of efficiency with area. For propellers lying within the more common ranges of pitch- ratio and area and operating at values of the slip between 20 and 30 per cent, it appears that the efficiency will vary but slightly over the mid-field range thus indicated, and that any proportions corresponding to operation within these general limits may be freely chosen without fear of any important drop in efficiency below the limits of 68 or 70 per cent. The highest values of the efficiency realized were about .73. Other experimenters have found values slightly higher,* but it will be safe to assume from .70 to .73, or .75, as the highest general range which values of the efficiency are likely to show. It should be noted that comparisons of experimental results made with propellers of different forms of blade cross-section, or with different proportions and arrangement of thickness, indi- cate a surprisingly large influence due to thickness, especially in shifting the relation of thrust, work, and efficiency relative to * Transactions Society Naval Architects and Marine Engineers, 1905, 1906. PROPULSION. I9 3 slip-ratio. In the propellers upon which the above general con- clusions are more directly based, the blade thickness was dis- tributed as above stated, and it must be understood that the conclusions as a whole only apply to this particular series of model propellers. It should be stated, however, that along general lines these conclusions are in accord with a large amount of other experimental data derived from model propellers of different sizes, shapes, and forms of blade, and with different proportions and arrangements of blade thickness. 37. ACTION OF A SCREW PROPELLER VIEWED FROM THE .STANDPOINT OF THE WATER ACTED ON AND THE ACCELERATION IMPARTED. In the treatment of the screw propeller in 35, 36, we have preferred the method there followed to that which involves a consideration of the amount of water acted on and its acceleration, because the former seems more fruitful prac- tically, and to furnish a more valuable means of relating theory to practice than the latter. The latter method, how- ever, has chiefly attracted the attention of mathematicians and theoretical investigators, and it may be well to consider briefly some of the leading features of the theory from this point of view, especially as they are of general interest as well. The water acted on must necessarily travel in helical paths. That is, its velocity will have a longitudinal and a rotary component. The longitudinal component is that which directly yields the reaction from which comes the thrust. The rotary component as well as the longitudinal absorbs work from the propelling agent. Its existence also 194 RESISTANCE AND PROPULSION OF SHIPS. gives rise to centrifugal force, due to which the pressure is less in the interior of the column than at its outer boundaries. The water in coming aft into the zone of influence of the propeller gradually has its velocity increased, and a certain amount of acceleration is therefore received before the water reaches the propeller itself. Mr. R. E. Froude * has shown in the limiting case where no rotation is produced, and where the propeller blades are so narrow that little or no sensible acceleration can be received in the time taken to pass through the propeller itself, that one half the acceleration will be received forward of the propeller and one half in the race aft of the propeller. With rotation, however, we shall have in and aft of the propeller the defect of pressure in the interior of the column due to centrifugal force. In the actual case, therefore, somewhat more than one half of the acceleration will be pro- duced forward of the propeller, a certain amount while pass- ing through, and the remainder aft of the propeller. It is also evident that the higher the pitch-ratio the more the rotation, and vice versa. Hence with propellers of high pitch-ratio we may expect that most of the acceleration will be received forward of the propeller, while with low values of pitch-ratio the proportion there received may be but little more than one half the total. The column as a whole must, for continuity of flow, decrease in cross-section as it passes through the propeller, and to such distance aft as the velocity increases. If the amount of water acted on could be known and the total acceleration produced, the thrust could be immediately determined, and knowing the speed, the useful work would follow. The total work would be the sum of the useful, plus * Transactions Institute Navai Architect*, voi. xxx. p 390. PROPULSION. IQ5 the work represented by the kinetic energy imparted to the water. The latter representing the waste is seen to depend on the amount of water acted on and on the initial and final velocities. It is also clear that we may divide the energy spent in the wake into three parts, as follows: (i) that due to longitudinal change of velocity; (2) that due to circular change of velocity or rotation; (3) that due to turbulence and eddies. The portion (i) is of course necessary to the produc- tion of a useful thrust, while (2) and (3), though not funda- mentally necessary, cannot be entirely avoided. The energy involved in the race rotation has been made the subject of examination by R. E. Froude,* who shows by certain ideal limit- ing cases that its influence on efficiency will be practically negligi- ble except with large values of the slip and high pitch-ratios. If in a simple ideal case M denotes the amount of water acted on and v the longitudinal change of velocity, 42, (4), then the thrust will be proportional to Mv and the energy involved to Mv*. Now for a given value of Mv it is readily seen that Mv 1 will be a minimum when M is a maximum and v a minimum. That is, for a given thrust the energy due to longitudinal acceleration is least when the mass acted on is large and the acceleration is small. For high efficiency, there- fore, this consideration points to the use of large propellers acting at small slip rather than the reverse. This corresponds * n 35 to the result consequent on the assumption of the decrease or absence of tangential forces. Actually the exist- ence of race rotation, eddies and turbulence due to skin-resist- ance, etc., as well as structural considerations, limit the size and slip for the best efficiency as already shown in 36. Compare also 49. It is thus possible to discuss the action of the screw pro- * Transactions Institute of Naval Architects, vol. xxxni. p. 265. 196 RESISTANCE AND PROPULSION OF SHIPS. peller from the standpoint of the present section. In the actual case, however, none of the terms here involved can be exactly related to known quantities, so that, as in the other mode of treat- ment, we are thrown upon experiment for the ultimate control of whatever formulae may be proposed. We give, however, in 38, as an illustration, the treatment of the paddle-wheel from this point of view.* 38. THE PADDLE-WHEEL TREATED FROM THE STANDPOINT OF 37. The paddle-wheel may be investigated in the same general way as the screw propeller in 35, 36. The lesser relative importance of this instrument of propulsion does not, however, seem to justify the necessary work, and furthermore we have no such experimental data for the final control of our equations as with the screw propeller. We will therefore consider it briefly from the standpoint men- tioned in 32 and referred to more especially in 37. The thrust is measured by the sternward momentum of the water imparted by means of the paddle. Let z^ be the initial mean velocity of the water in f. s. ; v^ be the final mean velocity of the water in f. s. ; A be the cross-sectional area of the stream affected ; u be the velocity of the ship in f. s. ; * For the more important details of the various theories which have been developed along the lines here referred to, reference may be made to the following: " The Mechanical Principles of the Action of Propellers." By Prof. Rankine. Transactions Institute of Naval Architects, vol. vi. p. 13. " The Minimum Area of Blade in a Screw Propeller necessary to Form a Complete Column." By Prof. J. H. Cotterill. Ibid., vol. xx. p. 152. "The Part played in the Operation of Propulsion by Differences in Fluid Pressure." By R. E. Froude. Ibid., vol. xxx. p. 390. "The Theoretical Effect of the Race Rotation on Screw-Propeller Effi- ciency." By R. E. Froude. Ibid., vol. xxxin. p. 265. "Marine Propellers." By S. W. Barnaby. London and New York. PROPULSION. 197 D be the diameter of the circle described by the center of pressure of the paddles (this point may be taken as sensibly at the center of the paddle radially) ; N be the number of revolutions per minute. ' Then nDNA is the volume acted on per minute and for sea- water: 6^.^nDNA -f- 60 is the mass in pounds acted on per second. v t z> o is the change in velocity per second. Hence 64.4nDN'A(v l nDNA thrust = T = 60X32.3 = -*T (V ' - ^ The thrust equals the resistance, and hence, solving for A, we have A ~ In this expression the value of (^ z> ) is uncertain. Most writers, following Rankine, have considered that z/, v = slip of propeller or paddle-wheel = s? . This is known to be quite inexact, but we may use it for the present illustrative purpose. We have, therefore, 60 X : Or we may put more generally and more safely r> And likewise, area of one paddle ~ A. Hence if we denote the area of one paddle by #, we may put r> 198 RESISTANCE AND PROPULSION OF SHIPS. For purposes of design we may preferably put the equa- tion in another form, as follows: We have R~a(DNJs\ ...... (3) . Useful work or Ru ~ a(DN)*s(i s). . . (5) Also Ru ~ total work or I.H.P. Denoting this by /, we have I ~ a(jDN?s(i s), ..... (6) By comparison with paddle-wheels which have performed well, constants may be found thus relating the area of blade to known and to previously assumed quantities. For radial paddles the slip s may be taken from 20 to 30 per cent, and for feathering paddles from 15 to 20 per cent. We have therefore a = K (D'NJs(i - s) The value of K will depend on the units chosen for )', N f , and /. For numerical convenience we may take these as follows: /= I.H.P. absorbed by one wheel; D' = (diam. of wheel in feet) -r- 10 = D -f- IO; N r = (revolutions per minute) -~ 10 = N ~ 10. We may then take K from 2.0 to 2.5. This fixes the area of one float. The number of floats PROPULSION. 199 may be made about .SZ> for radial wheels, and from .^D to ./D for feathering wheels. The proportions of floats are usually Length = 3 to 4 times breadth. The greatest immersion of the upper edge at mean draft should be from .3 to .8 the breadth, according as the boat is to navigate smooth or rough water and to vary little or much in draft. If u is in knots per hour, instead of (4) we shall have _ itDN(i -s) _ nDN(\ - s) ~~ 6080 H- 60 : 101.3 ' ' ' ' W Hence DN = J^L^ ....... ( IO ) To illustrate these formulae take the following data: /= 1000 on both wheels and 500 on one wheel; u = 1 8 knots; s = .20 per cent, assumed. Th- and D'N' = 7.255. Hence, taking K = 2.4, 2.4 X 500 = (7.255)' X.i6 = I9 ' 6 ^ ft " r S ^ 2 "I" ft ' The floats might therefore be 8' X 2'. 5. We may now divide DN in any proportion suitable to the circumstances of the case. Thus if we take N ' 40, we have D = 1 8'. i ; or if we put D = 15', we have N = 48.4. 2OO RESISTANCE AND PROPULSION OF SHIPS. Kinematic Arrangement of Feathering Paddles In Figs. 67 and 68 are shown the usual kinematic arrangements for FIG. 68. feathering paddles. In Fig. 67 the links similar to AB are all pivoted at A. This link actuates the lever BC and blade PROPULSION. 201 DE as shown, and similarly for the others. In Fig. 68 HB is a drive-link actuated by an eccentric with center at A. The remainder of the links are pivoted to the eccentric strap as shown, and are connected to blades similar to DE, as in Fig. 67. In both diagrams the plane of the blade may be beyond the pivot C as shown, or it may contain C, or it may lie between B and C. The arrangement of Fig. 68 is more commonly used, that of Fig. 67 being occasionally employed for small wheels. 39. HYDRAULIC PROPULSION. In this mode of propulsion the water is drawn into pumps or other similarly acting propelling agents situated within the boat. It is then acted on by the pump-vanes and delivered with an increased velocity toward the stern. The necessary reaction is thus obtained and the boat is moved. This mode of propulsion from its slight use does not merit any extended consideration. We may, however, instructively glance briefly at certain features. Let M be the water acted on per second in pounds. For simplicity we may assume the water when first brought under the influence of the propelling agent to be at rest relative to the surrounding still water. Let v l be its velocity sternward relative to the same datum when delivered by the propelling apparatus. The change of velocity per second is then z/,. The change of momentum which will equal the thrust will be therefore 202 RESISTANCE AND PROPULSION OF SHIPS. The useful work is therefore M W. = Tu = v,u. g The total work will be Tu plus that involved in the accel- eration and disturbance of the water, and possibly in raising it against gravity. If for simplicity we neglect the loss due to eddies and disturbance and suppose the water discharged under the same average head as that under which it enters, we shall have for the total work Hence e = This is sometimes called Rankine's ideal efficiency, since it is the limiting value of the efficiency under the suppositions made. In the actual case the additional losses due to eddies, Mv? etc., will raise the waste work to - or more, and decrease g the efficiency to e = - - or less. u-\-v^ Now for anything approaching moderate or high speeds the necessary thrust will require .correspondingly large values of M and v lt or of both. The first means heavy machinery and the second low efficiency. In any practical case the value of v l is so large as to reduce the efficiency below that of a properly installed screw propeller or paddle-wheel. Ideal conditions for e would require small v lt and hence large PROPULSION. 203 M\ but these are impossible to realize in practice, and the efficiency is therefore necessarily small. For special purposes where moderate speed is sufficient, where simplicity is an important element, or where an ordi- nary propeller might be liable to race violently or become fouled, or where great manoeuvring power is required as in a steam life-boat, this mode of propulsion may have advantages which will justify its use. The manoeuvring power is usually obtained by multiple-discharge nozzles so situated and so adjustable that the reaction may be directed in any line, and the boat propelled in either direction or transversely, or turned in either direction as on a pivot. 40. SCREW TURBINES OR SCREW PROPELLERS WITH GUIDE-BLADES. Thornycroft's screw turbine may be taken as an illustra- tive example of this type of propelling agent. The propeller proper consists of a hub ABC, Fig. 69, with blades ADEB very much longer axially and shorter radially than the common propeller-blade. In consequence the inclination of DE to the axis is slight. Aft of this is a projection FGHKLMN fixed to the ship. This consists of a prolonga- tion of the boss by the spindle-formed body FGH, a cylin- drical shell or casing KLMN, and a series of blades connecting 204 RESISTANCE AND PROPULSION OF SHIPS. this casing to the spindle-formed body as indicated at KG and HN. These blades have a slight inclination in the opposite direction to that of the propeller-blades ADEB. The pitch of the latter blades is variable. The product of the revolutions by the pitch at is intended to be equal to the supposed velocity of feed, or the velocity of the stern of the boat through the wake unaffected by the action of the propeller. The product of the revolutions by the pitch at D is likewise intended to be equal to the speed of delivery, and the increase of size in ABC and consequent reduction in area of stream are intended to be such as to give for this variable velocity a steady flow from BE to DA. The water on leav- ing the blades of the propeller has a considerable rotary component, due to which it impinges on the fixed blades KG. By their guidance it is brought gradually to an axial direction, and thus finally delivered. The long projection FGH is for the purpose of providing for something approach- ing steady flow after leaving the guide-blades, and to insure the gradual union of the various streams with each other and with the outside water. From the character of the actual wake as explained in 41, it is quite certain that while in a general way the inten- tion of the design as to the distribution of pitch and sectional area of stream to suit the velocities of feed and discharge may be partly realized, yet such result must be far from exact, and it is doubtful if such variation of pitch can have much influ- ence on the actual result. Remembering the relation of efficiency to the character- istics of a propeller, it is evident that there is no reason for expecting any marked difference in this feature between this form and the common propeller. If, then, there is any in- PROPULSION. 205 crease in ultimate efficiency it will probably be found in the relation of the propelling agent to the ship. Either the augment of resistance due to the propeller must be less or the wake factor greater ( 44), or both. There seems to be no reason for assuming any essential difference in the wake factor, but it may be fairly expected that due to the pressure on the blades KG, the normal direction of which will have a component forward, there may be some return for the increase in hull-resistance due to the action of the propeller. On the other hand the additional attachment furnishes a con- siderable increase of resistance, and hence it is doubtful if the net resistance is much less than with a propeller of the usual form. We should therefore expect about the same ultimate efficiency. Such conclusio is are borne out by experience. The presence of the surrounding casing does, however, prevent the indraught of air and the radial escape of the water, and thus increases the resistance of the blades to revolution, and hence the thrust. It results that the requisite thrust may be obtained from a propeller of smaller diameter than when of the usual form, especially if the draft is small and the tips of the blades of a common propeller would come near the surface of the water. In such cases there seems to be a field of usefulness for propellers of this character and many of the Thornycroft type have been installed with very satisfactory results. CHAPTER III. REACTION BETWEEN SHIP AND PROPELLER. 41. THE CONSTITUTION OF THE WAKE. IN our treatment of propulsion we have to this point assumed the propelling agent to act in undisturbed water. In the actual case this is far from ^correct, and we have now to examine the effect of irregularly disturbed water on the preceding results. We will first note the cause and nature of the disturbance. Taking the screw propeller as the typical propelling agent, its place of action is at the stern in what is termed the wake, The water in this immediate neighborhood is subject to the following disturbing causes: (i) Stream-line Motion. In 2 we have seen that the relative motion between the water and the ship is less at the stern than between the ship and the outlying undisturbed water, so that relative to the latter the stream-line action will give a motion forward. Due to the same general cause, the direction of the relative motion of the water and ship is aft, with an upward and inward component as it follows the contour of the vessel. This brings its direction of flow oblique to the plane of the propeller Due to the difference in their location, this is more strongly marked with twin- screws than with a single screw. The influence of this obliquity of flow will be discussed at a later point. This part 206 REACTION BETWEEN SHIP AND PROPELLER. 207 of the wake is frequently referred to the wave-motion due to the system of waves generated by the motion of the ship. We prefer, however, to refer it to stream-line motion, as a more fundamental aspect of the cause. This naturally in- cludes all features of the wake contained in the freely-flowing water, or in that not involved in frictional and dead-water eddies. (2) Skin-resistance or Frictional Wake. The nature of the action between the skin of the ship and the water has been discussed in /, from which it appears that the whole surface is accompanied by a layer of turbulent water partaking more or less of the forward movement. This motion is greater as we approach the after end, and at the stern we shall have a considerable mass of water moving forward rela- tive to the surrounding body of still water. As to the distribution of the resulting complex wake, we have the following considerations: Near the surface the form of the ship is relatively full and the convergence of the stream-lines is correspondingly more marked. The disturbance of the normal horizontal distribu- tion of the water by the resulting wave-motion is also rela- tively more marked near the surface. As to the velocity of the frictional wake, there seems little reason to expect any marked variation with depth for points near the ship's sur- face. The velocity will, however, rapidly decrease at points successively farther and farther from the surface. As to the actual distribution of wake-velocity, an interest- ing series of experiments has been carried out bv G. A. Calvert.* Across the stern of a model 28 feet long and representing a full-bodied cargo steamer was fitted a frame * Institution of Naval Architects, vol. xxxiv. p. 61. 208 RESISTANCE AND PROPULSION OF SHIPS. upon which were stretched several vertical wires extending from the deck to some distance below the keel. These wires carried Pitot tubes fitted with vanes and free to swing to the direction of flow. An outrigger extending into undisturbed water carried a similar tube. The heights recorded in these tubes were then translated into velocities by the usual formula, v = V2gh. This furnished the means of determining the velocity of the ship relative to any point in the wake and relative to still water, and hence the velocity of the water at this point of the wake relative to still water. The wake-velocities thus found were expressed as percentages of the velocity relative to still water. In the section of the wake including the screw aperture, the boss alone of the screw being fitted, the percentages in a vertical line near the stern-post varied from 64 to 17 per cent from the surface downward; in a horizontal line near the surface, from 64 to 7 per cent from the stern- post outward; and in a horizontal line near the keel, from 17 to 10 per cent from the stern-post outward. The average amount at this section was about 19 per cent. An attempt was next made to determine the portion of this due to the frictional wake. To this end a thin plank of the same length as the model was towed at the same speeds. Similar measuring appliances being fitted, the wake speeds at points near the surface of the plank at distances of I, 7, 14, 21, and 28 feet from the forward end were found respectively 16, 37, 45, 48, and 50 per cent of the speed of the plank. It was also shown that approximately the wake-velocities decreased in a geometrical progression as the distances from the surface increased in arithmetical progression. From various considerations, for the details of which the REACTION BETWEEN SHIP AND PROPELLER. 2OQ original paper may be consulted, the author concludes that of the 19 percent average wake about 5 per cent was due to frictional wake, 9 per cent to wave-motion, or stream-line motion as we prefer to term it, and the remaining 5 per cent unaccounted for otherwise is charged to the influence of " dead water" or eddying water about the stern due to the comparative fullness of form. It thus appears that the wake-velocity in general is ex- ceedingly variable throughout the cross-section of the wake stream. Also from this and other experimental determina- tions to be referred to at a later point, it appears that includ- ing simply the water immediately astern of the ship, and hence that likely to be influenced by propellers, its average value may vary from 6 or 8 per cent to 20 or 25 per cent of the speed of the vessel. In general it is found that the wake value is greater as the ship is longer and fuller, and less as it is shorter and finer. For empirical equations relating its value to the characteristics of the ship see 50, (9) and (10). 42. DEFINITIONS OF DIFFERENT KINDS OF SLIP, OF MEAN SLIP, AND OF MEAN PITCH. It will be remembered that, unless otherwise stated, all velocities are referred to the surrounding body of still water as datum. We will first suppose the wake to be uniform. Let u = speed of ship; v = speed of wake if propeller were not acting and the ship were towed at the speed u\ v l = actual speed of water at or just forward of the propeller with latter in operation; V2 = final speed of water beyond propeller with latter in operation ; 210 RESISTANCE AND PROPULSION OF SHIPS. v= pitch X revolutions = pN = speed of advance of propeller through wake if there were no slip. Then u v^ = speed of advance of propeller relative to the wake; u v l = speed of advance of propeller relative to the water immediately about it. v u = apparent slip of propeller = S 1 ; . . . (i) v (u z/ ) = (v u^-^-v^ S,-\-v Q true slip of propeller S^\ or 5, -j- V Q = true slip of propeller = S y ; (2) v v^ = total slip of water = S 3 (3) Usually v 9 is directed forward and v^ aft, and hence 5 3 will be numerically the sum of v and v^. It should be es"ecially noted that these three kinds of slip are separate and distinct, though of course not independent. In particular it should be noted that S y and 5 3 are not the same. In the simple ideal case of Fig. 55 we may readily relate the value of 5 a to 5 3 or s^ as follows: It is shown in mechanics that the action of a surface in deflecting a stream omitting skin-resistance and the forma- tion of eddies involves simply a change in direction without change in velocity, and the total force interaction between the stream and the surface is measured in direction and amount by the resultant change in momentum. In that diagram the water is considered as approaching the blade with velocity represented in amount and direction by EA, and as leaving with the same velocity parallel to the blade. This is repre- sented in amount and direction by making KA = EA. Then by the usual composition of motions the resultant change is REACTION BETWEEN SHIP AND PROPELLER. 211 represented in direction and amount by EK. The longi- tudinal component of this is EG, and this is therefore the acceleration which is directly useful in providing thrust. Now if the slip angle CAE is small we may put EG EC cos 2 a s^pN cos 2 a = 5 2 cos 2 a. . (4) It thus appears that the velocity ,S 8 is variable for a given value of S 2 , increasing as a decreases, and therefore increas- ing from the hub outward, at all points, however, being less than 5 3 . Effect on the Preceding due to the Irregularity of the Wake. From (2) we have S, = $, + .; ^0 whence s, = s, + (5) Now remembering the actual wake as described in 41 it appears that ? -r- v may vary from perhaps o to 50 per cent or more. The value of s l is usually found between 10 and 20 per cent. Hence s^ may vary from say 5 or 10 per cent to 70 per cent or perhaps even more. It thus appears that the value of the true slip is variable over the surface of the blade and from point to point in the revolution between the very wide limits noted above, and these may not perhaps be the widest extremes. It is indeed quite possible that for certain elements the slip might be o or even negative, while for others its value might rise possibly to nearly 100 per cent. It is thus seen that the idea of slip for the propeller as a whole, acting in the wake, has lost entirely the simplicity of meaning which we were able to give to it when dealing with a single element in undisturbed water. 212 RESISTANCE AND PROPULSION OF SHIPS. In spite, however, of the wide range of extreme values which it seems possible that the slip may have, the conditions for most of the blade and for most of the revolution are such as to give rise to a much narrower range of fluctuation, and most of the work is undoubtedly done between a compara- tively narrow range of variation. We are thus led to the definition of mean slip. This admits of definition in various ways. (a) It might be taken as the arithmetical mean for an entire revolution, of the various slips of the elements compos- ing the driving-face. (b) Since these elementary values are quite unequal in relative importance according to their location and the value of the slip, it might seem more fair to give to each element a weight corresponding to the gross work absorbed by it. Thus if w lt w^ w^ etc., denote the amount of work absorbed by each element, and s lt J a , s 9 , etc., the corresponding values of the slip, then Wi*i + *S* + mean slip = - - (c) Instead of giving to each elementary slip a weight proportional to its gross work, we may take its thrust or use- ful work. Denoting the elementary thrusts by /,, /,, etc., we should then have t t s. + t^s. 2ts mean slip = = =. An approximate method of applying the weights thus indi- cated in (b) and (c) will be given below under the discussion of mean pitch. (d) In 35, (12), the total work of an element is expressed REACTION BETWEEN SHIP AND PROPELLER. 21 3 as a function of the geometrical form of the propeller and the slip. For the entire propeller with variable slip the total work will be represented by the summation of such elements. Suppose now that the slip instead of variable is constant at a value j, all other conditions remaining as before, and that the corresponding summation for total work W is the same as before. Then evidently s may be considered relative to the variable distribution of slip as a mean or equivalent value. This is the same as defining mean slip as the slip at which the same propeller in a uniform stream at the same number of revolutions would absorb the same total work as in the actual case. (e) Instead of total work as in ( (8) and (6), will show that for usual values of the pitch ratio this would be approximately realized by giving to the pitch of each element a weight represented in the first case by the product of its area by the cube of its radius, and in the second case by the product of the area by the square of the radius. The following actual case of a four-bladed model propeller may be given as an illustration of the determination of the reduced mean pitch in the second manner. The mean pitch across the blades was taken at four radial distances nearly equal to .3, .5, .7, and .9 the radius, and the area was taken as proportional to the breadth. The work for one blade was then arranged as follows: Radius. r* Breadth. Pitch. r*b r*bp 1-77" 2.97 4.07 5-26 3-13 8.82 16.56 27.67 3-30" 3-60 3-30 2.16 16.13" 15.90 15.96 15-78 10-33 31-75 54-65 59-76 166.6 504-8 872.2 943-2 156.49 2486.8 15.89 The quotient of the sum of the column r*b into the sum r~bp gives for this blade the reduced mean pitch equal to 15.89". For the other blades the values similarly found REACTION BETWEEN SHIP AND PROPELLER. 215 were 15.45", 15.80", and 15. 54". This gives as the final mean for the entire propeller the value 15.67". A value of the mean pitch being thus found from either a simple or weighted mean, a definition of mean slip follows thus: Assume a propeller similar to the one given in all respects except as to pitch, which shall be uniform and of the mean value as above defined, and let this propeller working in a uniform stream at the same number of revolutions absorb the same amount of total work as the given propeller. Then the slip at which such conditions would be fulfilled would be a mean or equivalent slip for the given propeller. This is evi- dently similar to (d) above for the case of uniform pitch. We may also as in (e) take the useful work or thrust as the basis for a similar definition of mean slip. Instead of defining mean pitch on a purely geometrical basis, we may give it a dynamical definition as follows: Let the given propeller work in undisturbed water with given revolutions and speed. Let there be a propeller of uniform pitch t with the same diameter, area, and shape of blades, and let it work in undisturbed water at the same revolutions and speed. Then the pitch at which the latter propeller would have the same turning moment, or at which it would absorb the same work, as the first, may be consid- ered as the equivalent mean pitch of the former. Similarly the definition may be based on equivalent thrusts instead of equivalent turning moments. It may be remarked that the pitch thus determined would doubtless vary with the revolutions, and with the relation between revolutions and speed, so that it could not be con- sidered as a fixed dimension of the propeller. 2l6 RESISTANCE AND PROPULSION OF SHIPS. A dynamical definition of mean pitch having been thus taken, the definition of mean slip would most naturally follow according to (d) or (e) above. In all such cases where the mean pitch or slip of a pro- peller is based on its performance in comparison with that of a propeller of uniform pitch, the influence of the thickness or rounded back is virtually involved. To this we may briefly allude. Given a body of the cross-section of a propeller blade as in Fig. 70, moved relative to the water in the direction of the FIG. 70. face BA. Then the distribution of the stream-lines is such that relative to the pressure on AB there is an excess from A to C and a more or less pronounced defect between B and C. The result is in general represented by a pressure on the face AB having its center much nearer B than A, and hence tending to turn the blade about in the clockwise direc- tion as here viewed. If the direction KL is axial, there will result in such case a positive or forward thrust. That is, if the propeller be run in undisturbed water at such revolutions relative to the speed that the slip of the driving-face is o, there will in general result a slight positive or forward thrust REACTION BETWEEN SHIP AND PROPELLER. 21 J due to this distribution of the stream-line motion, and it is not until the slip of the driving-face is still further decreased and made slightly negative that the fore and aft components of the total surface forces exactly balance and give a zero thrust. It may also be noted that this zero thrust is actually the resultant of a component aft due to the tangential forces or those due to skin-friction and edge-resistance, and of a component forward due to the normal forces or those due to the stream-line pressures. If, therefore, the former could be reduced or eliminated the propeller would, under the condi- tions just assumed, still show a positive thrust, and it would require a still further increase of negative slip to reduce the longitudinal component of the normal pressures to zero. These results have been frequently noted experimentally, and have been made the subject of quantitative measurement in a series of experiments carried on by the author.* This is what is frequently referred to by the statement that the addition of thickness results in a virtual increase of pitch, because if the equivalent pitch be taken as (longi- tudinal speed for zero thrust) -f- (revolutions), the result will be greater than that derived by the measurement of the driv- ing-face, and the excess would be still greater could the tangential forces be eliminated. We have thus discussed various possible definitions of mean pitch and mean slip in order to show the variety of meaning which may be given to these terms, and the conse- quent inexactness of significance attending their use without some agreement as to the basis of definition. Our use of the equations of 36 will virtually assume the definition of mean slip as based on (e). That is, we shall assume, as will be * Carnegie Institution. Publication. Number 79. 2l8 RESISTANCE AA'D PROPULSION OF SHIPS. explained later, that for the actual variable slip may be sub- stituted some constant equivalent value for use in the equa- tions giving the value of the total useful work. In regard to propellers of variable pitch it may be noted that such design is usually intended to provide for some variable distribution of slip over the surface, assuming the propeller to work in a uniform stream. When, however, we remember the great variability of the stream or wake, it is quite evident that any attempt to secure any specified distri- bution would be entirely futile, and that in any given case the actual distribution will be quite different from that intended. Hence any effects resulting from a variable pitch will be quite accidental, and it is very doubtful if, in the present state of our knowledge, there is anything to be gained by introducing such a feature into our designs. These conclusions seem to be borne out by experience, for propellers of uniform pitch have shown themselves in practice to be equal in efficiency to those in which the pitch is variable according to various laws, We shall therefore pay no further attention to variable pitch as a feature of screw propellers, but shall in all cases assume them to be of uniform pitch over the entire driving-face. 43. INFLUENCE OF OBLIQUITY OF STREAM AND OF SHAFT ON THE ACTION OF A SCREW PROPELLER. In general the line of the shaft, the direction of the stream, and the direction of advance are all different. This is illus- trated in Fig. 71, where AC represents the direction and speed of advance of the ship relative to still water, BC the direction and speed of the stream relative to the same, OF the direction of the shaft, and OA the speed and direction of the element relative to the ship. Then by the composi- REACTION BETWEEN SHIP AND PROPELLER. 2 19 tion of relative motion 5 it follows that BA represents the direc- tion and speed of the stream relative to the ship, and BO that of the stream relative to the blade or element. Let ^ denote the velocity of the stream relative to still water as represented by BC, and denote the angle BCA by 77. Then, as in 33, we denote the angle between OG and the normal ON to the element by a. The inclination of z/ to the normal is then (a 77), and ^ cos (a tj) is the component of z/ in the direction of the normal, and hence the velocity in this direc- tion which would be impressed on the element by the stream of velocity v^ if there were no slip. The result of this in the direction of advance would be a velocity V Q cos (a rj) sec a. From 33 the longitudinal velocity without slip in still water would be v cos fi sec a. Hence denoting tlie total speed of advance if there were no slip by ', we shall have u' = v cos j3 sec a -\- v cos (a rf) sec a = (v cos (3 -\- v cos (a rf)} sec a. In the present case, with the shaft at an angle e with the fine of advance, the angle j3 for a given element is constant, 220 RESISTANCE AND PROPULSION OF SHIPS. while a varies. The exact values of a depend on the solution of a spherical triangle, as is readily seen, and these values will vary through a range 2e. In consequence, the values of the speed of advance without slip, u' ', will vary through a corre- sponding range. Denote the mean value of u' by / and the. range of its variation by 2Au' . Also, let 6 denote the angular location of the element reckoned from the position for which u' is maximum. Then it is found, if e and v are relatively small, that we have approximately u' = u 9 ' + Au' cos 6. The value of u' thus given is seen to vary from #/ + Au' through #/, #/ Au' , u to #/ + Au' for the successive quadrants of an entire revolution. The actual speed of advance being u, it follows that the difference u' u or the actual slip 5, and the percentage slip s may likewise be expressed in the same approximate form. Hence we shall have 5= S. + AScos 0; s = s + As cos 0. Denote the mean value of a by <*. This is seen to be the angle between the normal and the direction of the shaft. Then u 9 ' = (y cos ft -\- v cos (a rf) ) sec ), the speed of advance of the pro- peller through the water about the stern. Whence wu - i+w' From 42, (5), we have v w u w whence i s. (3) (4) 226 RESISTANCE AND PROPULSION OF SHIPS. Let us next consider the influence of the uniform wake on the expressions for useful work and efficiency in 36, (2) and (4). It is readily seen that the useful work must be obtained by multiplying the thrust by the speed of the ship relative to 'still water, and hence by (i s^)pN. In all other places, however, where slip enters into these expressions it will be the true slip s,. Hence we shall have in such case - fC}dA ; W = n*p*N*y\a$B + fE)dA . The ratio of these two will give an apparent efficiency. The true efficiency must, of course, be that for the propeller working at a slip of s 9 . The presence of the wake cannot change the efficiency of the propeller itself, while it may increase the amount of useful result L\. The increase in L\ must be credited to the wake rather than to the propeller. These points will be again referred to in 46. For these reasons we call the ratio of 7, to Wan apparent rather than a real propeller efficiency. This we may denote by e lt while the true efficiency, which we will denote by * a , will have the value as given in 36, (4). Hence we have (5) A6 -7' s and ^ a = er -' = -^ (6) 3 l T ^ C T I <7ffl V / REACTION BETWEEN SHIP AND PROPELLER. .22*] 45. AUGMENTATION OF RESISTANCE DUE TO ACTION OF PROPELLER. \Ye have thus far considered the influence of the ship, through the wake which it produces, on the action of the propeller. We now turn to the influence of the propeller on the resistance of the ship. \\ e have already in 37 considered the action of the pro- peller in producing in front of itself a defect of pressure and thus imparting a portion of the total acceleration produced, before the water acted on reaches the propeller itself. This defect of pressure is for the most part due to the centrifugal force consequent upon the rotation of the race by the pro- peller. The rotation will evidently be greater as the blades stand more nearly fore and aft, and hence as the pitch-ratio is higher. This defect of pressure will interfere seriously with the natural stream-line motion. The water instead of being able to close around the stern and form the natural stern wave will be more or less disturbed and drawn away to the pro- peller. The result of this is a diminution of the pressure which would naturally exist about the stern if the ship were towed at the same speed. The result of this is, of course, an increase in the amount of resistance to be overcome by the propelling agent. This increase when viewed from the standpoint of the resistance may be termed the augment or augmentation of resistance. Where paddle-wheels are used as the propelling agent, a like augmentation is experienced though it arises from some- what different causes. The action of the paddle-wheels at the sides and nearly amidships gives rise to a race of water moving faster relative to the ship than would be the case if 228 RESISTANCE AND PROPULSION OF SHIPS. the wheels were not there. The result is an increase in the skin-resistance. The wheels also undoubtedly exercise a disturbing influence on the natural stream-line motion, and they may thus introduce an additional element into the augmentation of resistance. The amount of augmentation due to paddle-wheels has not been as extensively determined experimentally as for screw propellers, but various experi- ments by Dr. Tidman, R. E. Froude, and Messrs. Denny indicate that it does not much differ in amount from that for a screw propeller under like conditions of speed. 46. ANALYSIS OF THE POWER NECESSARY FOR PROPULSION. Let W= the indicated horse-power; W f = the power absorbed by the friction of the engine and shafting, and by any attached pumps. Then W W f = W p power delivered to propeller. Let 7" = the necessary thrust, supposing the ship towed at the given speed u\ T the actual thrust = T -\- the amount of aug- mentation. Then T u is called the effective horse-power; and Tu is called *T* the thrust horse-power. The ratio ~ is called the propul- sive coefficient. The ratio ^-, which is much more definitely related to W t propulsive efficiency, has unfortunately received no special name. We shall here distinguish it as the coefficient //. As defined in 33, the apparent propeller efficiency is Tu REACTION BETWEEN SHIP AND PROPELLER. 229 while as in 44 the true propeller efficiency is T(u Q _ u v. _ e, e *~ W p * l u ~ i + w The efficiency e^ is the ultimate test of the performance of the propeller considered simply as a propeller. It is given a certain amount of,work W P . Working in undisturbed water at the same true slip, 5, = (v u -\- z/ ), it would develop the thrust Tj and deliver an amount of work T(v 5 a ) = 2\u v 9 ) as in the numerator of e t above. If, however, we consider the propeller simply as a means of getting the ship through the water, the useful work will be T 9 u, since this is the amount which would be required to tow the ship at the speed u. The ratio T 9 u -r- W p is therefore the final test of the value of the propeller as a means of actually propelling the ship. If the ship produced no wake and the propeller no augmentation of resistance, this ratio 7> -r- W p and the ratio e^ T(u v^ -r- W p would be the same. The relation between them, therefore, involves the mutual interaction of ship and propeller. The ratio express- ing this relation is hence termed the hull efficiency. We have therefore coefficient h Hull efficiency = true propeller efficiency ' or coefficient h = hull efficiency X true propeller efficiency. Hence TT 11 re T * u T ( u ~ *0 ^o u T, Hull efficiency == ^ - - -^- = y . ^^- - y (i + W ), and Coefficient h = $ ^ e, = ~\i + w)e t . ^ 230 RESISTANCE AND PROPULSION OF SHIPS. The hull efficiency is thus seen to be the product of two factors T -v- T and u -f- (it v ). The first is the ratio of the true to the augmented resistance, or the reciprocal of the coefficient of augmentation as above defined. This factor has been termed by R. E. Froude the " Thrust-deduction factor." We shall, however, prefer to consider it as the reciprocal of the coefficient of augmentation, T -i- T . The effect of this factor, as seen, is to decrease the hull efficiency in the ratio T -r- T. It therefore implies a loss in efficiency due to the augmentation of resistance. The other factor u -f- (u z' ) (i -f- /), as already defined in 44. The effect of this factor, as seen, is to increase the hull efficiency in the ratio (i -f- w). It therefore implies a gain in efficiency due to the location of the propeller in a forward wake instead of in undisturbed water. This gain in efficiency is seen to be due simply to the gain in the speed of advance. That is, if the propeller were working in undis- turbed water at the same revolutions and true slip S^ and consequent thrust T, the speed of advance would be (v 5 a ). In the actual case, due to the fact that the wake water is borne forward, so to speak, to meet the propeller, the same true slip 5 a is obtained with a speed of advance #. greater than (v ,) by the speed of the wake v . Hence the two speeds of advance are u and u z/ , and their ratio is (i -f- w), as defined. Of the two factors thus constituting hull efficiency, one is seen to be greater and the other less than I. Hence they tend mutually to offset each other, and the product or the hull efficiency will usually not vary widely from i. In fact many determinations by R. E. Froude indicate that the hull efficiency may usually be taken as I without sensible error, REACTION BETWEEN SHIP AND PROPELLER. 23! and that the causes which seem to increase one factor will correspondingly decrease the other, and vice versa. Hence the coefficient // may usually be taken as sensibly equal to the true propeller efficiency *. It should not, however, be for- gotten that the hull efficiency is by no means necessarily i, and that circumstances might arise in which such an assump- tion would involve a sensible error. This analysis of the total power and the various relation- ships involved may be illustrated by the diagram of Fig. 72. A C C, F E FIG. 72. The total power, revolutions, and thrust are supposed to remain constant. The subdivision is then shown both with and without wake. This is as follows: AE = I.H.P. = total power; AB = power absorbed by friction of engine and attached pumps; BE = power delivered to propeller = W p \ BC = power absorbed in the eddies, rotation, and sternward acceleration communicated to the water acted on, assuming the existence of a wake; BC l = power absorbed under the preceding head, assuming that there is no wake and that the propeller works in undisturbed water; CE thrust horse-power = 7*, assuming the existence of a wake ; C^E = thrust horse-power without wake = T(u z> ) = actual thrust T multiplied by the reduced speed (u i-j which would correspond to the assumed power, revolutions, and slip with no wake; 232 RESISTANCE AND PROPULSION OF SHIPS. FE power absorbed by the augmentation of resistance. This for convenience is assumed as the same with or without wake; CF = power absorbed by the true or towed resistance in the case with wake effective horse-power; C t F = power absorbed by the true or towed resistance in the case without wake.. Therefore with fixed power, revolutions, thrust, and re- sistance the wake would increase the speed from (u z/ ) to u or in the ratio (i -f- w), and hence the useful effect in the same ratio. The various ratios are also illustrated as follows: propulsive coefficient ; CF = coefficient h ; BE CE = apparent propeller efficiency ; Bh = true CE -^p = coefficient of augmentation ; Cr CE u u -=-= = = - - = (i + w) =. wake-return factor; C^E u l u ?/ CF = hull efficiency. It will be noted that the diagram of Fig. 72 is not appli- cable to the same ship with and without wake, for the power, revolutions, thrust, and resistance are supposed to remain constant, while the speed changes from (u v ) to u. The REACTION BETWEEN SHIP AND PROPELLER. 233 diagram is applicable, however, to the performance of the propeller with and without wake, under the conditions that revolutions, thrust, and resistance shall remain constant, and hence that the propeller at the two speeds (u v ) and u shall be opposed to the same resistance. This is usually the most useful mode of analysis, as it serves to connect the propeller in its actual surroundings with the same propeller in undis- turbed water, developing at the same revolutions and true slip the same thrust. We may, however, as in Fig. 73, illustrate the effect of the wake on the same ship at constant speed, resistance, and With F E Without A, B, C, F, E, FIG. 73. thrust of the propeller, but varying revolutions and power. In this diagram ABCFE denote for the ship with wake the same points as in Fig. 72, while A 1 1 C 1 F l t denote similar points for the same ship at the same speed and resistance, but without wake. The power represented by CE is of course equal to that represented by Cf.^. The propeller power BE and the total power AE are, however, less than the corresponding amounts B l E l and A^E^ This saving of power comes about as follows: As shown in 36, (i), thrust varies with revolutions and with slip. Now if there were no wake and the slip were S lt the necessary thrust would be developed by a certain number of revolutions. With the wake v a and the same speed u the slip becomes increased to S l + 7' 5,, and the revolutions necessary to develop the fixed thrust are correspondingly decreased. Now with con- stant thrust the turning moment or torque, and hence the 234 RESISTANCE AND PROPULSION OF SHIPS. mean effective pressure in the cylinders, will all remain very nearly constant. Hence the propeller horse-power W p and the indicated horse-power W will vary very nearly as the revolutions. Hence while the work CE will equal C^E^ the work BE and the total work AE will be less respectively than B^E^ and A 1 E 1 very nearly in the same ratio as that in which the revolutions are reduced. In this way, then, it is seen that the presence of the wake makes it possible to obtain the necessary thrust and propulsion of the ship with a lower number of revolutions and with a correspondingly decreased engine-power than in undisturbed water. The wake is due to the motion of the ship, and its kinetic energy is simply the energy which has been put into it by the ship in its movement through the water. The energy of the wake comes therefore from the engine, or more ultimately from the coal or fuel, as the source of energy. The reduc- tion of the engine-power and fuel-consumption, as above explained, may therefore be considered as a return from the wake, or as a reutilization of a small part of the energy which has been expended in its formation. We will now give a numerical illustration of the analysis illustrated in Fig. 72. Let s t = .16; s, = .27. This would mean that the propeller in undisturbed water with 27 per cent slip would give the same thrust as obtained in the actual case at 16 per cent apparent slip, the revolutions being the same in each case. Then from 44, I + l - 1 5 I ; w .151. REACTION BETWEEN SHIP AND PROPELLER. 235 Let^lE, the I. H. P., be denoted by i Let AB be 14 Then BE = W P . . . = .86 Let CE = Tu = .67 Then e l = apparent propeller efficiency. . = .779 And e. 2 true propeller efficiency = .677 Also C,E=CE-*r\-\-w = .582 Hence CC t = >o88 Let coefficient of augmentation be = 1 . 165 Then CF = E.H.P = .575 And FE = .095 Then coefficient // . . . = .669 Propulsive coefficient. = -575 Hull efficiency = The following table shows the results of certain experi- ments carried out on a Dutch tugboat."* During the trials the thrusts were measured at the thrust-block by a hydraulic dynamometer, while the values of the E.H.P. were deter- mined from model experiments. I.H.P. T.H.P. E.H.P. Rev. Speed. Principal Dimensions. 3I-03 50.56 80.24 132.35 170.83 230.58 260.32 19.76 33.16 53-22 89-43 118.85 161 .40 180.29 15-80 27.42 42-74 70.69 87-75 108.46 120.22 94 in 127. 148 160. 175 180. 6.97 8.07 9.02 10.07 10.47 10.84 II. 01 Length = 72 Beam = 14 Draft forward = 3 Draft aft = 7' 4^ Displacement = 69 tons 9 10" The ratio of T.H.P. to I.H.P. varies with increase of speed from .64 to .69, and that of E.H.P. to I.H.P. from * See Steamship for October, 1897. 236 RESISTANCE AND PROPULSION OF SHIPS. .54 to .46. The pitch of the propeller was 7.63 feet, and, 'as may be determined, the slip varied from about 1.5 to 19 per- cent. The increase of propulsive coefficient at low values of the speed and slip is somewhat abnormal, but whether due to errors in the data or to an increased value of the ship efficiency at these points does not appear. The series of experiments is a valuable one, and it may be hoped that the near future will give us many more of the same character. 47. INDICATED THRUST. This term, which frequently occurs in engineering litera- ture, is defined as follows: 33000 I.H.P. Indicated thrust in tons = - ^ . . (i) /7V2240 This is a thrust which would correspond to an absence of slip and a total efficiency of I. This is a wholly impossible set of conditions, but as it is considered convenient for the expression of certain relationships, we will show its connection with a much better known quantity. Let C be an engine constant defined by the equation L.P. cylinder area X 2 X stroke in feet zLA^ C = . (2) 33000 33000 V ' Then the mean effective pressure reduced to the L.P. cylin- der is the pressure defined by the equation I.H.P. 33000 I.H.P. (m.e. P .)= -ar^nxj?-- - (3 > Now comparing indicated thrust and reduced mean effec- tive pressure, it is readily seen that there is a constant ratio REACTION BETWEEN SHIP AND PROPELLER. 237 between them, and hence that the one is in constant pro- portion to the other. Hence we may remember that the indicated thrust is simply a quantity proportional to the reduced mean effective pressure. We will also show here another expression for the re- duced mean effective pressure. Let A lt A A, be the areas of the successive cylinders of a triple-expansion engine, A, being the value for the low-pressure cylinder. Let p iy / and /, be the successive actual mean effective pressures in the same cylinders. Then evidently , (4) The values in (3) and (4) are evidently the same. For multiple-expansion engines of any number of stages the same general method applies. For triple-expansion engines with initial pressures from 160 to 180 pounds absolute, or 145 to 165 by gauge, the reduced mean effective pressure is usually from 30 to 40 pounds per square inch. 48. NEGATIVE APPARENT SLIP. If there were no wake, the apparent slip s l and the true slip s t would be the same. With the development of the wake, however, the true slip remaining the same, the appa- rent slip decreases until some point is reached where an equilibrium of conditions is maintained. From 44 we have r, = *.(i + w = s t (i s,)w. Now if, as stated above, s, remains constant and w 238 RESISTANCE AND PROPULSION OF SHIPS. increases, si will continuously decrease. If w could reach a sufficient value, si would become o and then negative. The con- dition that SI=Q is w or s 2 = - . . . . (i) or The question of the possibility or otherwise or a o or negative value of si is then a question of the possibility or otherwise of a wake equal to or greater than this value s 2 +(is 2 ). When the slip is determined relative to the pitch of the driving- face of the propeller, it is not likely that such a wake can exist. With ideal blades consisting of a thin geometrical surface and acting in an ideal manner, it may be shown dynamically that such a value of the wake could not be formed. The distribu- tion of forces on a blade with thickness, however, may produce a considerable change in the effective pitch, as noted in 42 and 52, and it is entirely conceivable and practically demon- strable that with such a blade a zero or even negative true slip may exist relative to the pitch of the driving or after face of the blade. It is also clear since the apparent slip is less than the true, that if the propeller is large for its work, and hence calculated to realize the needed thrust with a moderate or small true slip, that the apparent slip may readily approach very small values and minor errors of observation might easily transform such a condition into an indication of negative apparent slip. Such a condition is not desirable or indicative of good pro- peller efficiency, since it means a low value of the true slip and REACTION BETWEEX SHIP AXD PROPELLER. 239 consequently a low efficiency, as shown by the type curve of Fig. 66. It is probable that most of the cases of negative apparent slip have arisen from errors of measurement, especially in the pitch. In particular might this be the case with propellers of variable pitch, in which the definition of mean pitch is neces- sarily arbitrary in character, according as we make it purely geometrical, or give it a dynamical basis by giving to the pitch of each element a weight proportional to the thrust which it develops or the work which it absorbs. Due to this necessarily arbitrary character of the definition of mean pitch with such propellers, the expressions mean pitch and apparent slip lose much of the significance which we are able to give to them with propellers of uniform pitch. See also 42. CHAPTER IV. PROPELLER DESIGN. 49. CONNECTION OF MODEL EXPERIMENTS WITH ACTUAL PROPELLERS. WE now take up the considerations relating to the appli- cation of experimental data derived from models to the design of full-sized propellers. We must remember that the actual data given by experi- mental research relate to certain model or small-sized propellers of known dimensions and geometrical characteristics and oper- ating at stated revolutions, slip, and speed of advance. If we may assume that the experiments are properly distributed in range, as regards variation in the various factors aside from mere size, then, as noted in Chapter II, we may by suitable methods of analysis derive results expressed in tabular or geo- metrical form which will represent the performance of all possible propellers of the one fixed diameter, and of varying values of the other characteristics and factors within the experimental range. Thus for illustration, if the experiments cover a series of model propellers all of the same diameter, general shape of blade, general law of- thickness and material of blade, but varying in pitch-ratio and area-ratio and operated at varying values of the slip ratio, then we may derive from such experiments a general 240 PROPELLER DESIGN. 241 law, whether expressed graphically or otherwise, which shall connect these various propellers, as well as all others of interme- diate values of pitch-ratio or area-ratio or operated at inter- mediate values of the slip. We have next to consider the justice of extending 'such results to propellers of other sizes, and more especially to full-sized propellers for actual ship pro- pulsion. The derivation of 36, (n), (12), shows that the whole question depends on the supposition that the forces P and Q vary as the area and as the square of the speed, or that the values of K, L, and K + L = e are independent of actual dimensions and are functions of geometrical proportion and of slip. The general factors upon which the performance of any propeller must depend are obviously the following: (1) Diameter as the distinctive or characteristic dimension determining size. (2) Pitch-ratio. (3) Developed area of blades or area-ratio taken in conjunc- tion with (i). (4) Number of blades. (5) Shape of blade or distribution of blade-area. (6) Thickness and the general schedule of its distribution. (7) Dimensions and proportions of hub. (8) Material employed. (9) Slip ratio at which operated. Turning now to 36, (n), it follows that the value of the function K would in the most general case depend conjointly on the various factors (2) -(9) inclusive, and that symbolically it might be displaced by a function of eight different variables, each one of which might be supposed to take care of the influence due to the corresponding characteristic of the propeller. Our 242 RESISTANCE AND PROPULSION OF SHIPS. present knowledge of the influences due to factors (4), (5), (6), (7), and (8) is, however, too meager and uncertain to admit of useful application in this manner, and we are therefore, aside from (i), reduced to three principal factors, (2), (3), (9), re- garding which sufficient is known to permit of use in a formula of this character. We therefore substitute for K the symbolic function klm and thus have for 36 (n): U = d 2 (pN)*klm, (i) where &= factor intended to account for influence due to slip-ratio; / = factor intended to account for influence due to pitch- ratio ; w=factor intended to account for influence due to area- ratio. These various factors are not independent, and in the form of a simple product, as in (i), they cannot each represent exclu- sively the influence of a single variable condition. The inter- relation of these factors will further depend somewhat on the form of analysis selected for the reduction of experimental results, as will appear below in connection with the numerical results given. This does not affect, however, the value of writing the formula in this manner, since the attention is thus directed to these factors as representing the principal conditions on which the performance will depend, and separate or combined judg- ments are invited regarding the influence which each may have. Returning now to the question of extending the results of model experiments to full-sized propellers, it is obvious that if we may assume efficiency independent of size and the factors klm likewise, then values of efficiency and of these factors, or of their joint product 7?, such as may be derived from model experiments, PROPELLER DESIGN. 243 may be immediately employed in equations such as (i) above for application to propellers of any dimensions. As the result of experience in such matters, it has come to be generally accepted that in extending efficiency values inde- pendent of absolute dimension, dependance may be placed on the general trend of change and on the values, especially in a relative sense, while absolutely some error may be involved. That is, we may depend on the experimental data to indicate the general conditions for good efficiency and how the efficiency will vary for given changes in the factors involved, while some error in absolute value may possibly or probably result from the exten- sion to full-sized propellers of efficiency values drawn from model experiments. Regarding the relation of the factors k, I, m to size, it must be admitted that experience is lacking as yet for any satisfactory determination of the problem. It is more than likely that the factor m should depend on size, but how much or according to what law remains for future determination. Such indications as are available point to some dependance though small in amount, and in the absence of more precise data it becomes necessary, in order to make use of model results, to neglect the influence of size on these factors and in effect, therefore, to assume equation (i) applicable to propellers of any size and with values of the factors as derived from an analysis of experiments on small-sized or model propellers. Certain further considerations relating to the factor m may with advantage be noted at this point. Suppose we start with very narrow, thin, elliptical blades of maximum width .our, or area-ratio h = .oi. These blades with given revolutions and slip will develop a certain thrust T and useful work U corresponding to certain average and 244 RESISTANCE AND PROPULSION OF SHIPS. distributed values of the coefficients a and /.* Now if the width and hence the area of these blades be doubled, the area- ratio being now .02, we shall undoubtedly find that the value of a will remain sensibly unchanged, and hence the total thrust will be doubled. If we continue thus to increase the width, and hence the area, we shall undoubtedly for a time find the same rate of increase in the thrust, indicating a sensibly constant value of a. If the path of the blade were rectilinear, this would undoubtedly hold true up to some increase of area beyond actual experience. With the screw propeller, however, the path is helicoidal ; and we readily see that as the width and area are increased the thrust cannot increase indefinitely, but must rather tend toward a limit. In other words, as the area is increased the thrust is increased at a slower and slower rate. This implies a gradual and continual decrease in the co- efficient a, while presumably the value of /remains nearly the same, or at least falls off much less rapidly than a. This de- crease in a is due to mutual interference in the streams acted on by the different blades. While therefore a decreases, f re- mains nearly the same, and the thrust per unit area falls off accordingly. In this way we shall finally reach a point where increase of area gives no increase of thrust. The value thus developed is therefore a maximum, and cannot be exceeded, no matter what the area. Indeed, certain indications seem to point toward a possible falling off of thrust with excessive increase of area. The ideal maximum thrust obtainable for given diameter, pitch, revolutions, and slip is that corresponding to the for- mation of what is termed a complete column ; that is, a col- * See 35 and 36. PROPELLER DESIGN. 245 umn of water in which each part has received the full acceler- ation which, with geometrically perfect action, the propeller is capable of imparting. The amount of this maximum thrust may be investigated as follows: Referring to Fig. 55, it is shown that the value of the longitudinal acceleration is given by EG EC cos 2 a = 5 2 cos 2 a = j,/7Vcos 2 a. Now for an element of the total column consisting of a shell of water of thickness dr, moving with this acceleration, the thrust will be dT = mass of water per second X acceleration EG. Hence dT = -7td . dr . velocity of feed X EG. \ & We consider the velocity of feed as represented by DG, or as the speed of advance DE -f- the acceleration EG. The value of EG above and found in 42 assumes the angle CAE small. On the same assumption we may also put CG = EC sin 2 a = 5, sin 2 a = s^Nsin* a. Hence DG = pN(\ - J 2 sin 2 a); and putting in general s for j,, we have dT = - nd . drsp*N* cos* a(i s sin* a). But tan a = p -f- n d, and d = 2r = py. Making these substitutions and reducing, we find 246 RESISTANCE AND PROPULSION OF SHIPS. Hence for the entire propeller the maximum ideal thrust will be given by the integration of this expression between the proper limits forj^. This is seen to be entirely independ- ent of the breadth of blade or area of its surface. The thrust thus found would be the maximum possible on the supposi- tions made, and would correspond to the perfect action of each element, and to the absence of skin-resistance. It is therefore greater than could be obtained in any actual case, no matter what the amount of surface. On the assumption that the value of a is constant, Cot- terill* has used the above expression for dT to find the breadth of blade necessary to form a complete column at various values of y. This was done by equating the thrust above to that in a form similar to 35, (6), omitting /and with an assumed value 1.7 for #, and solving for dA. Since, however, this value is not accurate for a, and since, more- over, it cannot remain constant but must fall off in marked degree with large increase of area, the results thus found can only be considered as rough approximations to the lower limits, beyond which we must go in order to approach the "complete column" condition. So far as these indications went, however, they showed that propellers of ordinary pro- portions do not form complete columns, and hence do not give the maximum thrust corresponding to their diameter, pitch, revolutions, and slip. This instead of being a fault is a decided advantage, for it is very sure that a propeller work- ing near the upper limit of the thrust would be less efficient than if the surface and thrust were less, all other conditions remaining the same. This arises from the fact that the last * Transactions Institute of Naval Architects, vol. xx. p. 152. PROPELLER DESIGN. 247 increments of thrust must be obtained by the addition of a disproportionate amount of surface with its accompanying skin-resistance. This will result in an increase of /relative to a, and in a corresponding loss in efficiency ( 36, (4)). Hence while such a propeller may give a large thrust for its diame- ter, pitch, slip, and revolutions, the proportion of useful to total work will be comparatively poor. Let us now return to the expression for the thrust in (i). By integrating the function of y between various values of the outer limit we obtain the values of the maximum ideal thrust for the entire propellers of corresponding pitch-ratios. A comparison of these with the values actually obtained by ex- periment under similar conditions of diameter, pitch, revolu- tions, and slips will be of interest. The integrations were effected by approximate methods, and the results are shown by CD, Fig. 74. These are plotted to represent the maximum ideal thrust in tons for propellers of 10 feet diameter at 100 revolutions and 20 per cent slip, and of various values of the pitch-ratio as given on the axis of abscissae. The curve AB in the same diagram gives similarly the actual values of the thrust as derived for the same condi- tions from experimental models, assuming geometrical simi- larity. This curve gives therefore the actual thrusts for pro- pellers the same as above, and of area-ratio ^ = .36. With other values of the slip the results were entirely similar, thus indicating still more clearly that the propeller of usual proportions is far from developing the maximum ideal thrust. The ratio of these two thrusts is shown by the curve EF. Returning now to the factors k, I, m, of the general equation (i), we note that the disposition of values among such factors is to some extent arbitrary, and in the following analysis we shall 248 RESISTANCE AND PROPULSION OF SHIPS. select some one propeller as an arbitrary standard for which we assume values of I and m=i.oo. The values of k for 10 I? 1.0 1.1 1.2 1.3 1.4 1.5 1.6 PITCH RATIO FIG. 74. 1.7 1.8 1.9 1.7 1.6 1.52 cc various values of the slip will then follow from experimental results. Then by appropriate methods of analysis, values of / and m for varying values of pitch-ratio and area may be found from suitable experimental results. PROPELLER DESIGN. 249 For the primary source of data for k, I, and m we shall use the results of the series of tests already referred to in 36. The shape of all the blades was elliptical, and as noted all had four blades, so that this series of experiments furnishes data for the determination of the k, I, and m factors only. We shall take arbitrarily as follows: / for pitch-ratio, i = i ; m for area-ratio, .4 = 1. Now in the practical application of this equation it is found convenient, in order to avoid the use of large numerical values, to use certain secondary units as follows: For N a unit of 10 revolutions per min.; d a unit of 10 feet; p a unit of 10 feet. Denote these respectively by NI, d\, p\ t and we have pi= p + io. Using these units and measuring U in horse-power, we find by applying this equation to the actual measured results for a propeller of pitch-ratio i and area-ratio .4, the series of values of k related to slip, as shown in Table I, 50. By suitable methods of analysis we then find values of / dependent on pitch-ratio and slip, as given in Table II, and of m dependent on area, pitch-ratio and slip, as given in Table III. Suitable treatment of the trial results gives also values of the efficiency for the model propellers, as in Table IV. Intermediate values may be determined by suitable interpolation. 250 RESISTANCE AND PROPULSION OF SHIPS. We may turn now to a brief consideration of the influence of shape and number of blades on the useful work developed by a propeller. The typical blade form being elliptical, it will be obvious that blades with wide tips or of a fan -shaped form with rounded corners should tend to show a higher average thrust per unit area or a higher thrust for the same a'rea than one of elliptical form. This excess is due to the higher velocity at which the outer elements move and to the lesser obliquity of their normal to the fore and aft direction. On the other hand, from the view point of the water laid hold of, it is apparent for an elementary strip of blade area lying at any given distance from the axis that there is only such a cylindrical ring of water to be laid hold of, and if the blade width was previously sufficient to give to such ring of water its full acceleration, no further advantage is to be gained by increase of width at this point. The influence of form can not therefore be properly considered inde- pendent of area, and while with narrow blades considerable advantage may be gained by increase at the tip, it must be ex- pected that such effect will decrease as the average width of blade is increased. Recent papers by Taylor * and Froude f give some data regarding this matter, but still further experimental investigation is needed. Regarding the difference between three blades and four blades for the same area, it may be said that with other elements of the problem equal the narrower blade within certain limits will show the higher thrust per unit area, and hence a four-bladed propeller should show higher thrust than a three-bladed of the same area. The difference, however, has usually been considered very small, * Transactions Society Naval Architects and Marine Engineers, 1906. f Transactions Institute of Naval Architects, 1908. PROPELLER DESIGN. 251 and in most cases of design, a given area has been considered of equal effect whether divided among three or four blades. Fur- thermore, from the view point of the water laid hold of, it will be evident that if the area is such that the three blades will give all the acceleration to be obtained under the geometrical conditions of pitch and slip, then four blades will give no more. Certain data relating to this point are given by Froude in the paper referred to above under the head of blade form. Influence due to the Thickness of Blades. Certain light has been thrown on this subject by experiments at the U. S. Naval Tank and reported by Taylor.* Various proportions of thick- ness and various forms of cross-section were tried with general results, as follows: For the usual form of section with flat face and rounded back, both thrust and work absorbed are increased up to a point beyond usual proportions, as the thickness is increased. Of these two rates of increase the latter is the greater, and thus the efficiency decreases with increase of thickness. The location of maximum efficiency relative to slip is also subject to change due to variation in thickness, but in manner too uncertain to admit of formulation into any definite law. Regarding influence of form of section, it was shown that the form of the following or rear half of the blade section seems to exercise a dominant influence over the results, and with equal thickness and similar rear halves, the performances of propellers with leading halves of various forms fell very close together. As regard efficiency, the modification shown in Fig. 81, 55 j g ave results superior to the more common form, especially at moderate or low values of the slip. * Transactions Society Naval Architects and Marine Engineers, 1906. 252 RESISTANCE AND PROPULSION OF SHIPS. In general, the influence due to thickness of section is sur- prisingly great. Thus to give one instance, doubling the thick- ness at 20 per cent slip produced an increase of 30 per cent in the thrust developed and of some 42 per cent in the power ab- sorbed. GENERAL REMARKS. We therefore consider equation (i), together with the tables relating to k, I, m, and e^ as our fundamental data for propeller design. As will appear more fully by illustrative examples, the method of design proposed does not take the place of judgment and experience. So far as the method itself is concerned, there may result an indefinite number of propellers, each more or less completely fulfilling the conditions imposed, and in the discrimination between these full scope may be found for the use of judgment and experience. It is intended rather as an aid to the intelligent use of experience, and to the systematic analysis of the results of trial data. 50. PROBLEMS OF PROPELLER DESIGN. Formula (i), 49, relates simply to the useful work of a propeller operating in undisturbed water. In the actual case the propeller operates in the forward wake. In Chapter III we have discussed this wake, and shown that we may substi- tute for the actual turbulent wake a uniform stream of velocity v giving a true slip S y = S t -f- v o , and a speed of advance of the propeller relative to the water in which it works, of PN S^ = PN 5, v = u v n . Now in order to connect the propeller in undisturbed water with the actual case, we assume that for a given propeller at given revolu- PROPELLER DESIGN. 253 tions working in the wake of velocity z/ and developing a thrust of T and a speed relative to the wake of (u z/ e ), the useful work and efficiency will be the same as if it were work- ing in undisturbed water at the same revolutions and true slip, and hence with an equal speed of advance (u z/ ). That is, referring to Fig. 75, we must consider the useful work of our formula as based on the true propeller efficiency and not the apparent, or on the speed of advance of the pro- peller through the water in which it works, and not relative to still water. Hence the useful work thus defined will be represented by C^E T(u v ), and not by CE = Tu. It is therefore the work Cf, which must be substituted in the formula, 49, (i). The following will therefore be the most natural order of procedure: The I.H.P. being given or AE, we assume AB. This gives us BE. We may then select or assume the true efficiency i 1 i 1 i A B c C, F E FIG. 75. at which we propose to work. Then BE multiplied by this efficiency will give C\E, the useful work to be substituted in our formula. Again, it may arise that instead of the I.H.P. we have the E.H.P., or assume the propulsive coefficient, and thus pass from AE to CF direct. Then CF multiplied by hull efficiency = CiE, the useful work, as before. As already noted, unless special information is available, the hull efficiency is usually taken as unity. 254 RESISTANCE AND PROPULSION OF SHIPS. We must next determine the value of pN. Let u = speed in knots. Then 101.3^ = speed in feet per minute. lOI.lU And P*f=T But from 44, (4), Hence pN=j- This determination calls for an estimate of the wake factor w. Unfortunately the information available as a basis for such estimate is somewhat meager. Fronde in his classic paper * on propellers gives a series of values for ships of various charac- teristics, but beyond this little information is available in the general literature of the subject. Adding to these a few other known values, Prof. McDermott f has connected them with the characteristics of the ships by empirical formulae, as follows: Let w denote wake factor; p prismatic or cylindrical coefficient; m midship-section coefficient; L " length in feet. Then w IP \ - .161 L* .6) for single-screw ships; I+W w (P \ = .13^ L i.i J for twin-screw ships. l+w * Transaction Institute of Naval Architects, 1886. f Transactions Society of Naval Architects and Marine Engineers, vol. p. 164. PROPELLER DESIGN. 255 These equations represent simply certain available data, and as this is relatively meager, care must be exercised in their use. Where no special information is otherwise obtainable, however, they will probably serve to give a fair approximation to the value desired. Further suggestions on this point, as well as on several others relating to the choice of values for the various quantities involved, will be given in the following section. We will now proceed to illustrate the application of our formulae and tables to actual problems of design. We first repeat in collected form, for convenience, the prin- cipal equations which we may have occasion to use: (i) (5) w Ip , \ ^-'^i- --6 j for single screw-ships; (7) w (p \ '=.131 L i . i ) for twin-screw ships ; . . (8) I+W The values of k, I, m, 62, and of 101.3^ may be taken from Tables I, II, III, IV, V, for the values of slip, pitch-ratio, area- ratio, and speed given, and by interpolation for intermediate values. 256 RESISTANCE AND PROPULSION OF SHIPS. TABLE I. VALUES OF FACTOR k. Slip Ratio. k Slip Ratio. k Slip Ratio. k .10 .110 .22 .168 -32 -195 .12 .121 .24 -175 -34 .199 .14 .132 .26 .181 -36 .202 .16 .142 .28 .186 -38 .204 .18 .151 -3 .191 .40 .205 .20 .l6o TABLE II. VALUES OF FACTOR /. PITCH-RATIO. Slip Ratio. .8 9 I .0 i . i i . 2 1-3 1.4 .10 i-35 -15 .00 . 5 7 i -7< >3 -675 -594 .12 -3 2 .14 .00 *75 -7' Ji .682 .610 .14 3 - J 3 .00 . *79 -7' j& .691 .621 .16 .29 -13 .00 *8 3 7 k .700 .630 .18 .28 .12 .00 386 7< )o .707 -637 .20 .28 .12 .00 . 38 9 7< >5 .714 .644 .22 .28 .12 .00 5 9 i 7< )9 .719 .649 .24 .28 .12 .00 *93 .8c )2 .724 -655 .26 .28 .12 .00 395 .8c )6 .727 .660 .28 .27 .12 .00 3 97 .8c >9 -73 1 .664 -3 .27 .12 .00 *99 S .81 [2 -734 .668 3 2 .27 .12 .00 m 900 .8 rOr -737 .670 34 .27 .12 .00 m 901 .8 :8 -739 -673 -36 .27 .12 .00 m 902 .8 [ 9 .741 -675 -38 .27 .12 .00 93 .8 20 .742 .676 .40 .26 .12 .00 903 .8 2O .742 .676 i Slip Ratio. 1.5 1.6 I -7 i . 8 1.9 2 .0 .10 .524 .465 .41 2 -3 58 . 3 2 9 - 2 93 .12 -539 .481 -43< 3 -3 *4 . 345 3i; .14 -55 -493 .44 [ -3 W . 359 -3 2 5 .16 .560 -503 -45 2 -4 28 . 369 -336 .18 -5 6 9 .512 .46 [ -4 i.7 B 379 -345 .20 -577 -521 . 4 6< ) -4 26 . 386 352 .22 .584 .528 -47< 5 -4 34 . 394 .360 .24 -590 .536 .48 \ -4 *l . 401 .366 .26 .596 -542 .49 i -4 *8 , 406 372 .28 .600 -547 .49 5 -4 54 412 -377 3 .605 -551 -5 i .4 58 . 417 .381 3 2 .609 555 -50 5 -4 52 . 421 -385 34 .612 .558 -50 ? 4 56 424 -389 -36 .614 .560 -5i 2 4 69 428 392 38 -615 .562 -5i *. 4 7i 43 394 .40 .616 .563 5* 5 4 72 432 .396 PROPELLER DESIGN. TABLE III. VALUES OF FACTOR m. PITCH RATIO .8. 257 Area-ratio. Slip Ratio. .20 .30 .40 .50 .60 .10 -73 .900 I. 00 1.05 1.07 . 20 .783 .927 I. 00 1.04 I- 06 3 .819 .942 I .00 1.03 1-04 .40 .840 -95 I. 00 1-02 I 02 PITCH-RATIO .9. Area-ratio. &. .20 30 .40 50 .60 .10 .660 .883 I.OO 1. 06 1.09 .20 .750 .914 I.OO 1.04 1.07 3 793 93 I.OO 1.03 1.05 .40 .820 944 I.OO 1. 02 1,03 PITCH-RATIO i.o. Slip Ratio. Area-ratio. Ratio. . 20 30 .40 .50 .60 .10 .620 .865 I.OO 1. 06 I.IO .20 .725 90S I.OO 1.05 1.07 3 .773 .927 I.OO 1.04 1. 06 .40 .802 .940 I.OO 1.03 1.05 PITCH-RATIO i.i. Sin Area-ratio. Ratio. .20 .30 .40 .50 .60 .10 595 .852 I.OO 1.07 1*12 .20 7i5 .895 I.OO 1.05 1. 08 '30 755 .919 I.OO 1.04 1,07 .40 .784 93 I.OO 1.04 1.06 258 RESISTANCE AND PROPULSION OF SHIPS. TABLE III. VALUES OF FACTOR m Continued. PITCH-RATIO i 2. Area-ratio. Slip Ratio. .20 30 .40 SO .60 .10 -575 .843 I.OO 1. 08 1-13 .20 .690 .890 I. 00 1. 06 1.09 30 743 .912 I.OO 1.05 1.07 .40 .770 .9 2 5 I.OO 1.04 1.07 PITCH-RATIO 1.3. Area-ratio. Ratio. .20 .3 .40 50 .60 .10 .560 .831 I.OO 1.09 1. 14 .20 .680 .880 I.OO 1. 06 I.IO 30 .732 905 I.OO 1.05 1. 08 .40 755 .919 I.OO 1.05 1. 08 PITCH-RATIO 1.4. Area-ratio. Ratio. .20 30 .40 50 .60 .10 552 .823 I.OO 1.09 1.14 .20 .674 .880 I.OO 1. 06 I.IO 3 .726 .903 I.OO 1. 06 1.09 .40 744 -9 J 3 I.OO 1.05 1. 08 PITCH-RATIO 1.5. Area-ratio. Ratio. .20 .30 .40 SO .60 .10 -549 .818 I.OO I.IO **$ .20 .670 .875 I.OO 1.07 I.IO 3 .720 .899 I.OO 1. 06 1.09 ,40 >736 .908 I.OO 1.05 1.09 PROPELLER DESIGN. 2 59 TABLE III. VALUES OF FACTOR m Continued PITCH-RATIO 1.6. Area-ratio. Ratio. .20 30 .40 .50 .60 .10 541 .813 I.OO I. id i. iS .20 .669 .873 I.OO 1.07 i. ii 3 .715 .898 I.OO i. 06 1.09 .40 73 .902 I.OO i. 06 1.09 PITCH-RATIO 1.7. Area-ratio. Slip Ratio. .20 30 .40 So .60 .10 540 .812 I.OO I.IO .*s .20 .668 .873 I.OO 1.07 I. II 3 ,712 .895 I.OO 1. 06 1.09 .40 .724 .901 I.OO 1. 06 1.09 PITCH-RATIO 1.8. Area-ratio. Ratio. .20 .30 .40 .50 .60 .10 .20 540 .669 .813 .872 I.OO I.OO I.IO 1.07 1.16 i. ii 3 .710 .892 I.OO 1. 06 I.IO .40 .719 .899 I.OO 1. 06 1.09 PITCH-RATIO 1.9. Area-ratio. Clip Ratio. .20 30 .40 50 .60 v .10 549 .815 I.OO I. II 1.16 .20 .670 -873 I.OO 1.07 i. ii 3 .710 .891 I.OO 1. 06 I.IO .40 .719 .899 I.OO 1. 06 1.09 PITCH-RATIO 2.0. Area-ratio. Slip Ratio. .20 30 .40 .50 .60 .10 .556 .819 I.OO I. II 1.16 .20 .674 .873 I.OO 1.07 I. 12 3 .711 .891 I.OO 1. 06 I.IO .40 .718 .899 I.OO 1. 06 1.09 360 RESISTANCE AND PROPULSION OF SHIPS. TABLE IV. VALUES OF EFFICIENCY. PITCH-RATIO .8. Area-ratio. Slip Ratio. .20 3 .4 .50 .60 .IO .58 .61 .63 63 .63 .20 .64 .65 .66 .66 .66 3 .62 .62 -63 .63 63 .40 .58 .58 .58 .58 .58 PITCH-RATIO .9. Slip Ratio. Area-ratio. .20 .30 .40 so .60 .10 .20 3 .40 ^64 .60 .61 - .67 .64 59 .64 .67 .64 -59 .64 -67 .64 -58 -63 .66 .64 .58 PITCH-RATIO i.o. Area-ratio. Slip Ratio. .20 30 .40 So .60 .10 -57 .61 .64 -64 -64 .20 .67 .68 .68 .68 -67 .30' .66 .66 -65 .64 -63 .40 .62 .61 .60 .58 .58 PITCH-RATIO i.i. Area-ratio. Ratio. .20 3 .40 .50 .60 .10 .56 .61 .64 .65 .65 .20 .69 .69 .69 .68 .68 3 .68 -67 .66 -65 .64 .40 .64 .62 .60 .59 58 PROPELLER DESIGN. 26l TABLE IV. VALUES OF EFFICIENCY Continued. PITCH-RATIO 1.2. Area-ratio. Slip Ratio. .20 -30 .40 50 .60 .10 0-6 .6l .64 65 .64 .20 .70 .70 .69 .69 .69 3 -7 .68 .68 .66 .65 .40 .65 -63 .62 .60 -59 PITCH-RATIO 1.3. Area-ratio. Ratio. .20 30 .40 50 .60 .10 55 .61 -64 .65 65 .20 -71 -7i .70 .70 .69 3 -71 .70 .69 .67 65 .40 .66 .65 .64 .61 .60 I PITCH-RATIO 1.4. Area-ratio. Slip Ratio. .20 30 .40 so 6 .10 -5 2 .60 -63 .64 .64 .20 -7 1 -73 -73 .72 .72 3 .72 -73 -72 -7i .70 .40 .68 .68 .67 -65 .65 PITCH-RATIO 1.5. Area-ratio. Slip Ratio. .20 .30 .40 50 .60 .10 -55 .60 .64 .65 .66 .20 -71 .72 71 .70 .70 3 .72 -71 .69 .68 .66 .40 .67 .65 .64 .62 .61 262 RESISTANCE AND PROPULSION OF SHIPS. TABLE IV. VALUES OF EFFICIENCY Continued. PITCH-RATIO 1.6. Area-ratio. Ratio. .20 30 .40 so .60 .10 -54 .60 .64 -65 .66 .20 .72 .72 .72 7 1 .70 3 73 .72 .70 .68 .68 .40 .68 .66 .64 -63 .62 PITCH-RATIO 1.7. Slin Area-ratio. Ratio. .20 .3 .40 5 .60 .10 -54 .60 .64 -65 .66 .20 .72 .72 .72 .71 .70 -3 .72 .72 69 .68 .40 .68 .67 -65 -6 4 -63 PITCH-RATIO 1.8. Qlj n Area-ratio. Ratio. .20 30 .40 so .60 .10 -53 .60 -64 -65 -65 .20 .72 -73 -73 .72 -71 3 -73 -73 -7 .69 .40 .69 .67 .66 .64 -63 PITCH-RATIO 1.9. Qij-n Area-ratio. Ratio. . .20 .30 .40 so .60 .10 -53 .60 -63 -65 -65 .20 .72 -73 -73 .72 .71 -3 .72 -73 .72 .70 7 .40 .69 .66 -65 .64 PITCH-RATIO 2.0. Qljn Area-ratio. Ratio. .20 30 40 so .60 .10 -5 2 .60 -63 .64 .64 .20 -71 -74 -73 .72 .71 3 .72 -73 .72 ,71 .70 .40 .68 .68 -67 .66 -65 PROPELLER DESIGN. 263 TABLE V. GIVING FEET PER MIN. CORRESPONDING TO KNOTS, OR VALUES OF IOI.3?*. u 101. 3. u ioi.3. 10 1013 25 2533 II IH5 26 2635 12 1216 27 2736 13 1317 28 2837 14 1419 29 2939 15 1520 30 3040 16 1621 31 3HI 17 1723 32 3243 18 1824 33 3344 19 1925 34 3445 20 2027 35 3547 21 2128 36 3648 22 2229 37 3749 23 2331 38 3851 24 2432 39 3952 EXAMPLES.* (a) As the first example let us take I.H.P. =4600; speed = u = 1 6 knots; friction power = 13 per cent = 598 H.P. (AB, Fig. 75). Hence propeller power = 4002 H.P. (BE, Fig. 75). Next from the data given suppose w found or estimated as .14. We will also select .26 as the proposed value of the true slip or s^ We then find from (6) I 5, = .844 and j, = .156. Hence and 16 X 101.3 pN = - Q - = 1922 ; .044 * In these examples we shall assume that the influence of number of blades as such may be neglected and that the influence of surface is sufficiently taken care of by the factor m. 204 RESISTANCE AND PROPULSION OF SHIPS. Mow supposing neither p nor N fixed, we may proceed as follows: We first select a pitch-ratio of, let us say, = 1.20. We will also assume in this problem that the blades are elliptical, with area-ratio .40, so that m = i. From the tables we then find = .181; / = .8o6; e 2 = .6S. Hence useful work # = .68X4002 = 2721 (CiE, Fig. 75). Then, substituting, we have 2721 = (i9.22)% 2 X.iSi X.8o6; whence di 2 = 2.62 t j- and d = i6.2i; ^ = I 9-45; N = 98.8. (b) With the same otherwise as in (a), let the area of blade be increased to area-ratio .48. Then from Table III we may take w = i.o4. This will produce a 4 per cent decrease in di 2 , and approximately a 2 per cent decrease in d. We shall have therefore ^ = 15.89; # = 19.07; TV =100.8. (;) Suppose with the same data as in (a), we fix both pitch nd diameter and propose to find the necessary amount of area. 1 hus let p = 24 feet and d = iS feet. Then with the same value .of pN we find AT = 8o.i. Also ^ = 1.33, and we have PROPELLER DESIGN. 26$ = .181, as before; 2 = -75; e 2 = .6() (for probable area-ratio); and 17 = 2761. Substituting and solving for the factor w, we find 2761 This may be realized with an elliptical blade of area-ratio about 35- (d) Given the same data as in (a), let us fix 2V at 130. Then with the same value of pN we have ^=1922-^130=14.8. Let us now assume for trial a diameter of 14 feet. Then c = 14.8 -7-14 = 1.06. and we have ft .181; / = .94; e 2 = .67 (minimum desired) ; and ^7 = 2682. Then solving for m as in (c), we have __ 2682 _ = W = 7iooXi.Q6X.i8iX.94~ This result is impracticable and shows that we must go back and modify the other conditions. We will therefore assume a true slip of .28 giving 1974; 2 66 RESISTANCE AND PROPULSION OF SHIPS. Then assume d = 14.50; then c = 1.05, and we have = .186; ^=95; 2 = .67, as before; and 7 = 2682. Then solving for m, we find which is easily realized by a slight increase in area-ratio above .40. The propellers thus far considered have been implicitly supposed to have 'four blades. To illustrate the application of the equations to propellers of two or three blades we may take the following: (e) Let LH.P.=5ooo; u = 20; JF P = . 87X5000 =4350; w=.o8; s 2 = .24. Then (i-,si)=.82i, 2027 and #^ = ^7 whence p\N\ = 24.69. Take, also, pitch-ratio = 1.4; number of blades = 3 ; area-ratio = .30. PROPELLER DESIGN. 26? Then we find = 175; ' = 655; ^=.89; ^ = .73; 17=3176. Then, substituting, we have 3*76 dl = 15051 X.I75X.655X.89" 2 - *' Whence we find d= 14.38; p = 20.13; N=I22.J. As an example of a two-bladed propeller we may take the following: *2 = -30. Then (i-si)=-77> and pN = g2i\ whence piNi = g.2i Take also pitch-ratio = i . i ; number of blades = 2, area-ratio = .20. 2 68 RESISTANCE AND PROPULSION OF SHIPS. Then we find = .191; / = .8 99 ; ^=.755; Then, substituting, we have whence ^ = 2.14; ^ = 2.35; (g) As a further illustration of special design let us take the case of a tugboat. If the design is based on the conditions of use we must naturally take the speed low and the slip high. Let I.H.P.= 3 oo; w = 7 ; ^ = .83X300 = 249; 1V = .I2', 5! = .40. Then i-5i=.6 and ^i7Vi = 10.59. Take also pitch-ratio = i . i ; number of blades = 4 ; area-ratio = .5 5. PROPELLER DESIGN. 269 Then we find = 205; .59; Then, substituting, we have ; 2 = _ 147 _ = fi 1 188 X. 205 X. 903X1.05 37 ' whence ^ = 7.98; TV =120.6. Propellers with Steam-turbines The design of a propeller to operate with a steam-turbine presents no new fundamental problems. The methods of treatment are the same as in any case of design where similar data are at hand. With turbine design, however, the I. H. P. is not one of the items of funda- mental data, but is rather inferred from the E.H.P. and the general conditions of operation. Most naturally, therefore, the E.H.P., represented by CF, Fig. 70, will be taken as the point of departure, and from which C\E directly follows on an assumption of hull efficiency. From this point the method of procedure will follow along lines similar to those noted above. The special feature of propellers used with turbines is the low pitch ratios necessitated by the comparatively high revolutions. Propellers with turbines have as a rule pitch ratios found between .7 and .9, and this low value should lead to caution with regard to the area-ratio employed. 270 RESISTANCE AND PROPULSION OF SHIPS. In general, experiment has shown that the best propeller efficiencies cannot be realized with extremely low values of the pitch-ratio, and furthermore that with such values of the pitch- ratio the efficiency is more dependent on area than with higher values. For this reason, there is danger of running into low values of the efficiency with increase in area beyond moderate proportions. While exact limits cannot be stated, it seems likely that having regard to efficiency the area ratio with such propellers should not exceed .40 to .45. On the other hand with the small propellers to which such proportions lead, there is likely to be trouble, especially at high speeds, with too little area and consequent cavitation see remarks on this subject in 52. In the face of these con- flicting tendencies, some compromise must be made, and this has often lead to the adoption of higher area-ratios than would be warranted by considerations of efficiency alone. Values of the constants in Tables I, II, III, are not extended below a value of pitch-ratio = .8. For carrying out design with still lower values of the pitch-ratio recourse may be had to methods by comparison, as referred to later, and also to the data derived by experiments on propellers of low pitch-ratio at the U. S. Naval Tank at Washington.* The usual operations involved in the preceding problems may be somewhat generalized as follows : Given I.H.P., speed u, friction, wake factor w y and true slip s 2 . Whence W P or BE, Fig, 75, si, pN, and k. Then (i) Given neither p nor N. Assume pitch-ratio c and area factor m. * Transactions Society of Naval Architects and Marine Engineers, 1905, 1906. PROPELLER DESIGN. 2 *Jl Thence find e 2 , U, and /. Thence by (i) find d. Thence p and N. (2) Given either p or N. Thence the other N or p. Assume d and the efficiency 62. Thence find c, U, and /. Thence by (i) find m and determine by judgment as to the practicability and consistency of the re- sulting value. (3) Given d. Assume c or p and 62. Thence find p or c and N, also U and I. Thence as in (2). By an inversion of the above processes these equations may be used for the analysis of a given set of trial data, and thus for the determination of the values of s 2 and w in the given case. The use of the equations for this purpose assumes them as applicable to the given propeller, and hence we should expect the results to be generally more reliable the more closely the actual propeller approaches to the standard form and pro- portions of blade. The operation is as follows: (h) Given I.H.P., p, N, d, and si, and assume engine-friction. (i) Thence find W P and pitch-ratio. ('2) Then assume a value of $2 and find resulting values of m, I, and e 2 . (3) From equation (i) find value of k. (4) By means of Table I compare the corresponding value of 52 with the value assumed in (2). (5) Correct the assumption of s 2 until by process as above a consistent set of results is obtained. The final result will thus determine s 2 and thence w by equation (5). 272 RESISTANCE AND PROPULSION OF SHIPS. Thus for example, taking the data and results of (a) except 52 and w, let us first assume $2 = .20. Then we have /= .795; w = i.oo; e 2 = .69. Then .69X4002 2.627X7IOOX-795 .i86. But .186 for k corresponds to s 2 =.2&, which shows that the first assumption was too low. The real value will lie between .20 and .28, and a second or third trial would locate it at .26, which is, of course, the correct value for this case, and from which w may be determined as in (a). As a further example we may take the following data from the U. S. cruiser Detroit: We have given as mean results for the two propellers: d= 1 1 ; /= is; pitch-ratio = 1 . 1 8 ; I.H.P. =2577; N = 1 70. 1 ; u = 18.71 ; s i = -I4J area-ratio = .305 ; Engine-friction is taken at 12 per cent. Hence ^-2270. Assume for trial s% = .25. Then/ = .822; ^ = .906; PROPELLER DESIGN. 273 Then 69X2270 !.2iXio8o8X.822X.9o6~ For k = .i6i, s z = .20 (see Table I). Hence correct value will lie near .21 or .22. Further trial will show that $2 = .22 will give consistent values for k. Hence we have, from equation (5), ^ = .077. The use of the equations in this manner in a large number of cases carried out under the direction of the author showed that it is somewhat difficult to satisfactorily determine the value of w from such analysis alone. This is not so much due to inaccuracy in the equations as to the fact that a slight change or error in the data will make a relatively much larger error in the value of w. This arises from the fact that the value of w is determined from the quantity (i -f- w). Now an error of moderate amount in the data or in the suit- ability of the equations to the data might give, for example, an error of 2 or 3 per cent in the value of (i + w )i making, for example, its value 1.12 when it should be i.io. This would make a difference of .02 in w t or an error of 20 per cent in excess of the true value. So far as ordinary applica- tions are concerned, however, and especially so far as the problem of design is involved, it is the factor (i -|- w) and not w which is used, and hence changes of relatively large percentage amount in w will have comparatively small influ- ence on the solution of such problems. It may be also men- tioned that a comparison of the values of w determined in this manner with those given by equations (9) and (10) and by general estimate showed in nearly all cases that a differ- ence of a few per cent in the friction of the engine, or more 274 RESISTANCE AND PROPULSION OF SHIPS. especially of a very small amount in the pitch of the pro- peller, would be sufficient to account for the difference in the values of w. The uncertainty as to the real value of the pitch in cases of variable pitch has been considered in 42 ; and bearing these facts in mind as well as the possibilities of slight errors qf measurement in all cases, and the further uncertainty regarding engine-friction, it would appear that the degree of fulfilment is quite as close as might naturally be expected, and that for problems of design such equations and methods may be used with a high degree of confidence in the results. The influence of an incorrect estimate of w in a problem of design may be illustrated by the following example : (*) In problem (a) above, suppose .14 to be the correct value of w, but let its estimated value be taken as .09. Then with the same assumed value of s^ we shall have I-^= .807; pN 2009. Following the work through as in (a)\ we should find the following results: d= 15.17, / = 18.20, instead of those found in (a). Now suppose this design accepted and the propeller fitted to the ship. We may then ask two questions: (i) What will be the change in efficiency and what the power necessary at the desired speed of 16 knots? or, (2) What will be the re- PROPELLER DESIGN. 2*J$ sultant speed if the power 4002 is delivered to the propeller as in (a)? We first assume the speed of 16 knots attained with a value of CiE = 2'j2ij as in (a). We then have, by substitution in (i), 2721 = OiNi) 3 (i.52) 2 .8o6. In this equation -(piNi) and k are both unknown, but both dependent on s 2 . Hence solving, we have We must now find by a process of trial a value of $2 such that the resulting values of k and p\N\ (using of course the true value .14 for w) will give for (piNi) 3 k the value 1461, or a sufficiently close approximation to such value. For the value S2 = .26 assumed in (a) we have k = .iSi and (piNi) 3 = jioo. Whence (piNi] 3 k = i2S^. The value 1461 required, indicates a value of 50 larger than .26, and one or two trials are suf- ficient to show that the desired result is very near to .283 for which (/>iJVi) 3 = i459. We may take, therefore, as the result = 283; = 1983; N= 109; 62 will be slightly decreased; W P will be correspondingly increased. According to these results, therefore, the revolutions would be slightly less than the 110.5 expected, the efficiency would be slightly decreased, and there would be required a correspond- ing small addition to the power. 276 RESISTANCE AND PROPULSION OF SHIPS. If we next suppose but 4002 H.P. delivered to this pro- peller and inquire as to the resulting speed, we have a problem requiring for its solution a resistance-curve for the ship. For all practical purposes, however, we may assume a loss of efficiency of, say .007, and that power varies as the cube of the speed. This would give for the ratio of the power in the two cases the value .68 -T- .673 = 1. 01, and therefore Ratio of speeds = t'l.oi = 1.003. Hence with 4002 H.P. at the propeller in the case assumed we should have a speed of u = 16 -f- 1.003 = 1 S-9St or a resulting loss of .05 knot. In a similar fashion if we suppose w overestimated by .05, or taken at . 19 instead of .14, we may carry the work through, finding for the propeller the following : pN ' 1840; d = 17-31; /= 20.77; Then, as before, we should find that such a propeller ap- plied to the ship in (a) with w = .14 instead of .19 would give rise to the following values : s t = .24 (app.) ; = 1872; PROPELLER DESIGN. JV = 90.i; e 2 will be sEghtly increased j W P will be correspondingly decreased. In the same manner as above we should also find that with 4002 H.P. delivered to this propeller the speed would be in- creased by about .04 knot or to 16.04 knots. These illustrations of the influence of an error in 'the esti- mation of w may perhaps be taken as extreme cases, for we should under all usual conditions be able to estimate w within .05 , and it thus appears that the errors resulting from an incorrect estimate of w are in themselues not likely to be serious, espe- cially if the propeller is designed to work at or near its maxi- mum efficiency. Some further discussion of these points will be found in the following section. Additional Methods of Propeller Design. The formulae and methods of design whose applications are shown in the present section are such as naturally result from the general method of treatment developed in this work. Similar data may be used in a variety of other ways as illustrated by Froude,* Barnaby,t Caird,J and McDermott. There have been long in use also briefer and less accurate methods of screw-propeller design than those involving the detailed considerations discussed in the present work. A brief reference to such methods may here be made. The value of pN is found from the given speed u and an * Transactions Institute of Naval Architects, vol. xxvu. p. 250. | Transactions Institution of Civil Engineers, vol. en. p. 74. J Transactions Institute of Engineers and Shipbuilders in Scotland, 1895-96, p. 21. ij Transactions Society of Naval Architects and Marine Engineers, vol. IV. p. 159- 278 RESISTANCE AND PROPULSION OF SHIPS. assumed value of the apparent slip s l (see (4) ). The values of /and TV are then selected by judgment. The value of d may then be found from the formula =^y I.H.P. in which K l is usually taken between 18000 and 25000. It is readily seen that this is equivalent to taking the value of the product k, /, m, (i) as constant, or assuming its value by judgment between limits corresponding to those specified for K v The helicoidal area A is sometimes determined by the formula 'I.H.P. A = in which K^ may vary from 8 to 15. The helicoidal area may also be determined by the formula Indicated thrust I.H.P. X 33000 , A = ~ - - ~ (S6e 47)> in which K t may vary from 1000 to 1500. Having thus determined A, the resulting values of the disk area and of the diameter d may readily be found by the use of standard proportions. Since helicoidal area should vary nearly as <^*, it is evident that the three equations above are mutually inconsistent in form, though when used with judgment and with abundance of data for comparison they may be made to yield good re- PROPELLER DESIGN. 279 suits. The third, relating helicoidal area to indicated thrust, is probably the most satisfactory of the three. The disk area may also be related to the wetted surface of the ship by the formula Wetted surface Disk area = - , in which K^ is usually found between 70 and 90, while for specially high-powered craft it may fall to from 50 to 70. The disk area may also be related to the area of midship section by the formula Area of midship section Disk area = f in which K^ is usually found between 2 and 2.5, while for specially high-powered craft it may fall to from 1.2 to 2. The disk area may also be related to the displacement D by the formula Disk area = , in which K t is usually found between .8 and i.i, while for specially high-powered craft it may fall to from .6 to .8. In formulae relating the disk area to the ship the result found is considered as the area for all the propellers, one, two, or three, as fitted. The value may then be divided as required, and the diameter immediately found. Normand has proposed * a formula for the area of the pro pulsive surface, the constants of which he has checked by appli- cation to a large number of actual cases, and which he con- * Bulletin de 1' Association Technique Maritime, 1899. 280 RESISTANCE AND PROPULSION OF SHIPS. siders may be taken as a safe guide for the determination of the necessary helicoidal area, in particular when the speed of the ship is not far from the normal maximum speed as discussed in 25. In English units this formula is nd 2 u 2 where r = area ratio as defined in 34; # = I.H.P.; n = number of propellers; d= diameter in feet; u = speed in knots; J = constant with values from 6 to 8. The values of the constant are to be selected according to cir- cumstances, less as the immersion is greater and the general con- ditions are favorable, and greater as the immersion is less and the conditions of operation are less favorable.* The methods and formulae briefly referred to in the various equations above involve necessarily a large element of judgment, with but slight opportunity for its orderly exercise. They have, furthermore, no provision for estimating in detail the probable results due to variations in the different charac- teristics of the propeller. With the more detailed methods presented and illustrated in the present chapter, the range of judgment is somewhat narrowed, its application is simpler, and full provision is made for the orderly estimate of the probable results due to changes in the more important char- acteristics. * In addition to the above formula for propulsive surface, the reader will find an excedingly interesting and highly suggestive series of papers on propeller theory and design under the following references: Rateau, Bulletin de 1' Association Tech nique Maritime, 1900; Drzewiecki, ibid., 1900-1901; Doyere, ibid. t 1901-1902 Brosser, ibid., 1903; Bassetti, ibid., 1906-1707. PROPELLER DESIGN. 2 8l 51. GENERAL SUGGESTIONS RELATING TO THE CHOICE OF VALUES IN THE EQUATIONS FOR PROPELLER DESIGN, AND TO THE QUESTION OF NUMBER AND LOCATION OF PROPELLERS. In the method of the preceding section the I.H.P. is taken as a part of the fundamental data. This implicitly involves values of the augmentation of resistance due to the propeller and of the wake factor. While, therefore, the augmentation of resistance is not explicitly involved in our equations, it is really represented by the assumed or determined I.H.P. The wake factor w is not only involved in the I.H.P., but we are also called on to assign to it a definite value. The principal source of information relative to both the wake factor and the factor of augmentation (or its reciprocal "thrust deduction," as termed by Froude) is in the experi- ments on models. These points are discussed by Mr. R. E. Froude in the paper* to which previous reference has been made, and to which the reader may be referred for details. The conclusions were that for both primary uses of such data, viz., the determination of the standing of a propeller on the scale of efficiency and the relation of its thrust to revolutions, such sources of error as may properly seem to exist appar- ently tend to neutralize each other, so that we have to deal not with a single error, nor with several tending in one direc- tion, but with a balance of errors in which, though the indi- vidual values may be sensible, it is quite likely that their combination will not form an error of serious amount. Com- paring the numbers of revolutions for the models with those for the full-sized ships, Mr. Froude states that the model data on the average were found to overestimate the revolu- * Transactions Institute of Naval Architects, vol. xxvii. p. 250. 282 RESISTANCE AND PROPULSION OF SHIPS. tions of twin-screws by about 2 or 3 per cent, and to under- estimate those of single-screws by about the same amount. As shown by example (i), 50, the result of an under- estimate of w when the design is carried through for a given pitch-ratio, as in (\ C \ gi ves tne propeller of uniform pitch with straight ele- ments bent back from the plane of revolution ; IJ>\ C \ gi ves the propeller of uniform pitch as in the last, but with the elements curved away from the plane of revo- lution ; ljb l c l gives the propeller of uniform pitch with elements bent or curved in the plane of revolution ; /i# t , gives the common expanding-pitch propeller, the pitch increasing from forward to after edge. We may also form these various combinations with c t or c and thus introduce radial variation of pitch in any way desired. It is also possible to greatly vary the appearance of a pro- peller-blade by varying the location of the part actually taken for the blade. Thus from a helicoidal surface of uniform pitch, by appropriately locating and shaping the contour, we may produce a blade which has the appearance of being bent back from the plane of revolution, and which on casual exam- ination may seem to be the same as that given by l, t b l c l or IJb^c^ though such is by no means the case. PROPELLER DESIGN. 309 While thus an indefinite variety of propellers are possible as regards shape of blade and distribution of pitch, it must be clearly understood that so far as experimental or actual results are concerned we are not in a position to select among them any one set of characteristics as the best, and especially any one distribution of pitch, or indeed to say that any distribu- tion of pitch will produce a propeller superior to that of uni- form pitch. It is possible that some distribution may be superior to the uniform, but we are not at present in a posi- tion to either confirm or disprove this by experimental data. We find therefore a very general tendency to consider the propeller of uniform pitch as the standard in type. The bent- back blades as given by /?,, has been much used on tor- pedo-boats, yachts, and small craft, the presumable reasons being a possibly better hold on the water, and a decrease in the so-called short-circuiting around the tips of the blades. In regard to the influence of any fanciful variation of pitch, we have but to remember the nature of the wake as discussed in 41. It is there shown that relative to a uniform pitch any propeller-blade must necessarily work with an excessively varying distribution of slip, both over its surface and during a revolution, and hence relative to a uniform slip it must act as though the pitch were correspondingly variable. A pro- peller of pitch variable according to some definite law will therefore not give a corresponding distributed variation of slip, and its actual variation of pitch is so small as to be en- tirely swallowed up in the greater wake variation of slip, so that its whole relation to the wake will be but slightly differ- ent from that of a uniform-pitch propeller, and in any event entirely different from that in a uniform stream. Any benefit supposed to arise from some particular distri- 310 RESISTANCE AND PROPULSION OF SHIPS. bution of pitch and slip will therefore be wholly fanciful or accidental, since in the actual case the variation of slip corre- sponding to that of pitch will not be even approximated to. See also 42 and 44. The Laying Down of a Screw Propeller. We take as the simple and standard case a propeller of uniform pitch as given by A?,^. We must have the following details given or assumed : Diameter. Pitch. Area. Diameter and Shape of Hub. Shape of Blade. We may also assume for simplicity that the blades are to be cast with the hub, and that they are to be symmetrical about a radial center line. Let XX, Fig. 79, be the axis and OY a perpendicular, on which the center line OB is taken equal to the radius of the propeller. The hub is then laid off as in the figure. The blade area being known, we readily find the mean half-width, and using this as a general guide sketch in a trial half- contour AGB. This area may then be checked by planimeter or otherwise, and adjusted until it is correct in amount and the contour is satisfactory. The length of the root-section is usually taken about .9 that of the maximum width. The distance CB is then divided into any convenient number of parts, measuring from either B, C, or , as may be most convenient. From (i) the value of tan a for any point F is tan a = = ~- r -2- -^ OF. 2nr 2n 2n PROPELLER DESIGN. If, therefore, we take a point E such that OE = / ~- 2n, the angle EFO will give the corresponding value of a. A similar relation holds for all other points, so that if from E a Y FIG. 79. bundle of lines be drawn to the several points on the radius, the inclinations of these lines to OB will give the correspond- ing values of a. 312 RESISTANCE AND PROPULSION OF SHIPS. Hence if EH is taken equal to FG, it is evident that El is the projection of FG on a longitudinal plane, and HI its projection on a transverse plane. If, therefore, FK is made equal to El, .AT will be the projected view of G, and making FL = FK, we have two points on the projected contour of the blade. In this way we may find the entire projected view of the blade as shown. The other projections are read- ily found in a similar manner; but as they involve only an exercise in descriptive geometry, the details need not be here explained. It is readily seen that, so far as the contour is concerned, this whole construction is approximate, and not accurate; and it may be mentioned that in the projection on a transverse plane the lines such as GG l may be taken as arcs of circles with OF as radius, and the distances such as HI would then be laid off along these arcs, or practically as chords. If the blade is not symmetrical about OB, it may be laid off in any form desired, and then the two parts must be pro- jected separately, but otherwise in the same general manner. For detachable blades the same general method is used, the form at the root being appropriately modified to suit the means of attachment to the hub. In any case the blade joins the hub, or, if detachable, its circular boss, by a well-rounded fillet. For blades of more complicated form and distribution of pitch the same general principles may be applied to find the approximate appearance of the projections. It may be ob- served, however, that at best these are but approximate, and moreover are of no practical value to the pattern-maker or moulder. The information of which he makes direct use is PROPELLER DESIGN. 313 much more restricted in amount, as will be explained at a later point. Returning to Fig. 79> the next step is to lay down the thickness CM at the root of the blade. A straight line, MB, is then drawn to the tip. This will give the middle-line thickness of the various sections. At the tip such line must be run into a curve as shown, in order to insure good casting. For bronze propellers such thickness may be taken at about ^ inch per foot of diameter. For cast iron it must be slightly thicker. The varying sections or thickness strips may now be put in. Thus SQ is made equal to ST, and a circular or parabolic arc is run through P, Q, and R. When the outer sections are reached, as in Fig. 80, such an arc would give too FIG. 80. thin an edge for reliable casting. We may, therefore, draw lines at the ends as shown, making the edge of the requisite angle, and then run in a smooth curve tangent to these lines at the ends, and parallel to AB at the centre D. This mini- mum angle may vary from, as little as 5 for small bronze propellers, such as are used on torpedo-boats and fast yachts, to 15 or 20 for cast-iron propellers of large size, such as are used in ordinary mercantile practice. The smaller angles are to be recommended. The thickness is usually placed on the back of the blade; that is, forward of the helicoidal surface which has been laid out as the driving-face. For the sections near the root, however, certain modifications are sometimes made, based on the following considerations: AB, Fig. 8i; denot- 3*4 RESISTANCE AND PROPULSION OF SHIPS. ing the face, we have by the usual construction EC = pitch and DC = slip. Then, considering the water as undisturbed and flowing axially to the propeller, we have DO as the direc- tion of relative motion. Draw HJ parallel to DO. Then, with the usual form of section, the back of the blade from B to J will be met by this stream, and the resultant pressure will have a component from forward aft, thus reducing the FIG. 81. effective thrust. If, however, the section is made as shown by the full line, we have the back merely exposed to frictional resistance, while the direct flow meets the face only, and may thus give a slightly increased thrust. This change in section amounts to a great increase of pitch on the driving-face from B' to K. The simplicity of this construction, however, is doubtless far from representing the path of the water in the PROPELLER DESIGN. 3 J 5 actual case, although experimental investigation as referred to in 49 seems to indicate some advantage from this form of section. We have above referred to the fact that the information actually needed by the pattern-maker or moulder does not include the various geometrical projections there referred to. The information strictly necessary is simply the following: The principal dimensions and characteristics of the propeller and hub; the expanded form or contour of the blade, as in AGBG V A^ Fig. 79; the shape and angular position of the generatrix, or else the form of a series of guides at successive radial distances; the distribution of the thickness. The necessary information may therefore be classified under these four heads : (1) The chief dimensions ; (2) The contour of the blade ; (3) The law of the pitch ; (4) The distribution of the thickness ; Thus for a uniform pitch propeller, the information given in Fig. 79 aside from the projection AKB is all that is needed to prepare a pattern in wood or sweep up the form in loam. Instead of the above method some designers prefer to lay down first the projection on a transverse plane, making the projected area a certain fraction of the disk area. See, for example, Fig. 53. The expanded form and the other projec- tion may then be found by the inverse of the methods given above. In Plate A is given a copy of the working drawing for the propellers of the U. S. battleships Indiana and Massa- 316 RESISTANCE AND PROPULSION OF SHIPS. chusetts, showing the information provided in these cases of naval war-ship design. Measuring the Pitch of a Screw Propeller. From (i) we have/ 2nr tan a. If therefore at a given value of r the SCREW PROPELLER BATTLE'S HIP NO 2, 3CALE PLATE A. value of tan a can be determined, the pitch is known. Such measurement requires necessarily the establishment of a transverse plane, or plane perpendicular to the axis of the propeller. If the propeller is lying on a floor, the latter may be used, care being taken that the propeller is so blocked up that its axis shall be perpendicular to the floor. Otherwise a plug carrying perpendicular to the axis a swinging arm may be fitted to the shaft hole, thus giving a movable line of PROPELLER DESIGN. 317 reference. If the propeller is in place on the ship, the arm may be attached to the hub or to the shaft nut, or so arranged in any way as to give a line of reference movable or adjustable in the transverse plane. We will first note an approximate method which consists in determining at what value of r we have a = 45, tan a = i, and hence / = 2nr. The transverse plane being established as above, a bevel or triangle with angle 45 or even an improvised sheet of metal may be applied, and the corresponding location of r approximately found. For more accurate measures we have to determine at a selected value of r two sides of a triangle, as in Fig. 82. AB FIG. 82. lies in a transverse plane and on the surface of a cylinder of radius r. AC lies on the face of the blade and on the same cylinder, being in fact their intersection. BC is parallel to the axis. The triangle ABC developed is evidently similar to ADC of Fig. 55 for the same radius and pitch, the relation being as shown in Fig. 83. If, therefore, we have an arm swinging about the axis, we may by running a line per- pendicular to this arm at a fixed value of r determine any desired number of points on the line AC. The length of AC may then be measured by a flexible rule or batten, or, if it is short, it will not differ sensibly from the straight line joining its extremities. BC is then determined as the difference i8 RESISTANCE AND PROPULSION OF SHIPS. between the axial distances from the points A and C to the transverse plane or swinging arm. We have then sm a = BC AC and and tan a = 2nrBC BC VAC* -- BC* . VAC 4 - BC FIG. 83. Still otherwise, we may measure the angle ft subtended at the axis by the arc AB. We then have AB = rft, tan a = - = -, and If the distance AC is small relative to the dimensions of the propeller it gives the pitch simply for that particular small part of the surface, and if the pitch is thus determined in various parts of the surface it will usually be found to vary very considerably even in a propeller intended to be of uniform pitch. This arises from minor departures from exactness in the pattern, mould, and casting. In rough work such variation may be as great as 10 per cent on either PROPELLER DESIGN. 319 side of the intended value, though with proper care the limit of variation should be much less. The details of the operation of measuring the pitch will frequently vary with the means at hand and the particular circumstances of the case ; but if the fundamental geometrical problem to be solved is kept clearly in mind, such necessary variations will present no difficulty. Table for connecting the Slip A ngle with the Slip Ratio s. This is deduced as follows: Referring to Fig. 55, we have tan a = and tan (a 0) = , 2nr 2nr whence, with the nomenclature there used, we readily find nys tan = a a , r . Table I gives the values of for varying values of y and s. Table for Determining the Modification of Pitch due to the Twisting of a Blade. In propellers with detachable blades it is customary to provide for twisting the blade slightly on the hub, and thus modifying the pitch. The amount of modifi- cation at any given radial distance or value of y, is found as follows: Referring to Fig. 55, we have in general p = 2nr tan a = n\.w\ a P ! whence tan a = --3 = nd ny 320 RESISTANCE AND PROPULSION OF SHIPS. Let ft be the amount of twist. Then, evidently, p, = 7T /?), and A = / tan a Since ft is always quite small, this may be put in the form /, _ i /? cot __ i ftny A "~ I =F ft tan " __ ' *r Table II gives values of/, -T-/ O for varying values of ft and j/. It is thus seen that the amount of change varies consider- ably for different portions of the blade. Hence if the pitch were uniform before twisting, it cannot be so afterward. The twist will have, in fact, transformed the surface from one of uniform pitch, to one with a more or less irregular distri- bution from root to tip of blade. This, of course, is by no means necessarily prejudicial, and so long as the angle of twist is small we have no reason experimentally to expect any particular effect either good or bad due merely to the irregu- larity of the pitch. The twisting thus provided for is for the purpose of changing the effective or mean pitch of the pro- peller as a whole, and the above considerations indicate that so long as the amount of twist is small, we have no experi- mental basis for objecting to this mode of obtaining such modification. As a final question we may ask what will be the mean pitch of the blade thus modified. This will depend on the definition of mean pitch as discussed in 42, to which refer- ence may be made. PROPELLER DESIGN. 321 TABLE I. VALUE OF THE SLIP ANGLE 0. Slip. 05 .10 15 .20 25 .30 35 .40 Diameter Ratio. . I o 51 I 48 2 51 4 oo 5 17 6 44 8 21 10 12 .2 21 2 47 4 20 6 oo 7 49 9 46 ii 53 14 II 3 28 3 or 4 39 6 22 8 n 10 06 12 O6 14 13 4 25 2 54 4 26 6 02 7 41 9 24 II IO 12 59 5 19 2 40 4 04 5 30 6 58 8 28 10 00 ii 35 .6 12 2 25 3 41 4 57 6 15 7 34 8 56 10 18 7 5 2 12 3 19 4 28 5 38 6 48 7 59 9 12 .8 00 2 00 3 oi 4 03 5 05 6 08 7 12 8 16 9 o 54 I 49 2 45 3 41 4 37 5 34 6 32 7 30 I.O o 50 I 40 2 31 3 22 4 14 5 06 5 58 6 5i TABLE II.* FACTOR FOR MODIFICATION OF PITCH DUE TO TWISTING OF BLADE. Twist of Blade. x- 2 3 4 5 6 Diameter Ratio. Factor for Factor for Factor for Factor for Factor for Factor for Increase. Decrease. Increase. Decrease. Increase. Decrease. 1 Decrease. Increase. Decrease. Increase. Decrease. .1 .2 3 4 5 .6 '.S 9 I.O .065 .040 .036 037 039 043 .047 .051 1-056 1. 060 943 963 .966 .965 .962 958 954 950 945 940 1.138 1.083 1.073 1.074 1.079 1. 086 1.094 1-103 1.113 1. 122 .891 .927 933 930 925 .918 .909 .900 .891 .880 .221 .127 .III .112 . I2O .130 I . 142 I.I56 I.I70 1.184 843 .893 .901 .897 .886 877 .864 .851 837 .821 315 175 152 .152 .162 .176 .192 .210 .229 247 .800 .861 .870 .864 .852 .837 .820 .802 783 763 425 .226 .194 193 .205 .222 .242 .264 .288 . 7 6l .830 .840 .832 .817 .798 777 754 730 705 553 .281 237 236 .249 .269 293 320 348 376 725 .800 .811 .801 .783 .760 734 .707 .678 .648 * Taylor's " Resistance of Ships and Screw-propulsion," Table XV. CHAPTER V. POWERING SHIPS. 56. INTRODUCTORY. THE various constituents of the power which it is neces- sary to develop in the cylinders of a marine engine in order to propel a ship at a given speed have been already discussed in 46. We have now to consider the various suppositions which will enable us to make an estimate of the total power thus required in any given case. These are of two general classes and lead to two general methods for the estimate of power. (i) An assumption of the necessary constants which will enable us to compute the resistance and the power required to overcome it, together with that required for the various other constituents going to make up the I.H.P. This also may be done in either of two ways: (a) We may make the' various assumptions necessary to compute the different constituents individually. This in- volves the following elements: the power for the resistance proper or the E.H.P. ; the amount involved in the augmen- tation of resistance; the wake factor; the propeller efficiency; and the amount absorbed by friction. By a proper combina- tion of these as indicated by Fig. 72 we readily work back to the total I.H.P. 322 POWERING SHIPS. 323 (&) We may make the assumptions necessary to compute the E.H.P. as above, and then assume the propulsive coefficient or ratio E.H.P. -f- I.H.P., whence the latter fol- lows directly from the former, and without special assump- tions as to its other individual elements. (2) The second general method involves certain assump- tions which justify the extension of the law of comparison from resistance to power, and thus enable us to compute the I.H.P. in any given case from that actually observed for similar ships at corresponding speeds. We shall discuss these methods in order. 57. THE COMPUTATION OF W p , OR THE WORK ABSORBED BY THE PROPELLER. The necessary information and assumptions requisite for the computation of R have been discussed in Chapter I and in 45. The amount of augmentation of resistance must be taken by judgment guided by such information relating to similar cases as may be available. Using the E.H.P. as a base, the additional amount of power thus required will usually be found between 10 and 25 per cent. Definite information with regard to the amount of this constituent of I.H.P. is only generally obtainable by model experiments as referred to in 30. With such appliances, the true and augmented resistances are readily found. Unless there is available some definite information based on model experiments, relative to the probable amount of augmentation, there is little use in attempting its independent estimate. The determination of the wake factor w has been already 324 RESISTANCE AND PROPULSION OF SHIPS. referred to in 44 and 49. With the wake factor and augmentation known or assumed, we readily pass, as ex- plained in 46, from CF to C t , Fig. 72. If, instead of estimating augmentation and wake factor separately we should consider the ship efficiency ( 46) as I, then we have, in Fig. 72, CF = C^E and propeller power or BE = CF divided by propeller efficiency. The considerations relating to propeller efficiency have been discussed in 49 to 51. Guided by such information as is available and as seems applicable to the case in hand, its value is assumed. This, as we have seen, may in usual cases be expected to lie between 65 and 73 per cent. 58. ENGINE FRICTION. We come next to the amount of power absorbed by fric- tion between the propeller and the cylinders. We may remark at the beginning that there is very little exact or satisfactory information on these points. Model experiments are here of no avail, at least for purposes of actual measurement. Such measurement would require at the various speeds the simultaneous determination of the power in the cylinders and of the power received at the pro- peller. If, for example, just forward of the propeller there could be inserted a transmission dynamometer which would measure the turning moment, then the product of this by 2n times the number of revolutions per minute would give the power transmitted to the propeller. Such determinations are obviously difficult and costly, and hence more or less indirect methods are resorted to. In the first place it is assumed that the total power POWERING SHIPS. 32$ absorbed by friction may be expressed by an equation of the form W f = /iN+lW, (i) where // is a constant for any given engine, N is the number of revolutions, and hN is the amount due to what is termed the initial friction, while W is the I.H.P. and /is likewise a constant for the given engine. The supposition that the total frictional power may be thus expressed is quite arbitrary, but seems to answer the purpose required as closely as present data can determine. It assumes simply that there will exist at the various joints and rubbing-surfaces a certain tangential resistance of which a part proportional to h is sensibly independent of revolu- tions or load, and represents what is termed the initial fric- tion. The existence of this factor is chiefly due to the pressure at the various stuffing-boxes, piston-rings, etc., and to the weight of the shafting and other members supported in bearings. This part of the total friction is also sometimes known as that due to the dead load. The work necessary to overcome such a constant resistance will vary directly as the revolutions, and hence will be represented by a term of the form JiN. It is further assumed that the remainder of the tangential resistance at the rubbing-surfaces will vary directly as the load on the engine, or as the mean effective pressure, and hence the work necessary to overcome this part will be proportional to the product of mean effective pressure and revolutions, or to total power developed. Hence this part of the frictional work will be represented by a term of the form IW. Both of these terms may be expected to vary with the type and condition of the engine, and with the presence or absence 326 RESISTANCE AND PROPULSION OF SHIPS. of attached pumps. In general the friction of horizontal en- gines is greater than that of vertical. The power absorbed in friction also increases in general with the number of cylin- ders and with the multiplication of rubbing-joints. The fric- tion of new engines is also naturally greater than when they have become somewhat worn. Again, incorrect adjustment or alignment either at the beginning or due to wear or work- ing of the ship may give rise to excessive frictional loss. In such cases the condition is likely to manifest itself by hot bearings and other signs that something is wrong at the points thus affected. The presence of attached air-pumps occasions an absorption of power not only to overcome the friction of the pump, but for its regular running, which amount must be subtracted from the I.H.P. in order to properly obtain that sent to the propeller. This item is not to be viewed as a loss, but simply in such cases as a constituent of the total power which must be eliminated in order to determine the net amount delivered to the propeller. With independent pumps this item, of course, does not exist as a constituent of the power of the main engines. The power required to operate pumps has been deter- mined by various special and individual trials. The power absorbed by the initial friction has been determined by slowly running the unloaded main engine by the turning engine, or by derivation from curves of revolution and power as described under the next sub-head. The power absorbed by the load friction has been estimated from various coefficients derived from stationary engines in which the different quan- tities are susceptible of measurement, and from analyses of the total I.H.P. in which the various efficiencies and ratios are derived by comparison from model experiments. POWERIXG SHIPS. 327 Determination of Initial Friction from Speed-trial Data. In 47 we have defined the reduced mean effective pressure, and we may evidently consider this as balanced against the mean total resistance to the movement of the engine. This total resistance may be considered as the sum of two parts, one due to the useful load and the other due to the friction load. We have already referred to the frictional or tangential resistance at the joints and rubbing-surfaces under two heads: that due to initial conditions and that due to the load, corre- sponding to initial and load friction. Evidently the frictional work will require an additional pressure on the pistons or an additional amount of reduced mean effective pressure as above defined, and such additional amount may naturally be considered in two parts, one due to initial friction and the other to the load friction. We may thus consider the entire reduced mean effective pressure as made up of three parts, one corresponding to the net or delivered power and two due to friction as above. Denoting these respectively by /,/,, /,, we have (2) From the nature of initial friction as above defined it follows that/j is constant and is in fact proportional to the factor h there used, while / 2 and also / will vary with the load. If now we have a series of corresponding values of /fand Nj we may derive by 47 (3) the values of/. Plotting these on ^Vas an abscissa, we shall have a curve as in Fig. 84. Now when N=o, /, and/ must necessarily be O, and hence/ will be equal to/j. If therefore the curve be extended back by 328 RESISTANCE AND PROPULSION OF SHIPS. judgment, it will cut the axis of Y at a point A which will give an approximation OA to the value of/,. Then for the VALUES OF N FIG. 84. general value of the initial friction power at any number of revolutions N we have 33000 = /iN, as above. Hence we see that 33000 In order that such an approximation may be of value, special care should be taken with the determinations relating to the lower points on the curve, and the revolutions should be reduced to the lowest number at which the engine can be kept in continuous movement. We may also for the same result use the same data some- what differently as follows: POWERING SHIPS. From the common horse-power formula we have a/7 _ 2pLA 'dN ~ 33000 " 3 2 9 Hence TV 33000 Hence if H be plotted on TV as an abscissa we shall have a curve as in Fig. 85. If this be extended back through O by VALUES OF N FIG. 85. judgment, it will have at O a certain inclination to OX, and the tangent of this angle or of BOX will therefore be an ?i //""I approximation to the value of ^ ; and from this the corre- sponding value of/, may be found. See also 78 for another method of finding the initial friction. The existing data relating to initial friction gathered in this and other ways indicate that in amount it may be expected to lie between 5 and 8 per cent of the full power of 330 RESISTANCE AND PROPULSION OF SHIPS. the engine, or that/, will be from 5 to 8 percent of the/ for full load. This would correspond to a value of from 2 to 3 pounds per square inch on the low-pressure piston for triple- expansion engines under usual conditions as above noted. Let this percentage ratio be denoted by i and denote full- power conditions by accents. Then we should have Work of initial friction at full load = hN' = iH', Jt f h=ijjr\ and work for initial friction in general = hN= , where, as above stated, we may expect i to be found between .05 and .08. These figures relate more especially to the main engines alone. Value of the Load Friction. We have above stated that this is taken as proportional to the whole power H. It might seem more correct to take it as proportional to the total power minus the initial friction. The difference involved is, however, quite negligible in view of the general uncertainty surrounding the whole question, and we naturally prefer, as the simpler of the two, the method as stated. We have therefore Load friction = IH. The existing data relating to load friction indicate that / may be expected to vary between nearly the same limits as i t . or from .05 to .08. General Equation for Friction. Combining the two parts ? we have Total friction = ^7 \- POWERING SHIPS. 331 If we plot the total power and the power absorbed by these two components of friction on revolutions as an abscissa, we shall have a diagram as in Fig. 86, where the ordinates VALUES OF N FIG. 86. between X and OA give the values of the initial friction, those between OA i.nd OB the values of the load friction, and those between OB and OC the values of the net or delivered power. This equation and diagram serve to illustrate the increasing relative importance of friction at reduced speeds. Thus suppose at full power the friction is taken at 14 per cent, equally divided between load and initial friction. Denoting full power by 100, we should then have for one- half speed approximately one-eighth power or 12.5. The load friction would be reduced in the same proportion, and would therefore become 7 -7- 8 = .875. The initial friction would be reduced in the ratio of the revolutions or approxi- 332 RESISTANCE AND PROPULSION OF SHIPS. mately to one half its full-power value, or to 3.5. Hence the total friction will be 4.375, and the percentage 4.375 -f- 12.5 = 35 per cent. This illustrates forcibly the wastefulness of running at reduced power, and the impossibility of obtaining good propulsive efficiencies under these conditions. At full power, therefore, we may expect that the friction of a marine engine will absorb an amount lying, perhaps, not far from 15 per cent as a mean value. With the best modern design and construction this figure may fall to 12 per cent or even possibly to 10 per cent, while with older engines or poorer construction or lack of adjustment it may rise to 1 8 or 20 per cent or even more. It should be understood that the division of the power absorbed by the friction as indicated in the present section is not altogether satisfactory, and the only excuse for its use is our ignorance of a more correct analysis. It is not to be expected that /, will be absolutely constant at all revolutions or that /, will vary exactly with the load. Further experi- mental investigation is much needed on these points. * 59. POWER REQUIRED FOR AUXILIARIES. . In the present work we are fundamentally concerned, of course, with the power needed for propulsion. Inasmuch, ' however, as we not infrequently meet with air-pumps attached to the main engine, as referred to in the last section, and also with estimates of power or power data, including that required for auxiliaries, it will be well to note briefly the power which the usual auxiliaries may be expected to absorb. There has been a great reduction in this amount within the past few years. Auxiliary machinery has been so im- POWERING SHIPS. 333 proved in design and construction that its cost in power required for operation is much less than it was from ten to twenty years ago. At the earlier period from 5 to 8 per cent of the I.H.P. was believed to be absorbed by the resistance of the various pumps, most of which were attached to the main engine. Blechynden,* writing with reference to this period gives the following estimate, based on data derived mostly from merchant steamers with triple-expansion engines: Dead load and air-pump. . . 7.8 Circulating-pump 1.5 Feed-pump 6 Bilge-pump $ j Per cent of I.H.P. If we take an estimate for the air-pump of about I per cent, which would not be far in error for the cases involved, we should have 3.6 per cent required for all auxiliaries. These figures agree generally with those obtained in this country some years ago, especially in the early vessels of the modern steel navy. Taking more recently the results of the latest practice, especially in war-ship design, we have the following results: Air-pump I to .2 Circulating-pump 2 to .5 Feed-pump 5 to .6 Blowers 5 to 1.5 :.H.P. Omitting the item for blowers, we have for the remaining three items from .8 to 1.3 per cent, or slightly over I per * Transactions N. E. Coast Inst. Engineers and Shipbuilders, vol. vn. p. 194. 334 RESISTANCE AND PROPULSION OF SHIPS. cent as a mean value. The chief improvement has been in air- and circulating-pumps, which together a few years ago absorbed from 2 to 3 per cent, while with the latest practice this figure has been reduced to about one quarter of this amount. The item for bilge-pumps in earlier estimates is not so often met with now, having decreased to almost a negli- gible amount probably less than .1 per cent. The amount absorbed by blowers is naturally variable between wide limits, being dependent on the degree of forced draught and other circumstances. It may be doubted whether the power of an attached air- pump can be brought to as small a figure as for an independ- ent pump. This is due chiefly to the fact that the attached pump usually runs faster than the independent, and faster than is needed to maintain a vacuum. In consequence the power is necessarily greater, and for such cases probably not less than .5 per cent will be required. 60. ILLUSTRATIVE EXAMPLE. We will now illustrate the application of the preceding sections to the computation of the I.H.P. in a given case, assuming where necessary the various values involved. Referring to Fig. 72, let true resistance = R. Then E.H.P. = CF = Ru. Let the factor of augmentation be 1.16. Then actual resistance = I. \6R. And thrust horse-power = CE = i.i6Ru. Let true propeller efficiency = .67. Let wake factor w .15. Then (i + w] = 1.15. POWERING SHIPS. 335 And horse-power C^E i.i6Ru -r- 1.15 = i.oiRu. Then propeller horse - power = BE = i.oiRu -f- .67 = Next, for the frictional loss let i and / each = .07. Then I.H.P. = AE = i.$\Ru-t> .86 = 1.756^*. Ru Propulsive coefficient == == .57- l These suppositions correspond to the following subdivision of the total I.H.P. in percentages. Initial friction ...................... = .07 Load friction ........................ == .07 1. 1.6 Thrust horse-power = - ^ .......... = .661 Propeller loss decreased by wake gain. = .199 Total = i .000 If, as in 56 (i) (), the propulsive coefficient were to be assumed directly, we should in the preceding example divide Ru by .57, thus giving i.f$6Ru at once. Unless special data are available for the estimate of the coefficients and ratios involved in the process above, it will be found usually quite as satisfactory to assume the propul- sive coefficient directly. This again can only be done intelli- gently when experimental data are at hand from somewhat similar ships under generally like conditions. The range of values, it will be remembered, is usually from .50 to .60. 336 RESISTANCE AND PROPULSION OF SHIPS. 61. POWERING WITH TURBINE MACHINERY. As already noted, with turbines the I.H.P. or the power corresponding to the I.H.P. with a reciprocating engine, can only be inferred. The power of turbines is usually estimated in terms of the delivered horse-power or power delivered on the turbine shaft. This correponds to the Wp or propeller power in the case with reciprocating engines, and in any event can only differ from what may be called, the I.H.P. by the friction of the rotor- shaft bearings, which should be small. In dealing with the problem of the determination of power for the case of turbine machinery, two methods are available, one by comparison, as discussed in 62-67, and the other by the use of the principles of 60, stoppng the determination at the propeller power Wp. If the method by comparison is employed, it will be well to use only such data as may be drawn from ships with turbine machinery. This results from the -fact that the general condi- tions of operation and the distribution of coefficients and efficiencies may vary considerably between turbine and recipro- cating machinery. If the method by analysis, as in 60, is employed, then care must be taken in the assumption of a pro- peller efficiency. It seems to be generally conceded, with the pitch-ratios, area- ratios and peripheral speeds which the designer is sometimes forced to employ with turbine machinery, that some loss of propeller efficiency is likely to be involved, and as the characteristics of the design become more widely removed from those known to give the best efficiency, proper caution must be exercised in the assumption of the value to be employed. POWERING SHIPS. 337 62. POWERING BY THE LAW OF COMPARISON. The assumptions involved are as follows: (1) It is assumed that the entire resistance is subject to the law of comparison. The propriety of this assumption has been considered in 26, to which reference may be made. As there shown, the error is small, and may perhaps be con- sidered as within the limit of general uncertainty surround- ing the entire problem. Further, with a slightly roughened bottom, as actually exists for most of the time, the index of the speed in the term for skin-resistance will approach the value 2.00, and the error will correspondingly decrease. (2) It is assumed that the term similar is extended to include ship, propeller, and machinery. This involves the general geometrical similarity between the ships and between the propellers, the ratio being, of course, the same for each. Geometrical similarity is not necessary for the machinery, but it should be of the same general type in each case, or at least there should be good ground for assuming an equal engine friction in each case. (3) It is assumed that the term corresponding speeds is extended to cover the speeds of the propellers relative to their ships and to each other, as well as the speeds of the ships relative to the water. Hence, as in 53, we shall have similar points on the two propellers describing similar paths with velocities in the ratio A>, and as there shown we shall have also , _ (A_V /2 _ JL ,''' \Lj ~ V 2 ' (4) It is assumed if (2) and (3) are fulfilled that the thrusts will bear the same relation to the resistance in each 338 RESISTANCE AND PROPULSION OF SHIPS case, that the slips true and apparent will be the same per cent in the two cases, that the efficiencies will be the same, and likewise the augmentation and wake factors. It is assumed, in short, that the entire relation of the propeller to the ship will be the same in the two cases. The propriety of these assumptions evidently depends on the same general basis as the application of the law of com- parison to the resistance of the ship itself. As a result of the preceding assumptions it follows that similar ships with similar propellers at corresponding speeds will have equal propulsive coefficients. Or otherwise we may less safely consider that indepen- dent of the fulfillment of (2) and (3), the propulsive efficien- cies and relations are the same in the two cases. Let the subscripts I and 2 denote the two cases. Then we have R t and ^ are the two resistances; u^ and u l are the two speeds; ^ and /j are any two similar dimensions; A, = / a -i- / a is the linear ratio ; a = the propulsive coefficient; //, and H l are the two powers. Then , * ** and Jr=RJTc Now we have in 26 a series of expressions for R^ -=- R^ in terms of various functions of the ship and of the speeds. Substituting these, and remembering that u^^-u l = y\ = POWERING SHIPS. 339 4// a _:_ i it vve readily find the following series of expressions for H^ -r- //,, the ratio between the powers necessary to propel similar ships at corresponding speeds: _ ('.y _ M.y - (y _ (Ay _ (A) (*y_ ( Ay^y H\-\IJ \Aj "\Vj \DJ "\Dj\lJ-\Dj\lJ A, (Ay (.y _ (Ay (,y _ (/,y(.y _ f ^v,y = A^ =: ^ W "W V "\7j\uJ '-\Aj\uJ > It is obvious that the list of expressions in which the law of comparison may be thus represented is by no means exhausted, and that all are equivalent and equally correct, and will give identical results, provided the fundamental condi- tions are fulfilled; i.e., if the ships are similar and the speeds corresponding. If therefore we know the data in any one case it is a sim- ple matter to derive the value for a similar ship at a corre- sponding speed. As an illustration, let A = 2000; ff l = 3000; * t s= 16; A = 3200. In 26 various expressions have been given for corre- sponding speeds. Here, having given only A anc * Ai we naturally , = **\ = M*Si = 17-4 Then ^ = ^^=3000X^X1.085 = TT In a similar way, taking other expressions for ~ t the same **\ 340 RESISTANCE AND PROPULSION OF SHIPS. value would be found. Hence the similar ship of displace- ment 3200 at 17.4 knots would require 5230 I.H.P. Assuming that the similarity between the two ships is perfect, the two chief sources of error in the use of this method are in the extension of the law of comparison to skin- resistance, and in the possibility of a difference in the propul- sive coefficients in the two cases. The nature of the error due to the first of these has been discussed above and in 26. In regard to the latter it may be seen that if the speed of the first ship, for example, is low or far from the normal speed the propulsive coefficient will probably be poor, and less than might be properly expected for the second ship if near her normal speed The use of the power of the first ship in such a case would therefore, other things being equal, result in an overestimate of the power for the second. In all such cases, therefore, judgment must be used as to what extent the necessary conditions are fulfilled, and as' to the amount of reliance to be placed on the values resulting from any given comparison. 63. THE ADMIRALTY DISPLACEMENT COEFFICIENT. Among the various formulae for powering which have been in use for many years, none has obtained so wide and general acceptance as that involving the use of the so-called Admiralty displacement constant or coefficient. This formula dates from a period long anterior to the introduction of the law of comparison, and we shall find it of interest to examine it in the light of that law. The formula is expressed by the equation POWERING SHIPS. 341 In practice it was found that there was a general tendency to constancy in the values of K, ranging as they usually did from 200 to 250. The use of the formula required simply the estimate of the constant suited to the ship and to the speed. This, of course, was a matter of considerable uncer- tainty except as experience in similar cases furnished data as a basis. The values of K naturally vary for the same ship at different speeds, or for different ships at the same speed. These values are also seen to vary directly as the efficiency of propulsion. That is, in any given case the greater the value of K the less the value of H, and hence .the greater the pro- pulsive efficiency. In numerous cases in which progressive speed trials have been made the results for the value of K are similar to those shown in Fig. 87. At low speeds the ^ ' _ - -. ^^ B ) 2 4 6 8 10 12 U 16 18 20 SPEED FIG. 87. value is low, due to the general mechanical inefficiency of both engine and propeller at powers and speeds far below those for which they were designed. As the speed increases the value of K increases, indicating that due to an increasing efficiency the total power is related to the speed by an index of u less than 3. This continues to some speed at which, on this basis, the propeller is the most efficient and K has its maxi- mum value. Then as the speed is increased the resistance begins to increase excessively, and perhaps the propulsive 342 RESISTANCE AND PROPULSION OF SHIPS, efficiency begins to decrease. The result is that H becomes related to u with an index greater than 3, and we have a corresponding fall in the value of K. So long as speeds were moderate and abundant informa- tion was available relating to nearly equal ships at nearly the same speeds, the use of this coefficient was reasonably satis- factory. Its assumption, however, was almost entirely a matter of judgment, and there were few guiding principles by means of which any definite value might be determined as suitable in any given case. The use of the formula in this way has therefore in recent years fallen into disfavor, and it is well understood that thus employed it can only be expected to give a rough approxi- mation to the power suitable in any proposed case. Let us now examine the relation of this formula to the law of comparison. We have in one case and in another : K, w , , , , , Whence _=_.__ (l) Now suppose that D l and D 2 are similar ships and u : and U 9 are corresponding speeds. Then from 61 (i) we have, by the law of comparison, Hence, comparing (i) with (2), it follows that K l ~- K^ i or K l = K y We therefore derive the important result that POWERING SHIPS. .343 Similar ships at corresponding speeds have the same Ad- miralty coefficient. This may be taken as a statement of the law of extended comparison, and as such it will be equally correct with any of the other statements implied in the general equation of 61, (i). This statement supplies therefore the principle necessary to guide in the choice of Admiralty coefficients, and once such principle is recognized and followed, this formula is instantly raised from its former condition of relative unre- liability and placed on the same level, or rather made iden- tical with the law of extended comparison, as already so numerously expressed above. As an illustrative example let us take the following: D % = 8000 ; ,= 1 8. Among the available data is a speed-power curve for a similar ship of displacement 6000. We have now to find the speed at which the latter ship will correspond with the former. Without involving any other data we may take the ratio as (D 9 -i- D$ = 1.049. Hence u, = 18 -=- 1.049= 17.16. Turning now to the given curve at a speed of 17.16, suppose we find H 6320. We have then D, = 6000, , =-17.16, and HI = 6320. Substituting this in the Admiralty formula we find K = 264. Now by the law of comparison as above shown, this coefficient is suitable for the similar ship of dis- placement 8000 at the corresponding speed 18. Hence, making the substitution in the formula, we find for the pro- posed case 344 RESISTANCE AND PROPULSION OF SHIPS. 64. THE ADMIRALTY MIDSHIP-SECTION COEFFICIENT. In addition to the Admiralty displacement formula, the equation (area of midship section)?/ 8 has been used to a considerable extent, especially in con- tinental Europe. Where the value of K^ is selected by judgment alone, it stands on the same basis as the D% formula, though it is hardly as satisfactory, since the length of the ship is entirely unrepresented in the formula, except as its value is taken into account in the judgment by which the value of K l is selected. Comparing it, however, with 61 (i), it is seen to be identical with the D% formula in its rela- tion to the law of extended comparison. If, therefore, it is used in this way for similar ships at corresponding speeds, it becomes simply another mode of application of this extended law. We may thus say, as before: Similar ships at corresponding speeds have the same midship- section power coefficient. 65. FORMULA INVOLVING WETTED SURFACE. Formulae have been proposed and used quite extensively of the general form As proposed, these formulae were to be used in a manner similar to that in which the two Admiralty formulae have usually been used. That is, the values of K^ were to be selected by judgment. They are, when thus used, open to POWERING SHIPS. 345 the same objections and are under the same lack of confi- dence as the two above described when used in the same way. In the method known as " Kirk's Analysis " an approxi- mation to the wetted surface was found by computing that of a substituted block model with prismatic middle body and wedge-shaped fore and after bodies. The appropriate value of K^ was then assumed from the data available in similar cases. This method was more especially intended for the powering of cargo steamers of moderate size and speed, with an abundance of data available to serve as a guide in the selection of the values of the constant. Under these circum- stances the method gave fairly satisfactory results, as indeed would any of the other methods here discussed when used in the same way. In Rankine's " augmented surface" method the wetted surface, or more commonly the reduced surface, 8, was multiplied by a factor called the coefficient of augmentation. This coefficient is a function of the form of the ship at the bow, or more specifically of the angles of maximum inclina- tion of the various water-lines at the bow. The area thus obtained was called the " augmented surface. " Its value was used for 5 in (i) above, and appropriate values of K t were assumed exactly as in other cases. The derivation of this formula proceeded on the assump- tion that the only resistance necessary to consider was the frictional, or at least that the resistance might be considered as varying with the square of the speed and hence the power as the cube of the speed ; and further, that the influence of the form on the velocity of gliding or the relative velocity of skin and water could be represented by this function of the angles of entrance. The method is interesting as containing an 346 RESISTANCE AND PROPULSION OF SHIPS. attempt to represent the influence of the form at the bow. It is entirely inadequate, however, to properly provide for either variation of form in general, or for varying speeds. It is in fact subject to exactly the same limitations and errors as the other formulae here discussed, and requires the same kind of judgment for the proper selection of the coefficient in- volved. The relation of such formulae to the law of extended com- parison is seen to be identical with that of the two previously discussed. In general it is seen that they are all special cases of the general term in which the ship is represented by an area or second-degree function, and the speed has the index 3. If therefore they are used as expressions of the laws of extended comparison they will have the same authority and will give the same results as any of the other expressions previously given. All such formulae may therefore be made expressions of the law of extended comparison by the simple rule: Similar ships at corresponding speeds will have the same coefficient in the formula for power. 6(>. OTHER SPECIAL CONSTANTS. The various forms for the law of extended comparison in 6 1 show that a large variety of " constants " or coefficients might be found which would be the same for similar ships at corresponding speeds. The constants considered in 62, 63, and 64 are all derived- from the function / V. Of the various constants which might be found in a similar manner we will only refer to those derived from the functions Du and POWERING SHIPS. 347 Denoting these by K z and K we have for the corre- sponding formulae for power and , Du whence A 3 = ~jr * and H If for similar ships at corresponding speeds the same values of the constant K 3 be taken, or the same values of the constant K^ the law of comparison will be fulfilled and the formulae may thus be used for the determination of power by means of this law in the same manner as with those discussed in 62, 63, and 64. We will now briefly discuss the use of these various con- stants in cases where the geometrical similarity is not com- plete. 67. THE LAW OF COMPARISON WHERE THE SHIPS ARE NOT EXACTLY SIMILAR. Strictly speaking, the law of comparison has no significance where the ships are not similar. Where they are, any and all of the various expressions for corresponding speeds, for resistance, and for power will give exactly the same results. When the similarity is not perfect, however, the various expressions for corresponding speeds, for resistance and for power will not give the same results ; and while strictly none are applicable, it is quite evident that they may all be consid- 348 RESISTANCE AND PROPULSION OF SHIPS. ered as approximations, more or less near according to the nature of the similarity and to the particular expression used. It is evident therefore that in such cases, and hence in most cases likely to arise in actual practice, it is not altogether a matter of indifference what expressions are used for the rela- tions between the speeds and between the powers in the two cases. We must first note clearly the two questions involved: (a) The relation between the powers; and (ft) the relation between the speeds, or the definition of " corresponding speeds." With regard to the latter, we are free to take any of the various expressions of 26, (7) and (8). Correspond- ing speeds are intended to be those for which the liquid sur- roundings of the two similar ships shall also be similar. These depend principally on the wave-formation, and this on the length of the ship more than on any other one feature. It therefore seems preferable to fix the corresponding speeds by the ratio of the lengths, and we take therefore For the power-ratio we will discuss the three functions Du, %*, D*u', or the functions which give rise to the three coefficients K K, K^ as above derived. Taking the first, its use is equivalent to the assumption that for similar ships at corresponding speeds the power varies as the function Du, or in symbols H - Du. Denoting the chief dimensions of the ship by Z, B, h, the block coefficient by b, and remembering that u ~ Z* we have for similar ships at corresponding speeds R ~ D ~ BhLb ~ -0' ~ Bhbu\ POWERING SHIPS. 349 If therefore we assume more broadly that resistance in general can be expressed as the product of some function of the ship by the square of the speed, the use of the constant K^ is equivalent to the assumption that such function is Bhb. Similarly we have Hence, likewise, if we assume that resistance in general can be expressed as the product of some function of the ship by the fourth power of the speed, the use of the constant K 3 is equivalent to the assumption that such function is j . Now taking a general equation of resistance involving the second and fourth powers of the speed, let us introduce those functions into the coefficients. We have then R = P(Bhby + 0' ..... (I) The equation of resistance in this form is thus seen to correspond to the use of the coefficient K 3 as above. In a similar manner we find for the form of the general equation which corresponds to the use of the coefficient K the following: R = Likewise for that which corresponds to the use of K^ R = P(DLfu* + Q(Bhb)*u* ..... (3) But in 8 we have seen that approximately (DL)^ ~ wetted surface or (DL)* ~ S. Whence we may write (3) in the form R = P(S)u* + Q(Bhb) V ...... (4) 350 RESISTANCE AND PROPULSION OF SHIPS. We may also show the relation of the three functions to the dimensions of the ship and the block coefficient as follows: For comparison the function Du is equivalent to L*Bhb ; From these last expressions it is readily seen how the different values would vary in cases where the similarity is not exact. Thus suppose that the second ship is relatively longer than the first. Then the length-ratio will be greater than the others, and the power as derived from the above expressions will increase in amount from the first to the last. Vice versa, if the second ship were broader, deeper, or fuller, the first of the above functions would give the largest value, and the others successively less. Looked at from the standpoint of the formula of resist- ance, it would appear as though (3), and hence the function Z% 4 , or the constant K t should be the most nearly correct. Experience seems to indicate, however, that there is no one form equally applicable to all modes of variation from exact similarity,* and sufficient data are lacking for the definite rela- tion of the appropriate form of function to the nature of the departure from similarity. Mention may also be made of the function Z>*, which is sometimes used, especially in connection with a speed-ratio (A * D$- Of these various functions, D*u* and D*u* or the constants K and K 4 may be recommended in preference to the others. The latter is perhaps likely to be the more correct, though the former, from the fact that it is the well-known Admiralty coefficient, has been much used ; and such use is likely to con- POWERING SHIPS. 351 tinue so long as no more definite information is available regarding the relative values of the various functions and coefficients available. We may illustrate the use of these expressions by an example as follows. Given the following data: L B h D u H (l) 30 40 20 3700 15 3129 (2) 400 48 24 7640 Then (^) =(1.33)*= 1.155. Hence u z = 15 X I.I55 = 17-32. We will now find the value of //,, using the following ratios, the results being as shown : TT TT M* H* H, D^ ~) = 2.32 7260 rZW "' V =-497 78H = 2.385 7464 TO- W =-'> 8 Where the variation from similarity is marked, the method of comparison, of course, fails entirely, and it must be remembered that its reliability rapidly decreases as departures from similarity increase. The discussion of the present sec- tion is therefore not intended to show a means of extending 352 RESISTANCE AND PROPULSION OF SHIPS. the method of comparison to widely dissimilar forms, but simply to show what forms of expression are most likely to take rational account of such slight variations as experience indicates may be admitted without sensibly affecting the application of the law as stated. 68. ENGLISH'S MODE OF COMPARISON.* Given two similar ships of displacement D l and Z> 2 at speeds Vi and V^, not in general " corresponding." Then it is always possible to make two models of different dimension such that at the same speed one shall correspond to D l and the other to Z>,. From the principles of 26 we should then have the following: Residual Resistance. Skin- resistance. Dis- placement. Speed. W, $ A f, w* s, A P. Let be the ratio of the total resistances of the models at the speed V\^-) . Then d^ and (w 9 + j.) = (/, + J,). Also ^, = w, -^. "a * Proceedings Institution of Mechanical Engineers, 1896, p. 79. POWERING SHIPS. 353 Substituting for these values as above, we find Then Total resistance of A = 5 a + W 7 ,. The values of S lt S t , s lt s t , are to be computed in the usual way, using appropriate values for the skin-resistance coefficient. Assuming also for the ship D l the ratio between thrust horse-power and I.H.P. (CE -r- AE, Fig. 72), we can find 5, + W lt and hence W^. The value of n is found by towing the models d^ and d^ simultaneously from the ends of a bar supported at an intermediate point. The point of sup- port is then adjusted until the bar remains at right angles to the direction of motion. In this condition the resistances are evidently in the inverse ratio of the lever-arms, and hence a measurement of the latter will give the value of n as desired. In this way it becomes possible to determine the value of W+ above, and hence of the total resistance 5, -f- W^ As an extension of the above method of treatment, let us assume the law of comparison to cover the total resistance, instead of the residual only. Let the experiment be made as before and the value of n similarly determined. Then denot- ing total resistances by R lt R r lt r we have . r A_ A - r '~ ' 354 RESISTANCE AND PROPULSION OF SHIPS Hence , r -R t = n \ TT I T/"\ 7 and therefore = ratio of powers = 72(77) i \"i This method being wholly comparative, no dynamometric apparatus is necessary, and the experimental determinations are comparatively simple. Since, however, the results are comparative and not absolute, its field of usefulness will be much restricted in comparison with the usual method as described in 26. 69. THE VARIOUS SECTIONS OF A DISPLACEMENT SPEED POWER SURFACE. Let us consider a general series of ships, all of the same type of form. The law of extended comparison furnishes the connecting law between all the ships of such a series, no matter what the displacement or speed. It is therefore evi- dent that any speed-power curve for a given displacement may be considered as the speed-power curve for the whole series by means of an appropriate modification of horizontal and vertical scales. Suppose, for example, that we have a curve for displacement D l plotted with scales as follows: Abscissa, I linear unit =/ knots. Ordinate, I linear unit = q I.H.P. Then the same curve will also represent the speed- power relation for D^ with the following scales : //M* Abscissa, I linear unit = /\TT~) knots ; M*V Ordinate, i linear unit = q\jy} I.H.P. POWERING SHIPS. 355 With such scale modifications, therefore, one curve would represent the whole series of values of power for the various values of displacement and speed. 8COO- 6000 4000 2000 10 11 12 11 10000 8000 6000 4000 2000 15 1(5 13 * SPEED FIG. 88. Relation between Speed and Power. Displacement constant for each curve, as shown on the right. aooo c. u 1 4000 6000 DISPLACEMENT 10,000 FIG. 89. Relation between Displacement and Power. Speed constant for each curve, as shown on the right. With a uniform scale such a series of values would require for their complete representation a surface of which the 356 RESISTANCE AND PROPULSION OF SHIPS ordinate is the value of //, located by abscissae corresponding to the given values of D and u. The various sections of this surface by planes parallel to the axes of H and u would repre- sent each a speed-power curve for a given value of D. The general values may therefore be represented by a single curve with varying scales, or with uniform scales and varying curves. The surface above referred to may also be cut by planes parallel to the axes of H and D or D and u, thus cutting out 16 15 14 gj 13 CO 12 11 10 \ \ \ \ \ *v >. "^ 7000 6000 5000 4000 3000 \ \ \ X X X,, \ ^-v -x^ \ \ >^ X "> s^ \ N \ \ ^""^ \ x *"" s \ \M 00 ^v. ^ ^ 2000 2000 4000 6000 8000 10000 DISPLACEMENT FIG. 90. Relation between Displacement and Speed. Power constant for each curve, as shown on the right. curves, one set showing the relation between H and D for a fixed value of u, and the other the relation between D and u for a fixed value of H. The various curves of section are shown in Figs. 88, 89, and 90, a study of the general charac- teristics of which may be recommended. 70. APPLICATION OF THE LAW OF COMPARISON TO SHOW THE GENERAL RELATION BETWEEN SIZE AND CARRYING CAPACITY FOR A GIVEN SPEED. In Fig. 91, let AB denote the curve of E.H*.P. for a given ship of 5000 tons displacement at speeds from 10 to 20 POWERING SHIPS. 357 knots. Now suppose this model taken as the type for a series of vessels of displacement from 5000 to 20 ooo tons. Then by the appropriate transformation the E.H.P. for each of these vessels may be determined from the given curve, as explained in 68. Again, assuming a constant propulsive coefficient of JOOOO 8000 6000 4000 10 11 12 13 14 15 16 17 SPEED IN KNOTS FIG. 91. 13 19 20 say .54, the values of the I.H.P. may be similarly obtained from a curve CD, the ordinates of which are equal to those of AB -=- .54. If a constant propulsive coefficient could be assumed for the 5 H 16,000 i 14,000 12,000 I 10,000 1 z 8,000 H" | 6,000 4,000 5 FIG. 95. g, etc., denote the tidal velocities at the successive points of observation laid off on a time abscissa. Then OP, PQ, etc., 370 RESISTANCE AND PROPULSION OF SHIPS. are the successive time intervals between the points of obser- vation. Now without attempting any refined analysis of the tidal components, we may quite closely obtain a continuous value of the tidal velocity at the ship at each successive instant by passing through the points A, B, C, etc., a smooth curve, keeping in mind the characteristics of a sinusoidal form. If now the area of this curve be found by a planimeter or by approximate integration, the result will give the total distance or set due to the tide, + or according to the cir- cumstances of the case. The total correction found in either of these ways is then applied to the total length of the course, thus giving the corrected length, from which in conjunction with the time the speed is directly found. We will next consider the elimination of tidal influence for short-course trials without special observations. For a measured mile the time required to make a double run with and against the tide will be so small that without large error the tidal velocity may be considered constant throughout. In such case the simple mean of the two runs will evidently give the true mean speed; thus: An extension of this, known as the method of " continued averages," has also been much used. This is as follows: Suppose four runs made, two in each direction, giving appa- rent speeds u lf u 91 & 4 . These are treated as shown by the form on the next page The third average is then taken as the correct speed, thus deduced from the speeds observed. This is equivalent to giving the second and third runs three times the weight of the TRIAL TRIPS. 371 first and fourth, and the final result may be more readily found by this formula than by the actual formation of the successive columns of averages. Speeds. ist Averages. ad Averages. 3d and Final Average. Ui Uy Us u t 1 + W 2 Ui -f 2W 2 -f 3 i + 3a + 3s -|- 4 2 a -j- W 3 4 2 + 2U 3 -f U^ 2 Ms + tit 8 4 2 The correctness of this method depends on certain assumptions \vhich we may briefly note. Let the true speed remain constant at u. Let the tidal velocity be expressed as a function of the time as follows: v = a -f- bt + cf. Let the runs be made at equal time intervals which may be denoted by t. Then for the apparent speeds we shall have u l = u-\- a ; w 2 = u a b c ; , u + a+ 2b -\- ^c\ u 4 = u a ^b gc. By substituting in the formula for 'continued averages above, we readily find 8 = u. It thus appears that if the true speed remains constant, and if the tidal influence varies as a quadratic function of the 372 RESISTANCE AND PROPULSION OF SHIPS. time, and if the time intervals are equal, this method will give the true speed. In the actual case, however, not one of these conditions will in general be fulfilled. It is therefore doubtful if the result of four runs thus reduced is any more accurate than the simple average, or than the average of two runs made with as small a time interval as possible. It may also be noted that when the tide is just on the turn either way the water is frequently affected by eddies and counter-currents, which though small in velocity are so confused in distribution as to defy any attempt at elimination. When, however, the tide is at about half-ebb or half-flow, the velocity is greater but quite regular. This results naturally from the sinusoidal character of tidal motion, as a result of which the change of velocity is least when the velocity is greatest at about half tide, and the change is greatest when the velocity is o at full tide or slack water. It follows that half tide with a clearly defined regular tidal velocity uniformly distributed over the course is in general preferable to slack or high water with the accompanying eddies and irregularities. Further methods of dealing with tidal influences will be found in the next section. 74. SPEED TRIALS FOR THE PURPOSE OF OBTAINING A CON- TINUOUS RELATION BETWEEN SPEED, REVOLUTIONS, AND POWER. i We have thus far been concerned with the actual true speed attained, without reference to the corresponding num- ber of revolutions or the I.H.P. necessary. For the investi- gation of the propulsive performance, however, it is quite desirable to obtain sufficient data to furnish a continuous rela- tion between these three quantities. For this purpose the TRIAL TRIPS. 373 short-course trial alone is suitable. The actual conduct of the trial is in no wise different from that already described, but it is continued through a decreasing series of speeds with their appropriate revolutions and powers, to a number of revolutions as low as can be maintained uniformly by the engines. The problem of the proper disposition of the data thus found is more complicated than when speed alone is desired. We now desire .to determine not only the true speed, but also the revolutions and I.H.P. which correspond to it. Now in practice it is found difficult, even for any one run, to maintain the revolutions and power constant. With care the variation may be slight, but at the best the question may always arise as to the proper interpretation of the data observed. Let //,, N tt u l be the power, revolutions, and true speed for the first run, and //,, N tt u % those for the second, v being the tidal influence taken as constant for the two runs. Then the actual observations furnish us with ff l9 N 19 (, + *); //, , N t , fa v). For two runs intended to be at the same speed, the varia- tion will be so slight that we may properly assume the slip of the propeller to be the same for both. As a result the true speed will vary directly with the revolutions, and the true mean speed will correspond to the mean number of revolu- tions. That is, , - corresponds to 374 RESISTANCE AND PROPULSION OF SHIPS. Now with power we have in general H=Bu n ........ (i) where B and n depend alike on the ship and on the speed. Where the variation in speed is not large, we may safely take the variation in B and n as negligible. In the present case, therefore, we should have Whence Jf, = Bu,\ ' A U * + " H * + H * and 2T and *^_^ = (_ _L_ L-; . . . . . ( 2 ) The left-hand member of this equation is in the same general form as (i) above, and it must therefore give the value of the power which corresponds to the true mean speed. The relation of this to the observed powers H^ and //, is then shown by the right-hand side of (2) above. The exponent n will usually be found between 3 and 4. As an illustration let I = 3200; I 3600. TRIAL TRIPS. Then by substitution we find 375 LJ H \ TJ~ n \n ^4^-) = 3396 - The value by simple average would be 3400, and as the uncertainties in the measurement of power are relatively far greater than the difference thus resulting, and as these values of HI and H t represent a variation greater than would be likely to occur under the conditions assumed, it seems fair to conclude that in such case we may properly take the mean REVOLUTIONS FIG. 96. power as corresponding to the mean speed. For two runs thus made we may take, therefore, the mean revolutions, mean speed, and mean power, as all corresponding. The next pair of runs being made at a lower power, another set of means is found, and so for the entire series. 376 RESISTANCE AND PROPULSION OF SHIPS. Still otherwise, we may effect the general averaging by graphical process as follows: Let power be plotted on revolutions as in Fig. 96, each pair of spots corresponding to a double run intended to be made at the same power. Then a fair curve passing through and between the spots may be taken as the best approxi- mation to the continuous relation between revolutions and power. We may then plot the mean of the speeds as an ordinate, on the mean of the revolutions as an abscissa, as in OB, Fig. 97, thus giving a continuous relation between speed and revolutions. A straight line OA shows similarly % O REVOLUTIONS FIG. 97. the speed if the apparent slip were o. Hence the vertical intercept between OA and OB shows the corresponding loss in speed at each point. From these diagrams we may then obtain and plot if desired a curve showing the continuous relation between power and speed. TRIAL TRIPS. 377 We may also derive the relation between revolutions and speed graphically by plotting each separate speed on revolu- tions as an abscissa, thus obtaining two curves, one for speeds with and the other for speeds against the tide, with pairs of points at nearly the same revolutions on each. A mean curve will then give the desired relation between revolutions and speed. A slightly different mode of graphically determining the relation between revolutions and true speed is that due to Mr. D. W. Taylor,* which is briefly as follows: The course is steamed over back and forth, going over each time the same route off as well as on the course, so that the time intervals between the middle points of the runs may vary inversely as the speeds. For each successive run the revolutions are decreased by as nearly as possible an equal decrement. This gives a series of runs at certain revolutions with the tide, and another series with other revolutions against the tide. Each of these is then plotted, giving curves of speeds plotted on revolutions, with and against the tide. A mean curve is then taken as the curve of true speed. The author then shows by means of an illustrative example that with a fulfilment of the above conditions well within practi- cable limits a variable tidal influence will be eliminated with quite sufficient accuracy for all practical purposes. The advantage claimed for this method is that a given range of speed can be satisfactorily covered with a smaller number of runs than when they are made in pairs as pre- viously described. * Journal Am. Soc. of Naval Engineers, vol. iv. p. 587. 3/8 RESISTANCE AND PROPULSION OF SHIPS. 75. RELATION BETWEEN SPEED POWER AND REVOLUTIONS FOR A LONG-DISTANCE TRIAL. While the intention may be to carry on such a trial at uniform power, in any actual case there will naturally arise considerable variation in the various quantities involved. The mean speed and the mean revolutions are readily found, and with any probable amount of variation we may presum- ably consider them as corresponding the one to the other. With regard to the power which corresponds to such mean speed or revolutions, however, there is more room for uncer- tainty. The data available will be a series of indicator-cards taken at intervals throughout the trial, and giving presumably the mean effective pressures in the cylinders at the instant in question. The revolutions are also taken at the same time, and thus each observation gives the data for determining a value of the power at that instant. The mean of these revo- lutions, however, will not be the true mean, due to individual errors of observation, and to the fact that they are merely a series of detached values of N, while the true mean is obtained from the continuous counter giving the total number for the whole run. On this account, therefore, and for finding the mean power, it seems preferable to use the mean effective pressures from the cards, and the true mean revdlutions from the total counter. The method for doing this we may develop as follows: In any given propeller it is readily seen that the relation between the mean effective pressure in the cylinders, or more definitely between the mean effective pressure reduced to the 1. p. cylinder ( 47), and the thrust of the propeller or resistance of the ship, must be very nearly TRIAL TRIPS. 379 constant except for very widely varying conditions. We may therefore properly assume that the reduced mean effective pressure will vary very nearly as the resistance. Denoting these pressures by / 1} /> etc., we may, with all significant accuracy, assume them expressible in the form /, bN*> etc. Hence, extending the method outlined in 73 for mean power, we should here use for the final mean pressure / the square of the mean square roots of the various values /!, /> 2 , etc. This pressure used with the true mean revolu- tions would then give an approximation to the corresponding power having all significant accuracy, at least so far as obtainable from the data at hand. A still greater refinement, but usually without practical value, would be the substitution for 2 of an index to be derived from a special examination of the relation between speed or revolutions and power. If the variation in the power is not unusually large, the square of the mean square root will not sensibly differ from the simple arithmetical mean, and in such case the latter may properly be used. As an illustration of the amount of the variation, let three values of/ be denoted by 64, 81, and 100. Then the square of the mean square root is 81, and the arithmetical mean is 81.66, or somewhat less than one per cent in excess. We therefore conclude that in all usual cases we may, for the power corresponding to the mean revolutions or speed, use the simple arithmetical mean of the pressures, as above, with the mean revolutions. At the same time the substitu- tion of the square of the mean square root for the arithmeti- cal mean will undoubtedly give a more correct result, and if the difference is significant, the latter method should be used. 380 RESISTANCE AND PROPULSION OF SHIPS. 76. LONG-COURSE TRIAL WITH STANDARDIZED SCREW. We have thus far in all cases assumed the measurement of distance to be effected by means independent of the ship's propeller. A method of long-course trial has been used to some extent, however, by means of which the propeller itself, having been standardized, is made the instrument for the measurement of the distance. This is known as the standardized screw method, and has been used in certain government trials at the suggestion of the Bureau of Steam Engineering of the Navy Department. The ship is first taken on the measured mile and the con- tinuous relation between speed and revolutions as discussed in 74 is carefully determined. The ship being in the same condition as regards draft, trim, condition of bottom, etc., then puts to sea in smooth water and calm weather, and runs such a distance or time as may be desired. For the best accuracy readings of the counter should be taken every ten or fifteen minutes. In any event the total time and total revolu- tions are known. Indicator-cards are, of course, taken at appropriate intervals throughout the run. Now if the revolutions have been uniform, or if they have varied somewhat, but not out of a range within which the apparent slip may be taken as sensibly constant, then, as in 74, the true mean speed and the mean revolutions will correspond, and the former may be determined directly from the latter by the aid of the curve of standardization. If in an extreme case the revolutions are widely variable and the slip is widely variable with revolutions, this correspondence will not be accurate. The cause of the error and a method for its elimination may be seen as follows: TRIAL TRIPS. 381 Let OX, Fig. 98, be a time abscissa on which are laid off as ordinates the successive numbers of revolutions per minute at the instants at which they are taken. A smooth curve AB drawn through the points thus found gives a continuous rela- tion between time and revolutions. Then from the curve of standardization the speed corresponding to each number of revolutions may be found. These being plotted at the same points will give a curve CD, showing the continuous relation between time and speed. The area OCDX being found by planimeter or by approximate integration, the result will be FIG. 98. proportional to the entire distance, and the time being known, the true mean speed is thus determined. If now the apparent slip were constant, or OB, Fig. 104, a straight line, the curves AB and CD would be similar and the integration of AB would serve the same purpose as that of CD. But the integration of AB is furnished by the revolution counter, and hence the mean revolutions thus determined would with constant slip give the true distance or true speed. Again, if the revolutions 382 RESISTANCE AND PROPULSION OF SHIPS. are constant, it is evident that both AB and CD become straight lines parallel to OX, and their constant ratio would be that between speed and revolutions given by Fig. 97, for i the particular value of the latter. If these conditions are not fulfilled, that is, if revolutions vary with time, and slip with revolutions, then the curves AB and CD would no longer be similar, and the use of the former, or, in other words, of the mean revolutions, would no longer be theoretically exact. With the usual amount of variation in these quantities, however, the error will be negligible 'and the mean revolutions may be used. The treatment of the power has already been considered in 74. As some of the advantages of this method mention may be made of the following: (a) There are no tidal corrections to consider. This arises from the fact that by means of the standardization the pro- peller is made the instrument for measuring distance. (&) The ship under trial is independent of the presence of other vessels for course-markers and for tidal observations. (c) The results are not invalidated by departure from a straight course. Any course desired may be followed so long as it is not a curve of small enough radius to give rise to a sensible retardation. (d) The results attained are constantly known during the trial, so that the exact status of the ship relative to the expected performance is known from beginning to end. (e) Any derangement to the machinery which might render a trial over a definite course not only inconclusive, but also devoid of any return of valuable data, will have no effect TRIAL TRIPS. 383 on the result, since the trial may be considered as ended at the expiration of any complete ten-minute interval, and whether the results are considered satisfactory or not, at least they are known, and valuable data are obtained. On the other hand the limitation of this mode of trial so far as relates to similarity of conditions between the standard- ization trial and regular trial must be carefully noted. The apparent slip will vary with the condition of the bottom, and with the draft, trim, and condition of the wind and sea; and since calm weather and a smooth sea are the only conditions which can be definitely described, it is necessary that both trials should be made substantially under such conditions. 77. THE INFLUENCE OF ACCELERATION AND RETARDATION ON TRIAL-TRIP DATA. If the revolutions and power or mean effective pressure are subject to rapid fluctuations, they will no longer have the same relation to speed as for uniform conditions. If the pressure rapidly increases, a portion of the resultant increase of thrust and work done goes toward accelerating the motion of the ship, and the mean pressure, revolutions, and power will all be greater than for the existing speed under uniform conditions. Similarly with a rapidly decreasing pressure the mean pressure, revolutions, and power will be less than for the existing speed under uniform conditions. The variation of the pressure is usually not sufficiently rapid to make this item of practical importance, though the possibility of its existence as a disturbing feature may be borne in mind. It could only be eliminated by a knowledge of the vary- 384 RESISTANCE AND PROPULSION OF SHIPS. ing acceleration, an experimental determination of which would be a matter of no small difficulty. A further problem involving somewhat similar considerations is that concerned with the time and distance necessary to effect a change of speed either of increase or decrease. This is of such general interest and importance in connection with the subject- matter of the present chapter that we may properly turn to its consideration at this point. 78. THE TIME AND DISTANCE REQUIRED TO EFFECT A CHANGE OF SPEED. Suppose the mean effective pressure in the cylinders to be dependent alone on the points of cut-off. With any fixed condition of these, therefore, and neglecting variations due to friction, the engine-turning moment will be constant, and hence its ^equivalent, the moment resisting the transverse motion of the propeller. Again, assuming that the distribu- tion of pressure over the surface of the propeller is sensibly the same for constant turning moment no matter what the speed of the ship, it follows that the relation between the actual thrust and the mean effective pressure is geometrical, and hence if the former remains constant, so will the latter. While these suppositions may not be quite exact, they must in any case be very near the truth. Suppose, therefore, the ship being at rest, that steam of a given pressure is turned on the engine with the cut-offs in such position that their joint result would give a mean effective pressure sufficient to give a thrust which would maintain the ship at a speed of u, knots. In consequence the ship will gradually gather headway and after a time attain sensibly the speed u lt during which a dis- TRIAL TRIPS. 385 tance s will have been traversed. In thus considering thrust and resistance it is evident that actual thrust and actual or augmented and not true or towed resistance must be taken. Let R l denote the resistance at speed u l and in general R that at speed u. We shall have then a constant thrust equal to R l acting against a variable hydraulic resistance R. The difference R l R will be a net force available for the accelera- tion of the ship. We must also remember that a certain amount of water is more or less closely associated with the motion of the ship, and partakes of its acceleration and retar- dation. We may therefore use i*D instead of D. We have then du _g(R l -R) df~ ~~ In the fundamental formulae of dynamics R is measured in pounds force and D in pounds mass. As their ratio is unchanged by a change in the unit, they may both be meas- ured in tons. We have therefore, as the units in this equa- tion, feet, seconds, and tons. Now represent R in the gen- eral form R = cD*u\ ...... (2) in which c will be later defined so that R and D may be in tons and u in feet per second or knots per hour. In this equation c is not necessarily a constant. This will provide therefore for any actual law of augmented resistance. We have then, from (i), 386 RESISTANCE AND PROPULSION OF SHIPS. ds Dividing by -;- = //, we have du __gc(u? - ') * = ~^u~ Hence from (3) and (4) du dt = - ...... (5) gc u* u* VD* udu ds- -- j ...... (6) gc ?// u If now to simplify integration we assume c a constant, we have, reckoning t and s from starting, From (7) and (8) it appears that when u = u lt both t and ^ become oo . This is simply a result of the form of the equations, and indicates that under the conditions assumed, the velocity would go on indefinitely increasing, but would never actually attain the amount u r This is the natural and du necessary result of the decreasing value of -j- as given in (3) when u approaches u lt and while it may be a suprising result, it is in exact accord with the physical conditions assumed. In the actual case, the mean effective pressure is never exactly constant, and hence the corresponding velocity u^ is con- stantly variable about a mean value. The important point practically is not how long or how far it will take to acquire TRIAL TRIPS. 387 the velocity u lt but rather a velocity sensibly equal to it, as for example .99^, or .999^. These relations are illustrated TF; 10,000 8, 7 2 9,000 8,000 7,000 6,000 5,000 g z < 4,000 3,000 2,000 1,000 100 500 600 300 300 400 TIME IN SECONDS FIG. 99. Time and Distance for gaining Speed. in Fig. 99, where values of (7) and (8) are plotted for succes- sive values of u. We will now explain the reduction of the equations to a form suitable for computation. 388 RESISTANCE AND PROPULSION OF SHIPS. Let Kbe the Admiralty displacement coefficient as defined in 62, and ^ the ratio between the thrust horse-power T.H.P. and the I.H.P. Then we have and T.H.P. = , . K. where the units are the horse-power, ton, and knot. To reduce to the units, foot-ton per minute, ton, and foot per second, we have eD%u * x 33000 and R = KX 2240 X (1.689)"' '' X 33000 KX 2240 X (1.689)' X 6o ^ Comparing this with the general form for R in (2), it is evident that c = .05 le -f- K. This may be taken as the defin- ing equation for c such that when substituted in (2) it will give R in tons when D is in tons and u in feet per second. Sub- stituting this value of c in (7) and (8) we should then have the values of t in seconds and s in feet, expressed in terms of D in tons and u in feet per second. While retaining the second and foot as units for t and s, it will be more conven- ient to introduce on the right-hand side of the equations the necessary factors, so that common logarithms may be used instead of hyperbolic, and speed may be measured in knots instead of feet per second. Making these substitutions in (7) and (8), and putting /* = i.i, we find , , t (sec.) -.457 log ; ... (9) (feet) = .772 log ._.. . . .(10) TRIAL TRIPS. 389 For illustration take K = 240 and K -r- e = 400, D 5000, w, = 15. Then we have * = 208.4 log ..... (ii) 225 and * = 5280 log _ ..... (12) The numerical values of (u) and (12) are plotted in Fig. 99, as above referred to. The curve AB giving the time is asymptotic to the 15 -knot line, thus showing geometrically the indefinite approach of the speed to the 15 knots as a limit. The curve CD shows likewise the relation between time and distance, approaching more and more nearly a straight line in form as the 15-knot speed is gradually approximated to. We will now suppose the vessel going ahead under uniform conditions at speed u lt and that the engine is suddenly reversed and driven with full mean effective pressure astern. In such case we shall have a different geometrical relation between the longitudinal force due to the screw and the mean effective pressure, than for motion ahead. In general, how- ever, we shall have in such case du_ g(P+R) dt ' where P = backward pull due to engine, and R = resistance to velocity of ship forward. We will now assume, similar to (i), R = cflu\ and P cTPu 390 RESISTANCE AND PROPULSION OF SHIPS. where for P, u, is the speed at which the given mean pressure would drive the ship ahead. We have then for (13) du g di = whence <& = - ^ ^T^TT^ (14) , _ r-^ udu g cji* -\-crf' Integrating in the usual manner, we have In selecting values of c^ and c^ we may bear in mind the following considerations: The coefficient , is less than the c for the go-ahead condition due to the rounded backs of the blades, and the consequent decreased longitudinal component of the distributed surface pressures. For a given mean pres- sure, therefore, the pull would be less than the thrust. As to the ratio between the two, definite information is lacking; and in default of such definite data we will take the pull as .8 the thrust. Again, the value of c 1 corresponds to the actual resistance of the ship to motion ahead when the propeller is backing. In this condition the latter is sending a stream of water forward against the stern, thereby producing an excess of pressure at this point instead of a defect as when working ahead. The actual resistance to motion ahead is therefore TRIAL TRIPS. 391 decreased instead of increased, or the augmentation may be considered as negative. The ratio between c t and the c for motion ahead at the same speed would be equal therefore to that between the true resistance minus this decrement, and the true resistance plus the usual augmentation. Taking the augmentation and decrement about the same, this ratio would be not far from .8, and we will therefore take c^ as equal to c iu Comparing also with the value of c, we have Taking jj. = i.i, substituting these various values, and reducing speed to knots and hyperbolic to common loga- rithms we have DK f l *\ ' = 493 (-785- t-'-); . . . ( l8 ) & K \ I ( u \\~\ s = -959-7- [-30103 - log ^1 + (-) JJ. . (19) If u = o, we have simply &K , = . 38 7_ ; . ....... (20) D-K * = .289 = .747*,* ..... (21) With the same numerical values as before, we have t=22$ (.785 - tan- ^); ..... (22) = 6s6o[. 30103- log (i + ()')] ; . (23) s and if u = o, /= 177, s= 1975. 39 2 RESISTANCE AND PROPULSION OF SHIPS. The varying values of / and s for any speed u are given in Fig. 100, which is thus a diagram of the extinction of forward JSWU 300 ^ ^ " B / ' 190 / / 180 ^/ / jcnn ^ / / fi 7 / 150 < & / 14U -ion 1 / <. 7 s>/ LO s 1000 / 4 / ri / / z on U ui L_ Z / / yu _ P bJ / / / < fe / / 70 Q crin / / / 60 / X / 50 / / 7 / / ^ ou / / 20 15 14 13 13 11 10 3 987 SPEED TIME AND DISTANCE FOR STOPPING FROM FULL SPEED FIG. ioo. elocity. AB gives the relation between speed and distance, ;d CD that between speed and time. For simplicity of operation we have assumed that the TRIAL TRIPS. 393 augmented resistance varies throughout as the square of the speed. This, of course, is far from being actually the case. The ratio K -r- e may be expected to vary considerably in the case of a vessel gaining speed from rest or stopping from full speed. If the law of variation of K -- e were known that is, the law of the variation of the actual resistance with speed, both with engines turning ahead and backing-^-the funda- mental equations could readily be solved by approximate or graphical integration. The difference in the result, however, would be slight, and would not change the general nature of the results obtained by the substitution of the constant value of K -i- e. It may be remarked, however, that in the selec- tion of a value of K a mean rather than maximum value should be taken. An interesting conclusion from (21) is that the distance in which a vessel may be brought to rest from motion ahead is sensibly independent of the speed, and depends only on the 7 size and the ratio , or if e is taken as practically constant, on D and K. The reason why there must actually be a ten- dency toward some such uniformity in the distance run may readily be seen. In the example taken this distance would be from five to six lengths. In the case of very full or poorly formed vessels, or with foul bottom, we might have an average value of K much less than 240. As an illustration, let K = 1 80 and K - e = 300. Then the distance traversed in such case would be only three fourths of that under previous con- ditions, and we should have for the same size of ship s = 1481. 394 RESISTANCE AND PROPULSION OF SHIPS. These figures agree with the general results of the experi- ments reported to the British Association,* by which it was found that ships could usually be stopped in from 4 to 6 lengths, and that this distance seemed practically independent of the power, or in other words, of the speed at which the experiment was made. We have assumed in all the foregoing that the engine is instantly reversed, and that the engine instantly gathers its motion either ahead or back, as the case may be. This is not actually possible, so that due allowance should be made in comparing experimental results with those given by the fore- going approximate theory. In any case the general nature of these results is correct, and it is readily seen that they have an important bearing on the manoeuvring of vessels and on the proper conditions for trial trips. In measured-mile trials especially the importance is clearly shown of a long start in order that the ship may attain sensibly her maximum speed before entering upon the course. 79. THE GEOMETRICAL ANALYSIS OF TRIAL DATA. Considering the three items of data, power, speed, and revolutions, we may plot the former on either of the latter as abscissa giving curves as in Figs. 101, 102, or we may plot speed on revolutions giving a curve as in Fig. 103. The various uses of these curves need no especial explanation. In using the power-speed curve for purposes of comparison, 61, however, it must be remembered that the given curve strictly applies only to the ship in the given condition of dis- * Reports for 1878, p. 421. TRIAL TRIPS. 395 placement, trim, condition of bottom, and of weather and water, and that it cannot be applied to the same ship for all SPEED FIG. 101. REVOLUTIONS FIG. io2. circumstances without due allowance having been made for such differences in condition as may exist. It is frequently of interest to know the index of the speed with which the power varies at any given speed, and similarly 396 RESISTANCE AND PROPULSION OF SHIPS. for the variation of resistance with speed, or either with revo- lutions, or in fact of any quantity which may be expressed in the general form y = ax n (i) For example, we may determine that at 10 knots the power varies as the cube of the speed and the resistance as u REVOLUTIONS FIG. 103. the square, while at 15 knots the power may vary as the index 3.7 and the resistance therefore as the index 2.7. We will now develop methods of deriving the value of this index. Let x l be the given value of the abscissa. Then the usual method has been to assume the index sensibly constant for a short range on either side of this value, say between the values x<> and x^ equidistant above and below x^ Then from the general equation we have TRIAL TRIPS. 397 whence ^ = and n = log , + log = Jo ^^o 2 - When, however, the curve is at hand showing the relation between y and x as is usually the case, the following geometrical method will be found much more rapid and satis- factory, besides being exact in principle for a continually varying value of n. We must first obtain a clear idea of the nature of the index with which we are here concerned. Let it be a question of the variation of any function y with x. Let the value at x^ be y 9 . Then let x be increased by a small increment such that its new value divided by its old is (i + e), where e is a quantity very small compared to i. This will result in a change in the value of y in a ratio which we may denote by (i +/"). Now obviously we may put (I +,)- = !+/). or = ...... (3) log (i + e) This we take as defining the nature of the exponent m with which we are here concerned. It would therefore follow that if m = 3 and the speed were increased i per cent it would result in an increase of the power in the ratio (i.oi) 3 . So long as the amount of increase is indefinite, the char- acter of our exponent will lack mathematical distinctness, and we are therefore naturally led to define m as the ratio of log (i +/) to log (i -\- e) when e is indefinitely decreased. 398 RESISTANCE AND PROPULSION OF SHIPS. Denoting the increments by dx and efy, we have then by defi- nition (4) By Maclaurin's theorem of expansion this becomes dy m j_ = or m = the quotient of the abscissa by the subtangent. In this determination the only uncertainty is that involved in drawing a tangent to a curve. This may, however, be done quite accurately as follows: We may evidently assume without sensible error that the curve in the vicinity of A may be considered as parabolic of the second degree. We then take two points E and F equidistant from B and note the corresponding points 5 and R on the curve. Then by a well- known property of such curves a line drawn through A parallel to RS will be tangent to the curve at A. The exponent n, which is essentially different in character from m, may be defined simply as the exponent which will satisfy the given fundamental equation y = ax n . Its value is then most readily found by taking logarithms, whence log x It may be readily seen that the value of n will depend on the particular unit used for x, but once this unit defined, the value of n becomes definite. Let x = i, or the unit. Then we have a=*y, or the value of the constant a = the ordinate for x = I. We may therefore put more generally y = y^x n . The exponent m may be seen to relate simply to the nature of the law at the point in question. We may therefore term it the index of instantaneous variation. It must neces- TRIAL TRIPS. 4OI sarily be the exponent involved when we say that at any point y varies as a certain power of x. Qn the other hand, the exponent n depends on the entire law of variation from the point where x = I to the point in question. It therefore represents the resultant effect, between these points, of the 10 entire series of values of m. If the law is such that m is con- stant, m and n become identical as in the common algebraic curves. The distinction between m and n for such curves as we are here concerned with must not be forgotten. The application of the preceding to the variation of power 402 RESISTANCE AND PROPULSION OF SHIPS. or resistance with speed is obvious. In Fig. 105 are given examples in which AB is the given fundamental curve, CD is the locus of the exponent m, and CE is that of the exponent n, where one mile is the unit. These diagrams are self- explanatory and will repay careful examination. Evidently the same investigation may be made with reference to the power and revolution curve. The speed and revolution curve gives the apparent slip, which may also, if desired, be plotted separately as AB, Fig. 106. If OA, Fig. 97, were a straight SLIP IN PER CENT o S 8 8 ^ ,B A . . -^ 5 10 15 SPEED IN KNOTS FIG. 106. line the apparent slip would be constant, AB would be hori- zontal, and the indices relating power to revolutions and speed would be the same. Practically this is never found, though frequently the slip will vary but slightly through a considerable range. We have in 58 given a method of finding the mean effective pressure corresponding to the initial friction of the engine. We will now give another due to Wm. Froude. In Fig. 107 let AB be a curve of mean pressure, propor- tional, as we have seen in 47, to the curve of indicated t TRIAL TRIPS. 403 thrust. Now if we assume that this pressure varies as the resistance, and the latter for speeds below tt l as speed with the constant index n, we have only to continue BA by a curve tangent to it at A and with reference to some at present unknown axis EX, fulfilling the condition that m = n at all points. Froude took m = 1.87, and the construction is readily seen to be an immediate result of the general method established above for the determination of m. The construction is as follows : At A draw a tangent to the curve. SPEED FIG. 107. Lay off u^C = u^O -r- m, erect a perpendicular CD, and draw EX. Then a curve EA run in tangent at E to EX and at A to AD will approximately fulfil the conditions required for the continuation of the pressure curve downward, and OE will therefore be the value of /> , or the value for the initial fric- tion. Due to the uncertainties of drawing a tangent at the end of a curve and the uncertainty that the pressure varies as the index 1.87, or indeed with any other constant index of the speed, the final result necessarily partakes of the uncer- 404 RESISTANCE AND PROPULSION OF SHIPS. tainty inherent in all attempts to extend a graphical law beyond the range covered by the observations. 80. APPLICATION OF LOGARITHMIC CROSS-SECTION PAPER. In logarithmic cross-section paper, the divisions are similar to those on a slide-rule, or in other words, the distances from 8 9 10 the origin are proportional to the logarithms of the numbers, as in Fig. 108. Cross-section paper ruled in this way has so TRIAL TRIPS. 405 many special applications to various problems in naval archi- tecture, that a brief description of the more important may be of interest. Given an equation y = ax*. Whence logj = log a + n log x. Now remembering that on this paper the abscissa and ordinate are not x and y, but log x and log y, the above equation is really ordinate = constant -f- n times abscissa. This for usual plotting is in the form y = Ax + B, and hence represents a straight line. It follows that any such equation will on this paper be represented by a straight line inclined to X at an angle whose tangent is n, and cutting Fat a point which gives a. This fundamental property has a number of important applications, of which we may note the following: (a) If a = i, we have y = x n . Hence a straight line on this paper will give a locus of roots or powers, of index whole or fractional, to determine which it is simply necessary to draw a line or a series of lines at an angle tan' 1 ;/ to X. As commonly made, this paper has but a single logarith- mic scale from I to 10, subdivided according to the size of the unit. By drawing on this, however, a number of lines dependent on the nature of the index, a single sheet may be made to give the ;/th power or root of any number from o to oo . For illustration we will take the case of a table of * 3 406 RESISTANCE AND PROPULSION OF SHIPS. powers. That is, we wish to express graphically the locus of the equation y x*. Let Fig. 109 denote a square cross- sectioned with logarithmic scales as described. Suppose that G FIG. 109 there were joined to it and to each other on the right and above, an indefinite series of such squares similarly divided. Then considering in passing from one square to an adjacent one to the right or above, that the unit becomes of the next higher order, it is evident that such a series of squares would, with the proper variation of the unit, represent all values of either x or y between o and oo . Suppose next the original square divided on the horizontal edge into 2 parts and on the vertical into 3, the points of division being at A, B, C, E, F y G. Then lines joining these points as shown in the diagram will be at an inclination to the horizontal whose tangent is f . Now beginning at O, OE will give the values of x* for values of x from I to 10. For greater values of x the line would run into the next adjacent square, and the location of this line, if continued, may be seen to be exactly similar to that of BC in the square before us. TRIAL TRIPS. 407 It follows that, considering the units as of the next nigher order, the line BC will give values of x* for x between O and G or 10 and 3i.6-|~ For larger values of x we should run into the adjacent square above with change of unit for jj/, but without change for x. We should here traverse a line similar to GF. Therefore by proper choice of units we may use GF for values of x* where x lies between 31.6+ and 100. We should then run into the next square on the right requiring the unit for x to be of the next higher order, and traverse a line similar to AD, which takes us finally to the opposite corner and completes the cycle, the last line giving us values of x% for x between 100 and 1000. Following this, the same series of lines would result for numbers of succeeding orders. A little consideration of the subject will show that the value of x* for any value of x between I and oo may thus be read approximately from one or another of these lines; and if for any value between I and oo , then likewise for any value between o and I. The location of the decimal point is readily found by a little attention to the numbers involved. A rule for its location might be derived, but is of little additional value in practice. The limiting values of x for any given line may be marked on it, thus enabling a proper choice to be readily made. The principles involved in this case may be readily ex- tended to any other, and it will be found in general that if the exponent be represented by , the complete set of lines may be drawn by dividing one side of the square into m and the other into n parts, and joining the points of divi- sion as in Fig. 109. In all there will be (m-\-n i) lines, and opposite to any point on X there will be n lines correspond- 408 RESISTANCE AND PROPULSION OF SHIPS. ing to the n different beginnings of the nth root of the mth power, while opposite to any point on Y will be m lines corresponding to the m different beginnings of the mth root of the nth power. Where the complete number of lines would be quite large, it is usually unnecessary to draw them all, and the number may be limited to those necessary to cover the needed range in the values of x. If, instead of the equation y = x n , we have a constant term as a multiplier, giving an equation in the more general m form y = Bx n , or Bx n , there will be the same number of lines and at the same inclination, but all shifted vertically through a distance equal to log B. If, therefore, we start on the axis of Fat the point B, we may draw in the same series of lines and in a similar, manner. It will be noted, of course, that the index m -f- n may be used either way for the same set of lines. That is, the same lines will give the f or f power of numbers, or the second or \ power, etc. (b) If in the general equation y = ax n the exponent n is variable, the locus plotted on logarithmic paper will be curved instead of straight, and the index of instantaneous variation m, at any given point ( 79), will be the tangent of the incli- nation of the tangent line at such point. This is readily seen from 79, (6). Also, n will be the tangent of the inclination of a line drawn from the point where the locus cuts Y (or where x i) to the given point. This readily follows from 79, (9). This serves also to clearly illustrate the difference in character between m and n. (c) Proportions of the form TRIAL TRIPS. are frequently met with. As an equation this is 409 or log j, = log y, + n(\og x 9 log x v ). Now in Fig. no let A and B denote the points x^ and x^ and C and D be at the heights denoted by y\ and y^ Then, remembering that the actual distances involved are the logarithms of the corresponding quantities, we readily see that BD = AC + nAB or AC + . Therefore, a line from (7 to D will be at an inclination to X whose tangent is n. If, FIG. no. therefore, we have any ready means of passing from C, a point determined by the coordinates x l y 1 , along an oblique line inclined to the axis of X at an angle whose tangent is n, it is evident that we shall pass along a continuous series of points x^y^ so related to x^y^ that the proportion above men- tioned will be fulfilled. To solve any such proportion, there- fore, we have simply to start at the given point, x l y lt and pass along the oblique line until we reach a point whose abscissa is x t . The corresponding ordinate will be the desired 410 RESISTANCE AND PROPULSION OF SHIPS. value of y^ Or, vice versa, if we stop at any given value of the ordinate y the corresponding abscissa will be x^, so related to the other quantities that the proportion is fulfilled. To provide the necessary means for moving in the right direction from any point whatever as x^y^, a series of equi- distant lines at the proper angle may be ruled. With this aid a very close approximation to the proper values of x^y^ may be made. As an instance of the use of this proportion, suppose that we wish to find the corresponding speeds for two vessels from the sixth roots of the displacements. We have - = Given the u l an^ D l we find the point, considering one axis as that of speed and the other as that of displacement. Then by passing along a line inclined at an angle tan' 1 \ to the axis of D, we shall pass through a continuous series of points which will give corresponding displacements and speeds. Stopping at the displacement Z> 2 we read off directly the desired speed # ; or, vice versa, stopping at a speed z/ 3 we read off the corresponding displacement Z> 2 . Having thus found the corresponding speeds, we may by another application of the same principle find the I.H.P. by using, for example, the equation ff. (D.\* As a further illustration we may take the determination of Admiralty coefficients from the equation z?V ~~~- All of the operations here involved may be carried out on logarithmic paper and the value of the coefficients rapidly determined. STEAMSHIP AND PROPELLER DATA. IN Tables A and B will be found a collection of data relating to ship and propeller performance. A few of the cases given in Table B are not represented in A due to the difficulty of obtaining satisfactory values of some of the data required. In Table B it should be noted that the value of the I.H.P. is that required for one propeller, and not that required for the ship as a whole in the case of twin or triple screws. The data presented in these tables have been gathered from various sources, as near as possible in all cases to the origi- nal publication. All ratios and derivative quantities have, furthermore, been independently computed from the funda- mental data, and while it can hardly be hoped that among so many figures all errors have been avoided, it is hoped that their number is small and their character not significant. In Table B, column headed Screw, the letters have the following significance: C center S starboard P , port M mean 4" 412 STEAMSHIP AND PROPELLER DATA. ID en n co o en I*** t** M o^ 0*0 o* en r^ o^ ""> ^ >H D en O O O o en >O D ""> too O *D t^O O O u-> u-> 1-g.S co co O ^t O i* O mo vr> O co i/^ ID ID r^ ID co ID r^ o *^ w w r^Ococoooo CT>cor^OOr> O O^O SI3 1J enOi^Nr^-^-w O Oi^- r^oo tDQenenOOO^QOOOOr^ Tt o O moo -to HH o O co t^co OcoW'^-O^MCoqrtr^qCJq C^C4>-< OMMC4I-I 8OOOOQOOOO OtHiDiDOOOOO O WOOOCO N O IDO W TfNOCOCO MCOOO 't rt f^ N DrfO en ^f D T-J- TJ- cno ^ en rfo co ^f ^ ^r w iDiDcni-H rj-cnenrrenM OC^iDVDr^MOcooOOOw^C^MMencococnt^OOOO an w eooo to ^ M * 9 | ; a h c C. > Ui i a E ^~ >- i vX- c ^ % ^- V I i- ^- g r? la i !. S-s c C G c W Ui " ' J3 !i c > ^4 c C I 5 n 1 5 c ^ 8 t? ^2 w< ^ rt rt X3 ^3 t 6 3 fc 5.S C : : 4- i'c 1 in en ^ * >. c w p TABLE A. 413 w oo c* r^^-o O O O O *t o r~ eno vr>O O O O to ir>O *^O oo o t^ '*'*''3'O>-' O u-> ir>O O O O tr>O en M co co o to S-cg lS-8 a ^i oo rococo r-^ r^ en "tf* * to o O "} tr>o i^O O >-i r^M MO -i o^ O <* N too TJ- en *l-co u->co -t-^-or^ino OO r^o^ O^oo r- t^ o> O^oo oo o^ O^oo r^co r-^ r^oo o r^oo t^oo oo oo B a MO r>t-M r^o O Ooo >H 1-1 enr^u^xntnenoo ^- o >-i TI-N woo T}- ^- o w OOO w N N w OOO ot^i-iO'^- i ^-O enNcn-^- 2S NOMOtnenOOt^-^-coOQ'oOooooOooOcnOOooOenO Oeno^woejqcoqqt-'qoo Ooo oo t^r^q *? "'f ^ O ""* "f 9. 8OOOOOootr> enoo loor^^Qoo O O enenO Otno ^ O toqqqoooctNooooooooooooenoowooo Tt to en a ^- en ^v qqqtnqqtnqqqq tno N N rtco qqqqqMqqqq dNco'Tt-cn6piiooddQNod6o6 i ^-o'doo O O OO O t"* tDOO 1-1 "^O O O Tf ~tO IO rf IO O OOO isplace- ment. Tons. ^ O O N o O O tntoenOoo xniootoenwoo O O ^ w tor^Q M oo O W W M Tj-oo Tf M O O O f N enoo N ^tO oo to OO CO Q O to Tfoo NOent-iONenNMinMMO OOOOOwenOO oiH-^-t^o>-ien 414 STEAMSHIP AND PROPELLER DATA. )5 2 - C^C^OOOOCOONNCOC>O (^ N O Tt- O r^O f> N vr> *t to O r~* ooo^coooco .c u-S.2 u .-> tn O m -^- O O 1 O "1" coo 1 SJ rf t>-Oo M coc o "- O^" r-O cco M S^M Mtnooom ^ cti | .' : : : : : | t \ D^ ' a >!X o a c .2 "J2 CX "4> l3 ,. rancisco. . . . . (/ : : ? !-.2-H^- 2 " i j c 2 ' c : be ! 3 5 -. TABLE B. 415 hH voco too q o M q o q w vr> q ooOO"^ O O -frf ei r^-i-i II cnao rr 1- TJ-O O O O O O co too tNWMwwwr^t-ii-i dwweiNco'Tl-tt OvC oovOooOOOOOO coco co tCO u->r>xou-)i-i too O too O CO \o t -TO oo P< C4 N K < H S-2 2 O r^ co co co Ooo cococo MvCvCOO tttOO to >- O O M O t t too cocott'-'OOOO>->-'--OOi~iCOcocoo Ooooooo coo O O O wir^r-^r^r^r^r-^r^i-i 1-1 r^. M <~< o^cc \n N c< a coq O mvnr^oooOvCOvCcocooo r^o t tttt>^ O O *+ ^ r>-cd oo co" oo oo* co 06 r^ r>>od t co co M O O O O ir>mu">u->u->OOOvCvO tttt^"tO u->ir>t tC> tN tCON tCOt^COCOCO S S C w-S < bo - - - - - PQPQ 0QD3 ffl 4i6 STEAMSHIP AND PROPELLER DATA. JJ 5 ^ < u o> O r~o6 0600 r^ oo coco cQiow M cocoO O !> ** to tx\O '**> W *^ W x M i> t^ o M 1-1 cncncnw M I-H >-> O o OOOOC>OO O r^oo co en en w M u->ir>mu-)u-)u-)tnu-)OOvO ~f-f OOOOvOOOOOOOOOOoocou-iooooOOO O O tnvooo O O voxovnvoxntoO O i^oooo O O O I.iM>f 1/3 "* O W " Ctf . , a a u u u o g ' ' |- u u u TABLE B. 417 eo 375 fc '- M in en en in $ Or>enMO*-t M citn^j-M eno co cno r~co \r> M en pi r- O O ^o" O od en M O vr>oo oo N Mi-ien MM McncnM MMMM u-> r^ w en O ^'r^OOOHi oco u->o oo 1 - w hJ CQ < H II "5 -2 S ir>O O O HI m vr>O *^oo i~ r^ ^ en O>O OO r)xn\r>tr>tnenc< O O O u->ir>o O r^o r^Tj-rrvr>o OOOO cncncnent^r^M en en eno encnen-3-cnw enenenenuitnintnTfr^-^^-N w en O enencn MOOoo Oooao ineno ^-rj-ior^vnO O O u->ir>u->u->co W O >-i)-ii-ii-iTj- rrco OO 3C34> oo UU UQQ QWfefe 4iS STEAMSHIP AND PROPELLER DATA. MminCr> Oco N o wxnm O o M u> O m O O 9 O O O oo cn<> in O c *^ O CG o "-"c O -< c inc M 10 r>< 10 M WCOMMCOt^r^OOONO^iOMOO'^'CnQ OcoOco ir>co M rr o^ N r>- ^co VDMOOCO VDM coOu-)n\r)ir>MMcocoNcocOMi-iQococococococoeofor^ u-)n\r)ir>MMcocoNcocOMi-i l-ii-.NNOO"">NCMOONN cocococococoOOOOOOOOO'^-'t'^- invoinvnmioOOr^OiHtnmmmtniovo 8 S SSsl- - TABLE B. 419 < flu ^ O oo r> O oo 00 vnoo coo O* >-> O N N "^ M coo N rf N m O i-t wcooo Ooocooooo \O O^ * O r> i^>> w u"> r^ r^ ^H in o w>o O O O oo ^^ HI r*% corj-cO'l-Ncnwi ____., i-iNMC4-iM Pu w 5 Sl en cno O -rr>crH-i cocoir>xnoo cnwoo r>o O^T COO O C^O W W t-i IH muicocoN u~, ^r -iM O O *^> 't O O cocncococncno O O O Q O O OOcoc r )cocncocoOOOOOOO en en O N co cooo ooooooaooo O O O O O O >cocococococo M HH xooooooooooooooooooo O O O O O O M tncococotnco W J3 V 1*1 fc 3j 420 STEAMSHIP AND PROPELLER DATA. Ill w O to O O w coco ^- O *^ c* f> m t^ t* coo vn ** co o ""> or^cicd cod ** ** w ti od t< t*jp vAooo cocod oo co^J-Tj-NNMMr^r^r^oOOvONOO NrtrtOOWNOOO'OOOO Ncoco O* O *r> too t^-oo oo co coo O cpcocococoQ^O^MvOvO CONGO O cio'cc^t'vOHHO> irjrt-MMiom t^. o^oo n co coo d^c5^ccor>vo r^oo M o O w w Ooo rxt-t OT|-COCOM\O w t^oo nr^covOtHOcovOci OO>n"^NiO < ^-r^r>. tntnovO ^ w NWW K 3 ^6 S 3 U5 ^ W O c^i co O O O^ O^ r^ r^ co o^ o^ O O *^ O O oo co co w M mu->iomvococoC4 C*d M H-tmind 06 d- oo co C4 c c< t tc co K g W3 | v - O tr> uo co co -- co c- rOO -< & M r> MMO MO <>oo oo M ^ oo vr> w xnoo PJvOcoooO O M c < O N cc<''cr-- M M CMXI MCO 5C.r JO M p tOCO vOmMOOMi^ OOCOOO^W OOCO rt o v-- u i co en en cn oo O co OQNNWNtotnOQ^OOOOOOOOOO r^oo O OOMMMMr^r^OOr^u^u^rr^iou^inxomu-) TOO O i ; . : I | : ; : : : : ; ' ' ; : : a .' : i 2 : C V 1=1 a - 2- - - 3. c o o, v - o - 'z X- i- U n c _ c _ a k > * "o : : : o B i M Ji rt SL 7 s 2iS3 J3 O O a,cu TJ bO O 3 u E 422 STEAMSHIP AND PROPELLER DATA. "> N en CM !> O*O -to M M r^. xn t>. O r-oo t^ CM H- en m "- en i r CM r M c M M en O *O - i o woo o in i. en O c i """"~ M u->r>.o>- I0 4, 127, 129, 154, 402 Gatewood 54 Greyhound 130, 154 Guide-blade propellers 203 Head-resistance coefficients 34 Helicoidal area denned 1 76 Hichborn 53 Hull efficiency ." 229 Hydraulic propulsion 201 Indicated thrust 236 Indraught of air 288 Initial friction 325, 327 Joessel's experiments 34 Kinematic similitude, law of 133, 147 Lagrange 3 Load friction 330 Location of propellers 286 Logarithmic cross-section paper, application of 404 INDEX. 425 PAG a Mas, M. de 119 Materials for propeller-blades 304 Maxwell , 19 McDermott 254 Mean pitch 209 Mean pitch, computation of 214 Mean slip defined 209, 21 2 Merkara 104 Middendorf 151 Negative apparent slip 237 New York, U.S.S no Normand 292 Number of propellers 284 Obliquity of stream and of shaft, influence of, on action of screw propeller. . 218 Paddle-wheel, action of element of 169, 172 Paddle-wheel, propulsive action of 196 Parsons 294 Pitch-angle 177 Pitch defined 176 Pitch, measurement of 316 Pitch, modification of, due to twisting of blade 319 Pitch ratio 1 76 Pollard and Dudebout 54 Powering ships, Chapter V 322 Powering ships, illustrative example 334 Powering ships by law of comparison 337 Powering ships by formulae involving wetted surface 344 Powering ships by use of special constants 346 Propeller design, Chapter IV 240 Propeller design, equations for 255, 278, 279, 280 Propeller design, problems 252 Propeller design, tables 256, 263 Propulsion, Chapter II 162 Propulsive coefficient 228 Propulsive element, action of 164 Projected area defined 176 Racing 290 Rankine 16, 54, 196, 202, 345 . Reduced wetted surface 48, 50 Resistance, Chapter I i Resistance, air 1 29 Resistance, eddy 29 426 INDEX. PAGE Resistance formulae 151 Resistance, head and tail 30 Resistance in canals 109 Resistance, increase of, due to banks and shoals 109 Resistance, increase of, due to change of trim 1 24 Resistance, increase of, due to irregular movement 107 Resistance, increase of, due to rough water 108 Resistance, increase of, due to slope of currents 1 23 Resistance, oblique, of ship formed bodies 157 Resistance of planes moving normal to themselves 32 Resistance of planes moving obliquely to their normal 37 Resistance, relation of, to density of liquid 104 Resistance, residual 31, 149 Resistance, skin or tangential, of planes 42 Resistance, skin or tangential, of ships 48 Resistance, skin or tangential, values of constants 50 Resistance, stream-line 28 Resistance, surface or skin 29 Resistance, wave-making 29, 93 Risbec 47, 102 Russell, J. Scott 54, 1 1 7, 1 18 Sardegna, bilge-keels on 1 28 Screw propeller, action of element of 177 Screw propeller, definitions 1 74 Screw propeller, geometry of 306 Screw propeller, laying down 310 Screw propeller, propulsive action of 182 Screw turbines 203 Shallow-water resistance 109 Sink, stream-line 18 Size, carrying capacity, and speed, relation between 356 Slip angle defined 1 78 Slip angle, table for connecting with slip ratio 321 Slip defined 168, 209 Source, stream-line 18 Speed, size, and carrying capacity, relation between 356 Speed-trial with standardized screw , 380 Springing of blades 295 Stokes 54 Stream-lines 7 Stream-lines defined 4 Strength of propeller blades 299 INDEX. 427 PAGE Stopping, distance and time required for 384 Sweet, Elnathan : 118 Taylor, D. W 26, 42, 114, 152, 321, 377 Thornycroft's screw turbine 203 Thrust horse-power 228 Tidal influence, elimination of 368 Tidal observations 366 Tidman -. 42 Trial trips, Chapter VI 362 True propeller efficiency 226 True slip 210 Units of measurement xiii Wake, constitution of 206 Wake factor 225 Wake, influence of, on equations of propeller action 224 Wake-return factor 232 Waves 54 Waves, combinations of 63 Wave-echoes, formation of 89 Wave-formation due to raotion of ship 75 Waves of translation 69 Waves, shallow-water 67 Waves, tabular data 73~7S Waves, train of, propagation of 87 Wetted surface 49, 50 Wetted surface, reduced 48, 50 Wetted surface, powering by formulae involving 344 White no, 124 Yarrow 125, 154 SHORT-TITLE CATALOGUE THE PUBLICATIONS OP JOHN WILEY & SONS, NEW YORK. LONDON: CHAPMAN & HALL, LIMITED. ARRANGED UNDER SUBJECTS. Descriptive circulars sent on application. Books marked \vith an asterisk (*) are sold at net prices only. All books are bound in cloth unless otherwise stated. AGRICULTURE HORTICULTURE FORESTRY. Annsby's Principles of Animal Nutrition 8vo, $4 oo Budd and Hansen's American Horticultural Manual: Part I. Propagation, Culture, and Improvement 12010, 50 Part II. Systematic Pomology i2mo, 50 Elliott's Engineering for Land Drainage i2mo, 50 Practical Farm Drainage 2d Edition Rewritten I2mo, 50 Graves's Forest Mensuration .. .8vo, oo Green's Principles of American Forestry . I2mo, 50 Grotenfelt's Principles of Modern Dairy Practice. (Woll) i2mo, oo * Herrick's Denatured or Industrial Alcohol 8vo, 4 oo Kemp and Waugh's Landscape Gardening. New Edition, Rewritten. (In Preparation.) * McKay and Larsen's Principles and Practice of Butter-making 8vo, Maynard's Landscape Gardening as Applied to Home Decoration i2mo, Quaintance and Scott's Insects and Disease? of Fruits. (In Preparation). Sanderson's Insects Injurious to Staple Crops i2mo, * Schwarz's Longleaf Pine in Virgin Forests i2mo, Stockbridge's Rocks and Soils 8vo, Winton's Microscopy of Vegetable Foods 8vo, 7 50 Woll's Handbook for Farmers and Dairymen i6mo, i 50 ARCHITECTURE. Baldwin's Steam Heating for Buildings i2mo, 250 Berg's Buildings and Structures of American Railroads 4to, 5 oo Birkmire's Architectural Iron and Steel 8vo, 3 50 Compound Riveted Girders as Applied in Buildings 8vo, 2 oo Planning and Construction of American Theatres 8vo, 3 oo Planning and Construction of High Office Buildings 8vo, 3 50 Skeleton Construction in Buildings 8vo, 3 oo Briggs's Modern American School Buildings 8vo, 4 oo Byrne's Inspection of Material and Wormanship Employed in Construction. i6mo, 3 oo Carpenter's Heating and Ventilating of Buildings 8vo, 4 oo * Corthell's Allowable Pressure on Deep Foundations i2mo, i 25 1 Freitag's Architectural Engineering 8vo 3 50 Fireproofing of Steel Buildings 8vo, 2 50 French and Ives's Stereotomy 8vo, 2 50 Gerhard's Guide to Sanitary Inspections 12010, i 50 * Modern Baths and Bath Houses 8vo, 3 oo Sani ntion of Public Buildings I2mo, i 50 Theatre Fires and Panics i2mo, i 50 Holley and Ladd's Analysis of Mixed Paints, Color Pigments, and Varnishes Large 121110, 2 5o Johnson's Statics by Algebraic and Graphic Methods 8vo, 2 oo Kellaway's How to Lay Out Suburban Home Grounds 8vo, 2 oo Kidder's Architects' and Builders' Pocket-book i6mo, mor. 5 oo Maire's Modern Pigments and their Vehicles i2mo, 2 oo Merrill's Non-metallic Minerals: Their Occurrence and Uses 8vo, 4 oo Stones for Building and Decoration 8vo, 5 oo Monckton's Stair-building 4to, 4 oo Patton's Practical Treatise on Foundations 8vo, 5 oo Peabody's Naval Architecture 8vo, 7 50 Rice's Concrete-block Manufacture 8vo, 2 oo Richey's Handbook for Superintendents of Construction i6mo, mor. 4 oo * Building Mechanics' Ready Reference Book: * Building Foreman's Pocket Book and Ready Reference. (In Press.) * Carpenters' and Woodworkers' Edition i6mo, mor. * Cement Workers and Plasterer's Edition. i6mo, mor. * Plumbers', Steam-Filters', and Tinners' Edition .... i6mo, mor. * Stone- and Brick-masons' Edition i6mo, mor. 50 50 50 50 Sabin's House Painting i?mo, oo Industrial and Artistic Technology of Paints and Varnish 8vo, 3 oo Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, i 50 Snow's Principal Species of Wood 8vo, 3 50 Towne's Locks and Builders' Hardware i8mo, mor. 3 oo Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo Sheep, 6 50 Law of Contracts 8vo, 3 oo Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, 5 oo Sheep, 5 50 Wilson's Air Conditioning i2mo, i 50 Worcester and Atkinson's Small Hospitals, Establishment and Maintenance, Suggestions for Hospital Architecture, wkh Plans for a Small Hospital. i2mo, i 25 ARMY AND NAVY. Bernadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose Molecule i2mo, 2 50 Chase's Art of Pattern Making i2mo, 2 50 Screw Propellers and Marine Propulsion 8vo, 3 oo * Cloke's Enlisted Specialist's Examiner 8vo, 2 oo Gunner's Examiner 8vo, ^ 50 Craig's Azimuth 4to, 3 50 Crehore and Squier's Polarizing Photo-chronograph. . . 8vo, 3 oo * Davis's Elements of Law 8vo, 2 50 * Treatise on the Military Law of United States 8vo, 7 oo Sheep, 7 50 De Brack's Cavalry Outpost Duties. (Carr) 24mo, mor. 2 oo * Dudley's Military Law and the Procedure of Courts-martial.. . Large i2mo, 2 50 Durand's Resistance and Propulsion of Ships 8vo, 5 oo 2 * Dyer's Handbook of Light Artiller> i2mo, 3 oc. Eissler's Modern High Explosives Svo, 4 oo * Fiebeger's Text-book on Field Fortification Large i2mo, 2 oo Hamilton and Bond's The Gunner's Catechism iSmo, i oo * Hoff's Elementary Naval Tactics. 8vo, i 50 Ingalls's Handbook of Problems in Direct Fire 8vo, 4 oo * Lissak's Ordnance and Gunnery 8vo ? 6 oo * Ludlow's Logarithmic and Trigonometric Tables Svo, i oo * Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II.. Svo, each, 6 oo * Mahan's Permanent Fortifications. (Mercur) 8vo, hah' mor. 7 50 Manual for Courts-martial i6mo, mor. i 50 * Mercur's Attack of Fortified Places I2mo, 2 oo * Elements of the Art of War 8vo, 4 oo Metcalf's Cost of Manufactures And the Administratioa of Workshops. .8vo, 5 oo Nixon's Adjutants' Manual. 24mo, i oo Peabody's Naval Architecture Svo, 7 50 * Phelps's Practical Marine Surveying Svo, 2 50 Putnam's Nautical Charts Svo, 2 oo Sharpe's Art of Subsisting Armies in War iSmo, mor. i 50 * Tupes and Poole's Manual of Bayonet Exercises and Musketry Fencing. 24mo, leather, 50 * Weaver's Military Explosives Svo, 3 oo Woodhull's Kates on Military Hygiene i6mo, i 50 ASSAYING. Betts's Lead Refining by Electrolysis Svo, 4 oo Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. i6mo, mor. i 50 Furman's Manual of Practical Assaying Svo, 3 oo Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. . . .Svo, 3 oo Low's Technical Methods of Ore Analysis Svo, 3 oo Miller's Cyanide Process I2mo, i oo Manual of Assaying I2mo, I oo Minet's Production of Aluminum and its Industrial Use. (Waldo) i2mo, 2 50 O'Driscoll's Notes on the Treatment of Gold Ores Svo, 2 oo Ricketts and Miller's Notes on Assaying Svo, 3 oo Robine and Lenglen's Cyanide Industry. (Le Clerc) Svo, 4 oo Ulke's Modern Electrolytic Copper Refining Svo, 3 oo Wilson's Chlorination Process I2mo, I 50 Cyanide Processes , i2mo. i 50 ASTRONOMY. Comstock's Field Astronomy for Engineers Svo, 2 50 Craig's Azimuth 4 to, 3 50 Crandall's Text-book on Geodesy and Least Squares Svo, 3 oo Doolittle's Treatise on Practical Astronomy Svo, 4 oo Gore's Elements of Geodesy Svo, 2 50 Hayford's Text-book of Geodetic Astronomy Svo, 3 oo Merriman's Elements of Precise Surveying and Geodesy Svo, 2 50 * Michie and Harlow's Practical Astronomy Svo, 3 oo Rust's Ex-meridian Altitude, Azimuth and Star-Finding Tables Svo, 5 oo * White's Elements of Theoretical and Descriptive Astronomy i2mo, 2 oo 3 CHEMISTRY. * Abderhalden's Physiological Chemistry in Thirty Lectures. (Hall and Defren) 8vo, 5 oo * Abegg's Theory of Electrolytic Dissociation, (von Ende) i2mo, i 25 Alexeyeff's General Principles of Organic Syntheses. (Matthews) 8vo, 3 oo Allen's Tables for Iron Analysis 8vo, 3 oo Arnold's Compendium of Chemistry. (Mandel) Large i2mo, 3 50 Association of State and National Food and Dairy Departments, Hartford, Meeting, 1906 8vo, 3 oo Jamestown Meeting, 1907 8vo, 3 oo Austen's Notes for Chemical Students iimo, i 50 Baskerville's Chemical Elements. (In Preparation.) Bernadou's Smokeless Powder. Nitro-cellulose, and Theory of the Cellulose Molecule I2mo, 2 50 Bilts's Chemical Preparations. (Hall and Blanchard). (In Press.) * Blanchard's Synthetic Inorganic Chemistry I2mo, i oo * Browning's Introduction to the Rarer Elements 8vo, i 50 Brush and Penfield's Manual of Determinative Mineralogy 8vo, 4 oo * Claassen's Beet-sugar Manufacture. (Hall and Rolfe) 8vo, 3 oo Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood). . .8vo, 3 oo Cohn's Indicators and Test-papers I2mo, 2 oo Tests and Reagents 8vo, 3 oo * Danneel's Electrochemistry. (Merriam) i2mo, i 25 Dannerth's Methods of Textile Chemistry i2mo, 2 oo Duhem's Thermodynamics and Chemistry. (Burgess) 8vo, 4 oo Eakle's Mineral Tables for the Determination cf Minerals by their Physical Properties 8vo, i 25 Eissler's Modern High Explosives 8vo, 4 oo Effront's Enzymes and their Applications. (Prescott) 8vo, 3 oo Erdmann's Introduction to Chemical Preparations. (Dunlap) i2mo, i 25 * Fischer's Physiology of Alimentation Large i2mo, 2 oo Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. I2mo, mor. i 50 Fowler's Sewage Works Analyses i2mo, 2 oo Fresenius's Manual of Qualitative Chemical Analysis. (Wells) 8vo, 5 oo Manual of Qualitative Chemical Analysis. Part I. Descriptive. (Wells) 8vo, 3 oo Quantitative Chemical Analysis. (Cohn) 2 vols 8vo, 12 50 When Sold Separately, Vol. I, $6. Vol. II, $8. Fuertes's Water and Public Health i2mo, i 50 Furman's Manual of Practical Assaying 8vo, 3 oo * Getman's Exercises in Physical Chemistry i2mo, 2 oo Gill's Gas and Fuel Analysis for Engineers i2mo, i 25 * Gooch and Browning's Outlines of Qualitative Chemical Analysis. Large i2mo, i 25 Grotenfelt's Principles of Modern Dairy Practice. (Woll) i2mo. 2 oo Groth's Introduction to Chenvcal Crystallography (Marshall) I2mo. i 25 Hammarsten's Text-book of Physiological Chemistry. (Mandel) 8vo, 4 oo Hanausek's Microscopy of Technical Products. (Winton) 8vo, 5 oo * Raskins and Macleod's Organic Chemistry i2mo, 2 oo Helm's Principles of Mathematical Chemistry. (Morgan) i2mo, i 50 Bering's Ready Reference Tables (Conversion Factors) i6mo, mor. 2 50 * Herrick's Denatured or Industrial Alcohol 8vo 4 oo Hinds's Inorganic Chemistry 8vo, 3 oo * Laboratory Manual for Students i2mo, i oo * Holleman's Laboratory Manual of Organic Chemistry for Beginners. (Walked i2mo, i oo Text-book of Inorganic Chemistry. (Cooper) 8vo, 2 50 Text-book of Organic Chemistry. (Walker and Mott) 8vo, 2 50 4 Eolley and Ladd's Analysis of Mixed Paints, Color Pigments, and Varnishes. Large i2mo, 2 50 Hopkins's Oil-chemists' Handbook 8vo, 3 oo Iddings's Rock Minerals 8vo, 5 oo Jackson's Directions for Laboratory Work in Physiological Chemistry. .8vo, I 25 Johannsen's Determination of Rock-forming Minerals in Thin Sections . .8vo, 4 oo Johnson's Chemical Analysis of Special Steel. Steel-making. (Alloys and Graphite.) Large i2mo, 3 oo Keep's Cast Iron 8vo, 2 50 Ladd's Manual of Quantitative Chemical Analysis i2mo, i oo ^andauer's Spectrum Analysis. (Tingle) 8vo, 3 oo * .uangwurtny and Austen's Occurrence of Aluminium in Vegetable Prod- ucts, Animal Products, and Natural Waters 8vo, 2 oo Lassar-Cohn's Application of Some General Reactions to Investigations in Organic Chemistry. 'Tingle) i2mo, I oo Leach's Inspection and Analysis of Food with Special Reference to State Control 8vo, 7 50 Lob's Electrochemistry of Organic Compounds. (Lorenz) 8vo, 3 oo Lodge's Notes on Assaying and Metallurgical Laboratory Experiments 8vo, 3 oo Low's Technical Method of Ore Analysis . .8vo, 3 oo Lunge's Techno-chemical Analysis. (Cohn) I2mo, i oo * McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50 Maire's Modern Pigments and their Vehicles lamo, 2 oo Mandel's Handbook for Bio-chemical Laboratory i2mo. i 50 * Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe . . i2mo, 60 Mason's Examination of Water. (Chemical and Bacteriological.). . i2mo, i 25 Water-supply. (Considered Principally from a Sanitary Standpoint. 8vo, 4 oo * Mathewson's First Principles of Chemical Theory 8vo, i oo Matthews's Textile Fibres. 2d Edition, Rewritten 8vo, 4 oo * Meyer's Determination of Radicle? in Carbon Compounds. (TingleX . tamo, i 25 Miller's Cyanide Process I2mo, i oo Manual of Assaying I2mo, i oo Minet's Production of Aluminum and its Industrial Use. (Waldo) i2mo, 2 50 Mixter's Elementary Text-book of Chemistry I2mo, i 50 Morgan's Elements of Physical Chemistry I2mo, 3 oo Outline of the Theory of Solutions and its Results I2mo, i oo * Physical Chemistry for Electrical Engineers i2mo, i 50 Morse's Calculations used in Cane-sugar Factories i6mo, mor. i 50 * Muir's History of Chemical Theories and Laws 8vo, 4 oo Mulliken's General Method for the Identification of Pure Organic Compounds. Vol. I Large 8vo, 5 oo O'Driscoll's Notes on the Treatment of Gold Ores 8vo, oo Ostwald's Conversations on Chemistry. Part One. (Ramsey") i2mo, 50 Part Two. (Turnbull) 12 mo, oo * Palmer's Practical Test Book of Chemistry i2mo, oo * Pauli's Physical Chemistry in the Service of Medicine. (Fischer^ i2mo, 25 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8vo, paper, 50 Tables of Minerals. Including the Use of Minerals and Statistics of Domestic Production g VOt T oo Pictet's Alkaloids and their Chemical Constitution. (Biddle) 8vo, 5 oo Poole's Calorific Power of Fuels 8vo, 3 oo Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis I2mo, i 50 * Reisig's Guide to Piece-dyeing g VOj 25 oo Richards and Woodman's Air, Water, and Food from a Sanitary Standpoint.. 8 vo, 2 oo Ricketts and Miller's Notes on Assaying 1 . Svo, 3 oo Rideal's Disinfection and the Preservation of Food 8vo, 4 oo Sewage and the Bacterial Purification of Sewage 8vo. 4 oo 5 Riggs's Elementary Manual for the Chemical Laboratory 8vo, i 25 Robine and Lenglen's Cyanide Industry. (Le Clerc) 8vo, 4 oo Ruddiman's Incompatibilities in Prescriptions 8vo. 2 oo Whys in Pharmacy , I2mo, i oo Ruer's Elements of Metallography. (Mathewson) (In Preparation.) Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 oo Salkowski's Physiological and Pathological Chemistry. (Orndorff) 8vo, 2 50 Schimpf's Essentials of Volumetric Analysis I2mo, i 25 * Qualitative Chemical Analysis 8vo, i 25 Text-book of Volumetric Analysis I2mo, 2 50 Smith's Lecture Notes on Chemistry for Dental Students 8vo, 2 50 Spencer's Handbook for Cane Sugar Manufacturers i6mo, mor. 3 oo Handbook for Chemists of Beet-sugar Houses i6mo, mor. 3 oo Stockbridge's Rocks and Soils 8vo, 2 50 * Tillman's Descriptive General Chemistry 8vo, 3 oo * Elementary Lessons in Heat 8vo, i 50 Treadwell's Qualitative Analysis. (Hall) 8vo, 3 oo Quantitative Analysis. (Hall) 8vo, 4 oo Turneaure and Russell's Public Water-supplies 8vo, 5 oo Van Deventer's Physical Chemistry for Beginners. (Boltwood) i2mo, i 50 Venable's Methods and Devices for Bacterial Treatment of Sewage 8vo, 3 oo Ward and Whipple's Freshwater Biology. (In Press.) Ware's Beet-sugar Manufacture and Refining. Vol. I Small 8vo, 4 GO " " " Vol.11 SmallSvo, 500 Washington's Manual of the Chemical Analysis of Rocks 8vo, 2 oo * Weaver's Military Explosives 8vo, 3 oo Wells's Laboratory Guide in Qualitative Chemical Analysis 8vo, i 50 Short Course in Inorganic Qualitative Chemical Analysis for Engineering Students i2mo, i 50 Text-book of Chemical Arithmetic I2mo, i 25 Whipple's Microscopy of Drinking-water 8vo, 3 50 Wilson's Chlorination Process I2mo, i 50 Cyanide Processes i2mo, i 50 Winton's Microscopy of Vegetable Foods , , . 8vo, 7 50 CIVIL ENGINEERING. BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEER- ING. RAILWAY ENGINEERING. Baker's Engineers' Surveying Instruments 12 mo, 3 oo Bixby's Graphical Computing Table Paper 19^X24} inches. 25 Breed and Hosmer's Principles and Practice of Surveying. 2 Volumes. Vol. I. Elementary Surveying 8vo, 3 oo Vol. II. Higher Surveying 8vo, 2 50 * Burr's Ancient and Modern Engineering and the Isthmian Canal 8vo, 3 50 Comstock's Field Astronomy for Engineers 8vo, 2 50 * Corthell's Allowable Pressures on Deep Foundations i2tno, i 25 Crandall's Text-book on Geodesy and Least Squares 8vo, 3 oo Davis's Elevation and Stadia Tables 8vo, i oo Elliott's Engineering for Land Drainage i2mo, i 50 Practical Farm Drainage I2mo, r oo *Ficbeger's Treatise on Civil Engineering 8vo, 5 oo Flemer's Phototopographic Methods and Instruments 8vo, 5 oo Folwell's Sewerage. (Designing and Maintenance.) 8vo, 3 oo Freitag's Architectural Engineering 8vo, 3 50 French and Ives's Stereotomy ; 8vo, 2 50 Goodhue's Municipal Improvements I2mo, I 50 Gore's Elements of Geodesy 8vo, 2 50 * Hauch's and Rice's TaDles ot Quantities for Preliminary Estimates . . I2mo, i 25 Hayford's Text-book of Geodetic Astronomy 8vo, 3 oo Bering's Ready Reference Tables. (Conversion Factors') i6mo, mor. 2 50 Howe's Retaining Walls for Earth i2rno, i 25 * Ives's Adjustments of the Engineer's Transit and Level i6mo, Bds. 25 Ives and Hilts's Problems in Surveying i6nio, mor. i 50 Johnson's ( J. B.) Theory and Practice of Surveying Small 8vo, 4 oo Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo, 2 oo Kinnicutt, Winslow and Pratt's Purification of Sewage. (In Preparation. : Laplace's Philosophical Essay on Probabilities. (Truscott and Emory) i2mo, 2 oo Mahan's Descriptive Geometry 8vo, i 50 Treatise on Civil Engineering. (1873.) (Wood) 8vo, 5 oo Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 Merriman and Brooks's Handbook for Surveyors i6mo, mor. 2 oo Nugent's Plane Surveying 8vo, 3 50 Ogden's Sewer Construction 8vo, 3 oo Sewer Design i2mo, 2 oo Parsons's Disposal of Municipal Refuse 8vo, 2 oo Patton's Treatise on Civil Engineering 8vo. half leather, 7 50 Reed's Topographical Drawing and Sketching 410, 5 oo Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 4 oo Kleiner's Shaft-sinking under Difficult Conditions. (Corning and Peele). . 8vo, 3 oo Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, i 50 Smith's Manual of Topographical Drawing. (McMillan) 8vo, 2 50 Soper's Air and Ventilation of Subways Large i2mo, 2 50 Tracy's Plane Surveying i6mo, mor. 3 oo * Trautwine's Civil Engineer's Pocket-book i6mo, mor. 5 oo Venable's Garbage Crematories in America 8vo, 2 oo Methods and Dtvices for Bacterial Treatment of Sewage 8vo, 3 oo Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo Sheep, 6 50 Law of Contracts 8vo, 3 oo Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, 5 oo Sheep, 5 50 Warren's Stereotomy Problems in Stone-cutting 8vo, 2 50 * Waterbury's Vest-Pocket Hand-book of Mathematics for Engineers. 2X si inches, mor. i oo Webb's Problems in the Use and Adjustment of Engineering Instruments. i6mo, mor. i 25 Wilson's (H. N.) Topographic Surveying 8vo, 3 50 Wilson's (W. L.) Elements of Railroad Track and Construction i2mo, 2 oo BRIDGES AND ROOFS. Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 oo Burr and Falk's Design and Construction of Metallic Bridges 8vo, 5 oo Influence Lines for Bridge and Roof Computations 8vo, 3 oo Du Bois's Mechanics of Engineering. Vol. II. . . , Srrall 4:0, 10 oo Foster's Treatise on Wooden Trestle Bridges 4 to, 5 oo Fowler's Ordinary Foundations 8vo, 3 50 French and Ives's Stereotomy 8vo, 2 e O Greene's Arches in Wood, Iron, and Stone 8vo, 2 50 Bridge Trusses 8vo, 2 50 Roof Trusses 8vo, i 25 Grimm's Secondary Stresses in Bridge Trusses 8vo, 2 -o Heller's Stresses in Structures and the Accompanying Deformations 8vo, 3 oo Howe's Design of Simple Roof- trusses in Wood and Steel 8vo, 2 oo Symmetrical Masonry Arches 8vo, 2 50 Treatise on Arches 8vo, 4 oo 7 Johnson Bryan, and Turneaure's Theory and Practice in the Designing of Modern Framed Structures.. Small 4to, 10 oo Merrlman and Jacoby's Text-book on Roofs and Bridges: Part I. Stresses in Simple Trusses 8vo, 2 50 Part II. Graphic Statics 8vo, 2 50 Part III. Bridge Design 8vo, 2 50 Part IV. Higher Structures 8vo, 2 50 Morison's Memphis Bridge Oblong 4to, 10 oo Sondericker's Graphic Statics, with Applications to Trusses, Beams, and Arches. 8vo, 2 oo Waddell's De Pontibus, Pocket-book for Bridge Engineers i6mo, mor, 2 oo ** Specifications for Steel Bridges 12010, 50 Waddell and Harrington's Bridge Engineering. (In Preparation.) Wright's Designing of Draw-spans. Two parts in one volume Svo, 3 50 HYDRAULICS. Barnes's Ice Formation 8vo, 3 oo Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an Orifice. (Trautwine) 8vo, 2 oo Bovey's Treatise on Hydraulics 8vo, 5 oo Church's Diagrams of Mean Velocity of Water in Open Channels. Oblong 4to, paper, i 50 Hydraulic Motors 8vo, 2 oo Mechanics of Engineering 8vo, 6 oo Coffin's Graphical Solution of Hydraulic Problems i6mo, mor. 2 50 Flather's Dynamometers, and the Measurement of Power 12 mo, 3 oo Folwell's Water-supply Engineering 8vo, 4 oo Frizell's Water-power 8vo, 5 oo Fuertes's Water and Public Health i2mo, i 50 Water-filtration Works ... i?.mo, 2 50 Ganguillet and Kutter's General Formula for the Uniform Flow of Water in Rivers and Other Channels. (Hering and Trautwiue) Svo, 4 oo Hazen's Clean Water and How to Get It Lar:e ismo. T so Filtration of Public Water-supplies 8vo, 3 oo Hazlehurst's Towers and Tanks for Water- works Svo, 2 50 Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal Conduits Svo , 2 oo Hoyt and Grover's River Discharge Svo, 2 oo Hubbard and Kiersted's Water-works Management and Maintenance Svo, 4 co * Lyndon's Development and Electrical Distribution of Water Power. . . .Svo, 3 oo Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) Svo, 4 oo Merriman's Treatise on Hydraulics Svo, 5 oo * Michie's Elements of Analytical Mechanics Svo, 4 oo * Molitor's Hydraulics of Rivers. Weirs and Sluices Svo, 2 oo * Richards's Laboratory Notes on Industrial Water Analysis Svo, o 50 Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- supply. Second Edition, enlarged Large Svo, 6 oo * Thomas and Watt's Improvement of Rivers 4to, 6 oo Turneaure and Russell's Public Water-supplies Svo, 5 oo Wegmann's Design and Construction of Dams. 5th Ed., enlarged 4to, 6 oo Water-supply of the City of New York from 1658 to 1895 4to, 10 oo Whipple's Value of Pure Water Large i2mo, i oo Williams and Hazen's Hydraulic Tables Svo, i 50 Wilson's Irrigation Engineering Small Svo, 4 oo Wolff's Windmill ^s a Prime Mover Svo, 300 Wood's Elements of Analytical Mechanics < Svo, 3 oo Turbines Svo, 2 50 8 MATERIALS OF ENGINEERING. Baker's Roads and Pavements 8vo, 5 oo Treatise on Masonry Construction 8vo, 5 oo Birkmire's Architectural Iron and Steel 8vo, 3 50 Compound Riveted Girders as Applied in Buildings 8vo, 2 oo Black's United States Public Works Oblong 4to, 5 oo Bleininger's Manufacture of Hydraulic Cement. (In Preparation.) * Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 50 Byrne's Highway Construction 8vo, 5 oo Inspection of the Materials and Workmanship Employed in Construction. i6mo, 3 oo Church's Mechanics of Engineering 8vo, 6 oo Du Bois's Mechanics of Engineering. Vol. I. Kinematics, Statics, Kinetics Small 4:0, 7 50 Vol. II. The Stresses in Framed Structures, Strength of Materials and Theory of Flexures Small 4to, 10 oo Eckel's Cements, Limes, and Plasters 8vo, 6 oo Stone and Clay Products used in Engineering. (In Preparation.) Fowler's Ordinary Foundations 8vo, 3 50 Graves's Forest Mensuration 8vo, 4 oo Green's Principles of American Forestry I2mo, i 50 * Greene's Structural Mechanics 8vo, 2 50 Ho ly and Ladd's Analysis of Mixed Paints, Color Pigments and Varnishes Large izmo, 2 50 Johnson's C. M.) Chemical Analysis of Special Steels. (In Preparation.) Johnson's < J . B.j Materials of Construction Large 8vo, 6 oo Keep's Cast Iron 8vo, 2 50 Kidder's Architects and Builders' Pocket-book i6mo, 5 oo Lanza's Applied Mechanics 8vo f 7 50 Maire's Modern Pigments and their Vehicles , I2mo, 2 oo Martens's Handbook on Testing Materials. (Henning) 2 vols 8vo, 7 50 Maurer's Technical Mechanics 8vo, 4 oo Merrill's Stones for Building and Decoration. 8vo, 5 oo Merriman's Mechanics of Materials 8vo, 5 oo * Strength of Materials i2mo, i oo Metcalf's Steel. A Manual for Steel-users i2mo, 2 oo Morrison's Highway Engineering 8vo, 2 50 Patton's Practical Treatise on Foundations 8vo, 5 oo Rice's Concrete Block Manufacture 8vo, 2 o o Richardson' j Modern Asphalt Pavements 8vo, 3 oo Richey's Handbook for Superintendents of Construction i6mo, mor. 4 oo * Ries's Clays: Their Occurrence, Properties, and Uses 8vo, 5 oo Sabir.'s Industrial and Artistic Technology of Paints ard Varnish 8vo, 3 oo *Schwarz'sLong1eafPinein Virgin Forest izmo, i 25 Snow's Principal Species of Wood 8vo, 3 50 Spalding's Hydraulic Cement i2mo, 2 oo Text-book on Roads and Pavements i2mo, 2 oo Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 oo Thurston's Materials of Engineering. In Three Parts 8vo, 8 oo Part I. Non-metallic Materials of Engineering and Metallurgy 8vo, 2 oo Part IT. Iron and SteeL 8vo, 3 50 Part HI. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8v 2 SO Tillson's Street Pavements and Paving Materials 8vo, 4 oo Turneaure and Maurer's Principles of Reinforced Concrete Construction.. .8vo, 3 oo Waterbury's Cement Laboratory Manual i2mo, i oo Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on the Preservation of Timber 8vo, 2 oo Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel 8vo, 4 oo RAILWAY ENGINEERING. Andrews's Handbook for Street Railway Engineers 3x5 inches, mor. i 25 Berg's Buildings and Structures of American Railroads 4to, 5 oo Brooks's Handbook of Street Railroad Location i6mo, mor. i 50 Butt's Civil Engineer's Field-book i6mo, mor. 2 50 Crandall's Railway and Other Earthwork Tables 8vo, i 50 Transition Curve i6mo, mor. i 50 * Crockett's Methods for Earthwork Computations 8vo, i 50 Dawson's "Engineering" and Electric Traction Pocket-book i6mo. mor. 5 oo Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 oo Fisher's Table of Cubic Yards Cardboard, 25 Godwin's Railroad Engineers' Field-book and Explorers' Guide. . . i6mo, mor. 2 50 Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- bankments 8vo, i oo Ives and Hilts's Problems in Surveying, Railroad Surveying and Geodesy i6mo, mor. i 50 Molitor and Beard's Manual for Resident Engineers i6mo, i oo Nagle's Field Manual for Railroad Engineers i6mo, mor. 3 oo Philbrick's Field Manual for Engineers i6mo, mor. 3 oo Raymond's Railroad Engineering. 3 volumes. Vol. I. Railroad Field Geometry. (In Preparation.) VoL II. Elements of Railroad Engineering 8vo, 3 50 Vol. III. Railroad Engineer's Field Book. (In Preparation.) Searles's Field Engineering i6mo, mor. 3 oo Railroad Spiral i6mo, mor. i 50 Taylor's Prismoidal Formulae and Earthwork 8vo, i 50 *Trautwine's Field Practice of Laying Out Circular Curves for Railroads. i2mo. mor. 2 50 * Method of Calculating the Cubic Contents of Excavations and Embank- ments by the Aid of Diagrams 8vo, 2 oo Webb's Economics of Railroad Construction Large i2mo, 2 50 Railroad Construction i6mo, mor. 5 oo Wellington's Economic Theory of the Location of Railways Small 8vo, 5 oo DRAWING. Barr's Kinematics of Machinery 8vo, 2 50 * Bartlett's Mechanical Drawing 8vo, 3 oo * " " " Abridged Ed 8vo, i 50 Coolidge's Manual of Drawing. . . '. 8vo. paper, i oo Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- neers Oblong 410, 2 50 Durley's Kinematics of Machines 8vo, 4 oo Emch's Introduction to Projective Geometry and its Applications 8vo, 2 50 Hill's Text-book on Shades and Shadows, and Perspective 8vo, 2 oo Jamison's Advanced Mechanical Drawing Svo, 2 oo Elements of Mechanical Drawing 8vo. 2 50 Jones's Machine Design: Part I. Kinematics of Machinery Svo, i 50 Part H. Form, Strength, and Proportions of Parts Svo, 3 oo MacCord's Elements of Descriptive Geometry Svo, 3 oe Kinematics; or, Practical Mechanism Svo. 5 oo Mechanical Drawing 4to, 4 oo Velocity Diagrams Svo, i 50 10 McLeod's Descriptive Geometry.. Large i2mo, i 50 * Mahan's Descriptive Geometry and Stone-cutting 8vo, i 50 Industrial Drawing. (Thompson ) 8vo, 3 50 Meyer's Descriptive Geometry for Students of Engineering 8vo, 2 oo Reed's Topographical Drawing and Sketching 4to, 5 oo Reid's Course in Mechanical Drawing 8vo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo Robinson's Principles of Mechanism 8vo, 3 oo Schwamb and Merrill's Elements of Mechanism 8vo, 3 oo Smith's (R. S.) Manual of Topographical Drawing. (McMillan) 8vo, 2 50 Smith (A. W.) and Marx's Machine Design 8vo, 3 oo * Titsworth's Elements of Mechanical Drawing Oblong 8vo : i 25 Warren's Drafting Instruments and Operations i2mo, i 25 Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 50 Elements of Machine Construction and Drawing 8vo, 7 50 Elements of Plane and Solid Free-hand Geometrical Drawing i2mo, i oo General Problems of Shades and Shadows 8vo, 3 oo Manual of Elementary Problems in the Linear Perspective of Form and Shadow i2mo, i oo Manual of Elementary Projection Drawing i2mo, i 50 Plane Problems in Elementary Geometry i2mo, i 25 Problems, Theorems, and Examples in Descriptive Geometry. 8vo, 2 50 Weisbach's Kinematics and Power of Transmission. (Hermann and Klein) 8vo, 5 oo Wilson's (H. M.) Topographic Surveying 8vo, 3 50 Wilson's (V. T.) Free-hand Lettering 8vo, i oo Free-hand Perspective 8vo, 2 50 Woolf's Elementary Course in Descriptive Geometry Large 8vo, 3 oo ELECTRICITY AND PHYSICS. * Abegg's Theory of Electrolytic Dissociation, (von Ende) i2mo, i 25 Andrews's Hand-Book for Street Railway Engineering 3X5 inches, mor. i 25 Anthony and Brackett's Text-book of Physics. (Magie) Large i2mo, 3 oo Anthony's Theory of Electrical Measurements. (Ball) i2mo, i oo Benjamin's History of Electricity 8vo, 3 oo Voltaic Cell 8vo, 3 oo Betts's Lead Refining and Electrolysis 8vo, 4 oo Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood). .8vo, 3 oo * Collins's Manual of Wireless Telegraphy i2mo, i 50 Mor. 2 oo Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 oo * Danneel's Electrochemistry. (Merriam) i2mo, i 25 Dawson's "Engineering" and Electric Traction Pocket-book .... i6mo, mor. 5 oo Dolezalek's Theory of the Lead Accumulator (Storage Battery), (von Ende) i2mo, 2 50 Duhem's Thermodynamics and Chemistry. (Burgess) 8vo, 4 oo Flather's Dynamometers, and the Measurement of Power i2mo, 3 oo Gilbert's De Magnate. (Mottelay) 8vo, 2 50 * Hanchett's Alternating Currents i2mo, i oo Bering's Ready Reference Tables (Conversion Factors) i6mo, mor. 2 50 * Hotart and Ellis's High-speed Dynamo Electric Machinery 8vo, 6 oo Holman's Precision of Measurements 8vo. 2 oo Telescopic Mirror-scale Method, Adjustments, and Tests. .. .Large 8vc , 75 * Karapetoff's Experimental Electrical Engineering 8vo, 6 oo Kinzbrunner's Testing of Continuous-current Machines 8vo, 2 oo Landauer's Spectrum Analysis. (Tingle) 8vo, 3 oo Le Chatelier's High-temperature Measurements. (Boudouard Burgess).. i2mo, 3 oo Lob's Electrochemistry of Organic Compounds. (Lorenz) 8vo, 3 oo * London's Development and Electrical Distribution of Water Power . . . ,8vo, 3 oo 11 * Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II 8vo, each, 6 oo * Michie's Elements of Wave Motion Relating to Sound and Light 8vo, 4 oo Morgan's Outline of the Theory of Solution and its Results i2mo, i oo * Physical Chemistry for Electrical Engineers i2mo, i 50 Niaudet's Elementary Treatise on Electric Batteries. (Fishback). . . . I2mo, 2 50 * Norris's Introduction to the Study of Electrical Engineering 8vo, 2 50 * Parshall and Hobart's Electric Machine Design 4to, half mor. 12 50 Reagan's Locomotives: Simple, Compound, and Electric. New Edition. Large i2mo, 3 50 * Rosenberg's Electrical Engineering. (Haldane Gee Kinzbrunner). .. .8vo, 2 co Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 2 50 Swapper's Laboratory Guide for Students in Physical Chemistry i2mo, i oo * Tillman's Elementary Lessons in Heat 8vo, i 50 Tory and Pitcher's Manual of Laboratory Physics Large I2mo, 2 oo Ulke's Modern Electrolytic Copper Refining 8vo. 3 oo LAW. * Brennan's Handbook : A Compendium of Useful Legal Information for Business Men i6mo, mor. 5 . oo * Davis's Elements or Law 8vo, 2 50 * Treatise on the Military Law of United States 8vo, 7 oo * Sheep, 7 50 * Dudley's Military Law and the Procedure of Courts-martial . . . .Large i2mo, 2 50 Manual for Courts-martiaL i6mo, mor. i 50 Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo Sheep, 6 50 Law of Contracts 8vo, 3 oo Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo 5 oo Sheep, 5 50 MATHEMATICS. Baker's Elliptic Functions 8vo, i 50 Briggs's Elements of Plane Analytic Geometry. (Bocher) i2mo, i oo *Buchanan's Plane and Spherical Trigonometry 8vo, i oo Byerley's Harmonic Functions 8vo, i oo Chandler's Elements of the Infinitesimal Calculus i2mo, 2 oo Coffin's Vector Analysis. (In Press.) Compton's Manual of Logarithmic Computations i2mo, i 50 * Dickson's College Algebra Large i2mo, i 50 * Introduction to the Theory of Algebraic Equations Large i2mo, i 25 Emch's Introduction to Projective Geometry and its Applications 8vo, 2 50 Fiske's Functio -,s of a Complex Variable 8vo, i oo Halsted's Elementary Synthetic Geometry 8vo, i 50 Elements of Geometry 8vo, i 75 * Rational Geometry i2mo, i 50 Hyde's Grassmann's Space Analysis 8vo, i oo * Jonnson's (J B.) Three-place Logarithmic Tables: Vest-pocket size, paper, 15 100 copies, 5 oo Mounted on heavy cardboard, 8X10 inches, 25 10 copies, 2 oo Johnson's (W. W.) Abridged Editions of Differential and Integral Calculus Large lamo, i vol. Curve Tracing in Cartesian Co-ordinates i2mo, Differential Equations 8vo. Elementary Treatise on Differential Calculus Large i2mo, Elementary Treatise on the Integral Calculus Large I2mo, Theoretical Mechanics I2mo, 3 oo Theory of Errors and the Method of Least Squares i2mo, i 50 Treatise on Differential Calculus Large i2mo, 3 oo 12 Johnson's Treatise on the Integral Calculus. Large i2mo, 3 oo Treatise on Ordinary aad Partial Differential Equations. . Large i2mo, 3 50 Karapetoffs Engineering Applications of Higher Mathematics. (In Pre- paration.) Laplace's Philosophical Essay on Probabilities. (Truscott and Emory). .i2mo, 2 oo * Ludlow and Bass's Elements of Trigonometry and Logarithmic and Other Tables 8vo, 3 oo Trigonometry and Tables published separately Each, 2 oo * Ludlow's Logarithmic and Trigonometric Tables 8vo, i oo Macfarlane's Vector Analysis and Quaternions 8vo, i oo McMahon's Hyperbolic Functions 8vo, i oo Manning's Irrational lumbers and their Representation by Sequences and Series I2mo, i 23 Mathematical Monographs. Edited by Mansfield Merriman and Robert S. Woodward Octavo, each i oo No. i. History of Modern Mathematics, by David Eugene Smith. No. 2. Synthetic Projective Geometry, by George Bruce Halsted. No. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyper- bolic Functions, by James McMahon. To 'j. Harmonic Func- tions, by William E. Byerly. No. 6. Grassmann's Space Analysis, by Edward W. Hyde. No. 7. Probability and Theory of Errors, by Robert S. Woodward. No. 8. Vector Analysis and Quaternions, by Alexander Macfarlane. No. 9. Differential Equations, by William Woolsey Johnson. No. 10. The Solution of Equations, by Mansfield Merriman. No. u. Functions of a Complex Variable, by Thomas S. Fiske. Maurer's Technical Mechanics 8vo, 4 oo Merriman's Method of Least Squares 8vo, 2 oo Solution of Equations 8vo, i oo Rice and Johnson's Differential and Integral Calculus. 2 vols. in one. Large i2mo, i 50 Elementary Treatise on the Differential Calculus Large i2mo, 3 oo Smith's History of Modern Mathematics 8vo, i oo * Veblen and Lennes's Introduction to the Real Infinitesimal Analysis of One Variable 8vo, 2 oo * Waterbury's Vest Pocket Hand-Book of Mathematics for Engineers. zjXsi inches, mor. i oo Weld's Determinations 8vo, I co Wood's Elements of Co-ordinate Geometry 8vo, 2 oo Woodward's Probability aid Theory of Errors 8vo, i oo MECHANICAL ENGINEERING. MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. Bacon's Forge Practice I2mo 1 i 50 Baldwin's Steam Heating for Buildings I2mo, 2 50 Bair's Kinematics of Machinery 8vo, 2 50 * Bartlett's Mechanical Drawing 8vo, 3 oo * " " Abridged Ed 8vo, i 50 Benjamin's Wrinkles and Recipes i2mo, 2 oo * Burr's Ancient and Modern Engineering and the Isthmian Canal 8vo, 3 50 Carpenter's Experimental Engineering 8vo, 6 oo Heating and Ventilating Buildings 8vo, 4 oo Clerk's Gas and Oil Engine. Large i2mo, 4 oo Compton's First Lessons in Metal Working i2mo, i 50 Connton and De Groodt's Speed Lathe I2mo, i 50 Coolidge's Manual of Drawing 8vo, paper, i oo Coolidge and Freeman's Elements of General Drafting for Mechanical En- gineers Oblong 4to, 2 50 13 Cromweil's Treatise on Belts and Pulleys i2mo, i 50 Treatise on Toothed Gearing I2mo, i 50 Durley's Kinematics of Machines 8vo, 4 oo Flather's Dynamometers and the Measurement of Power, i2mo, 3 oo Rope Driving izmo, 2 oo Gill's Gas and Fuel Analysis for Engineers I2mo, i 25 Goss's Locomotive Sparks 8vo, 2 oo Greene's Pumping Machinery. (In Preparation.) Hering's Ready Reference Tables (Conversion Factors) i6mo, mor. 2 50 * Hobart and Ellis's High Speed Dynamo Electric Machinery 8vo, 6 oo Button's Gas Engine 8vo, 5 oo Jamison's Advanced Mechanical Drawing 8vo, 2 oo Elements of Mechanical Drawing 8vo, 2 50 Jones's Gas Engine. (In Press.) Machine Design: Part I. Kinematics of Machinery 8vo, i 50 Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo Kent's Mechanical Engineers' Pocket-book i6mo, mor. 5 oo Kerr's Power and Power Transmission 8vo, 2 oo Leonard's Machine Shop Tools and Methods 8vo, 4 oo * Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean).. .8vo, 4 oo MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oa Mechanical Drawing 4to, 4 oo Velocity Diagrams 8vo, i 50- MacFarland's Standard Reduction Factors for Gases 8vo, i 50 Mahan's Industrial Drawing. (Thompson) 8vo, 3 50 Oberg's Hand Book of Small Tools Large i2mo, 3 oo * Parshall and Hobart's Electric Machine Design Small 4to, half leather, 12 50 Peele's Compressed Air Plant for Mines 8vo, 3 oo Poole's Calorific Power of Fuels 8vo, 3 oo * Porter's Engineering Reminiscences, 1855 to 1882 8vo, 3 oo Reid's Course in Mechanical Drawing 8vo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo Richard's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism 8vo, 3 oo Schwamb and Merrill's Elements of Mechanism 8vo, 3 oo Smith's (O.) Press-working of Metals 8vo, 3 oo Smith (A. W.) and Marx's Machine Design 8vo, 3 oo Sorel ' s Carbureting and Combustion in Alcohol Engines . (Woodward and Preston) . Large 12 mo, 3 oo Thurston's Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, i oa Treatise on Friction and Lost Work in Machinery and Mill Work... 8vo, 3 oo Tillson's Complete Automobile Instructor i6mo, i 50 mor. 2 oo Titsworth's Elements of Mechanical Drawing Oblong 8vo, i 25 Warren's Elements of Machine Construction and Drawing 8vo, 7 50 * Waterbury's Vest Pocket Hand Book of Mathematics for Engineers. 2j X 5i inches, mor. i oo Weisbach's Kinematics and the Power of Transmission. (Herrmann Klein) 8vo, s oo Machinery of Transmission and Governors. (Herrmann Klein). . .8vo, 5 oo Wood's Turbines 8vo> 2 50 MATERIALS OF ENGINEERING * Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 50' Church's Mechanics of Engineering 8vo, 6 oa * Greene's Structural Mechanics . . 8vo, 2 50 14 Holley and Ladd's Analysis of Mixed Paints, Color Pigments, and Varnishes. Large i2mo, 2 50 Johnson's Materials of Construction 8vo, 6 oo Keep's Cast Iron 8vo, 2 50 Lanza's Applied Mechanics 8vo, 7 50 Maire's Modern Pigments and their Vehicles 12010, 2 oo Martens 's Handbook on Testing Materials. (Henning) 8vo, 7 50 Maurer's Technical Mechanics 8vo, 4 oo Mernman's Mechanics of Materials 8vo, 5 oo * Strength of Materials i2mo, i oo Metcaif f s Steel. A Manual for Steel-users i2mo, 2 oo Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 oo Smith's Materials of Machines I2mo, i oo Thurston's Materials of Engineering 3 vols., 8vo, 8 oo Part I. Non-metallic Materials of Engineering and Metallurgy. . .8vo, 2 oo Part II. Iron and Steel Svo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 oo Treatise on the Resistance of Materials and an Appendix on the Preservation of Timber 8vo, a oo Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel 8vo, 4 oo STEAM-ENGINES AND BOILERS. Berry's Temperature-entropy Diagram. Second Edition, Enlarged. . . .i2mo, 2 oo Carnot's Reflections on the Motive Power of Heat. (Thurston) i2mo, i 50 Chase's Art of Pattern Making i2mo, 2 50 Creighton's Steam-engine and other Heat-motors 8vo, 500 Dawson's " Engineering" and Electric Traction Pocket-book i6mo, mor. 5 oo Ford's Boiler Making for Boiler Makers i8mo, i oo * Gebhardt's Steam Power Plant Engineering 8vo, 6 oo Goss's Locomotive Performance .... 8vo, 5 oo Hemenway's Indicator Practice and Steam-engine Economy i2mo, 2 oo Button's Heat and Heat-engines 8vo, 5 oo Mechanical Engineering of Power Plants 8vo, 5 oo Kent's Steam boiler Economy 8vo, 4 oo Kneass's Practice and Theory of the Injector 8vo, i 50 MacCord's Slide-valves -. Svo, 2 oo Meyer's Modern Locomotive Construction 4to, 10 oo Meyer's Steam Turbines Svo, 4 oo Peabody's Manual of the Steam-engine Indicator i2mo. i 50 Tables of the Properties of Saturated Steam and Other Vapors Svo, i oo Thermodynamics of the Steam-engine and Other Heat-engines Svo, 5 oo Valve-gears for Steam-engines Svo, 2 50 Peabody and Miller's Steam-boilers Svo, 4 oo Pray's Twenty Years with the Indicator Large Svo, 2 50 Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. (Osterberg) i2mo, i 25 Reagan's Locomotives: Simple, Compound, and Electric. New Edition. Large 12 mo, 3 50 Sinclair's Locomotive Engine Running and Management i2mo, 2 oo Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50 Snow's Steam-boiler Practice Svo, 3 oo Spangler's Notes on Thermodynamics i2mo, i oo Valve-gears Svo, 2 50 Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 oo Thomas's Steam-turbines Svo, 4 oo 15 Thurston's Handbook of Engine and Boiler Trials, and the Use of the Indi- cator and the Prony Brake Cvo, 5 oo Handy Tables 8vo, i 50 Manual of Steam-boilers, their 7 esigns, Construction, and Operation.. 8vo, 5 oo Thurston's Manual of the Steam-engine 2 vols., 8vo, 10 oo Part I. History, Structure, and Theory 8v:>, 6 oo Fart II. Design, Construction, and Operation 8vo, 6 oo Steam-boiler Explosions in Theory and in Praci.ice i2mo, i 50 Wehrenfenning's Analysis and Softening of Boiler Feed-water (Patterson) Svo, 4 oo Weisbach's Heat, Steam, and Steam-engines. (Du Bois) Svo, 5 oo Whitham's Steam-engine Design Svo, 5 oo Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. ..Svo, A oo MECHANICS PURE AND APPLIED. Church's Mechanics of Engineering Svo, 6 oo Notes and Examples in Mechanics Svo, 2 oo Dana's Text-book of Elementary Mechanics for Colleges and Schools. .i2mo, i 50 Du Bois's Elementary Principles of Mechanics: Vol. I. Kinematics Svo, 3 50 Vol. II. Statics Svo, 4 oo Mechanics of Engineering. Vol. I Small 4to, 7 50 Vol. II. Small 4to, 10 oo * Greene's Structural Mechanics Svo, 2 50 James's Kinematics of a Point and the Rational Mechanics of a Particle. Large i2mo, 2 oo * Johnson's (W. W.) Theoretical Mechanics isrno, 3 oo Lanza's Applied Mechanics Svo, 7 50 * Martin's Text Book on Mechanics, Vol. I, Statics i2mo, i 25 * Vol. 2, Kinemdu^s and Kinetics . .i2mo, 1 50 Maurer's Technical Mechanics Svo, 4 oo * Merriman's Elements of Mechanics 1200, i oo Mechanics of Materials Svo, 5 oo * Michie's Elements of Analytical Mechanics Svo, 4 oo Robinson's Principles of Mechanism Svo, 3 oo Sanborn's Mechanics Problems Large i2mo, i 50 Schwamb and Merrill's Elements of Mechanism '. . . .Svo, 3 oo Wood's Elements of Analytical Mechanics Cvo, 3 oo Principles of Elementary Mechanics i2mo, i 25 MEDICAL. * Abderhalden's Physiological Chemistry in Thirty Lectures. (Hall and Defren) Svo, 5 oo von Behring's Suppression of Tuberculosis. (Bolduan) i2mo, i oo * Bolduan's Immune Sera i2mo, i 50 Bordet s Contribution to Immunity. (Gay). (In Preparation.) Davenport's Statistical Methods with Special Reference to Biological Varia- tions i6mo, mor. i 50 Ehrlich's Collected Studies on Immunity. (Bolduan) Svo, 6 oo * Fischer's Physiology of Alimentation Large i2mo, cloth, 2 oo de Fursac's Manual of Psychiatry. (Rosanoff and Collins) Large i2mo, 2 50 Hammarsten's Text-book on Physiological Chemistry. (Mandel). . .' Svo, 4 oo Jackson's Directions for Laboratory Work in Physiological Chemistry. ..Svo, i 25 Lassar-Cohn's Practical Urinary Analysis. (Lorenz) i2mo, i oo Mandel's Hand Book for the Bi~-Chemical Laboratory . . i2mo, i 50 * Pauli's Physical Chemistry in the Service of Medicine. (Fischer) i2mo, i 25 Pozzi-Escot's Toxins and Venoms and their Antibodies. (Cohn) I2mo, i oo Rostoski's Serum Diagnosis. (Bolduan) I2mo, i oo Ruddiman's Incompatibilities in Prescriptions 8vo. 2 oo Whys in Pharmacy "mo, i oo 16 Salkowski's Physiological and Pathological Chemistry. (Orndorff) 8vo, 2 SO * Satterlee's Outlines of Human Embryology I2mo, i 25 Smith's Lecture Notes on Chemistry for Dental Students 8vo, 2 50 Steel's Treatise on the Diseases of the Dog 8vo, 3 50 * Whipple's Typhoid Fever. . Large tamo, 3 oo Woodhull's Notes on Military Hygiene i6mo, i 50 * Personal Hygiene I2mo, i oo Worcester and Atkinson's Small Hospitals Establishment and Maintenance, and S ggestions for Hospital Architecture, with Plans for a Small Hospital i2mo, i 25 METALLURGY. Betts's Lead Refining by Electrolysis 8vo, 4 oo Holland's Encyclopedia of Founding and Dictionary of Foundry Terms Used in the Practice of Moulding I2mo, 3 oo Iron Founder i2mo, 2 50 " " Supplement lamo, 2 50 Douglas's Untechnical Addresses on Technical Subjects i2mo, i oo Goesel's Minerals and Metals: A Reference Book i6mo, mor. 3 oo * Iles's Lead-smelting i2mo, 2 50 Keep's Cast Iron 8vo, 2 50 LeChatelier's High-temperature Measurements. (Boudouard Burgess) i2mo, 3 oo Metcalfs Steel. A Manual for Steel-users i2mo, 2 oo Miller's Cyanide Process i2n?o, i oo Minet's Production of Aluminium and its Industrial Use. (Waldo) . . .12010, 2 50 Robine and Lenglen's Cyanide Industry. (Le Clerc) 8vo, 4 oo Ruer's Elements of Metallography. (Mathewson) (In Press.) Smith's Materials of Machines I2mo, i oo Tate and Stone's Foundry Practice i2mo, 2 50 Thurston's Materials of Engineering. In Three Parts 8vo, 8 oo Part I. Non-metallic Materials of Engineering and Metallurgy . . . 8vo, 2 oo Part II. Iron and SteeL 8vo, 3 50 Part HI. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Ulke's Modern Electrolytic Copper Refining 8vo, 3 oo West's American Foundry Practice i2mo, 2 50 Moulder's Text Book i2mo, 2 50 Wilson's Chlorination Process i2mo, i 50 Cyanide Processes i2mo, i 50 MINERALOGY. Barringer's Description of Minerals of Commercial Value Oblong, mor. 2 50 Boyd's Resources of Southwest Virginia 8vo, 3 oo Boyd's Map of Southwest Virginia Pocket-book form. 2 oo * Browning's Introduction to the Rarer Elements 8vo, i 50 Brush's Manual of Determinative Mineralogy. (Penfield) . . 8vo, 4 oo Butter's Pocket Hand-Book of Minerals i6mo, mor. j oo Chester's Catalogue of Minerals 8vo, paper, i oo Cloth, i 25 * Crane's Gold and Silver 8vo, 5 oo Dana's First Appendix to Dana's New " System of Mineralogy. .". .Large 8vo, i oo Manual of Mineralogy and Petrography i2mo 2 oo Minerals and How to Study Them i2mo, i 50 System of Mineralogy Large 8vo, half leather, 12 50 Text-book of Mineralogy 8vo, 4 oo Douglas's Untechnical Addresses on Technical Subjects I2mo, i oo Eakle's Mineral Tables 8vo, i 25 Stone and Clay Products Used in Engineering. (In Preparation.) 17 Egleston's Catalogue of Minerals and Synonyms 8vo, 2 SO Goesei's Minerals and Metals: A Reference Book i6mo, mor. 3 oo Groth's Introduction to Chemical Crystallography (Marshall) 12010, i 23 * Iddmgs's Rock Minerals 8vo, 5 oo Johannsen's Determination of Rock-forming Minerals in Thin Sections 8vo, 4 oo * Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe. 12010, 60 Merrill's Non-metallic Minerals: Their Occurrence and Uses 8vo, 4 oo Stones for Building and Decoration 8vo, 5 oo * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8vo, paper, 50 Tables of Minerals, Including the Use of Minerals and Statistics of Domestic Production 8vo, i oo * Pirsson's Rocks and Rock Minerals I2mo, 2 50 * Richards's Synopsis of Mineral Characters I2mo, mor. i 25 * Ries's Clays: Their Occurrence, Properties, and Uses 8vo, 5 oo * Tillman's Text-book of Important Minerab and Rocks 8vo, 2 oo MINING. * Beard's Mine Gases and Explosions Large 12010. 3 oo Boyd's Map of Southwest Virginia Pocket-boo* form 2 oo Resources of Southwest Virginia 8vo, 3 oo * Crane's Gold and Silver 8vo, 5 oo Douglas's Untechnical Addresses on Technical Subjects i2mo i oo Eissler's Modern High Explosives 8vo, 4 -o Goesei's Minerals and Metals : A Reference Book i6mo, mor. 3 oo Ihlseng's Manual of Mining 8vo, 5 oo * Iles's Lead-smelting i2mo, 2 50 Miller's Cyanide Process i2mo, i oo O'Driscoll's Notes on the Treatment of Gold Ores Svo, 2 oo Peele's Compressed Air Plant for Mines Svo, 3 oo Riemer ' s Shaft Sinking Under Difficult Conditions . ( Corning and Peele) ... Svo , 300 Robine and Lenglen's Cyanide Industry. (Le Clerc) 8vo, 4 oo * Weaver's Military Explosives Svo, 3 oo Wilson's Chlorination Process iimo, i 50 Cyanide Processes xarno, i 50 Hydraulic and Placer Mining. 2rf edition, rewritten i2mo, 2 50 Treatise on Practical and Theoretical Mine Ventilation 12 mo, i 25 SANITARY SCIENCE. Association of State and National Food and Dairy Departments, Hartford Meeting, 1906 Svo, 3 oo Jamestown Meeting, 1907 Svo, 3 oo * Bashore's Outlines of Practical Sanitation I2mo, i 25 Sanitation of a Country House .i2mo, i oo Sanitation of Recreation Camps and Parks I2mo, i oo Folwell's Sewerage. (Designing, Construction, and Maintenance) Svo, 3 oo Water-supply Engineering Svo, 4 oo Fowler's Sewage Works Analyses i2mo, 2 oo Fuertes's Water-filtration Works i2mo, 2 50 Water and Public Health i2mo, i 50 Gerhard's Guide to Sanitary Inspections I2mo, i 50 * Modern Baths and Bath Houses 8vo, 3 oo Sanitation of Public Buildings i2mo, i 50 Hazen's Clean Water and How to Get It Large i2mo, i 50 Filtration of Public Water-supplies Svo, 3 oo Kinnicut, Winslow and Pratt's Purification of Sewage. (In Press.) Leach's Inspection and Analysis of Food with Special Reference to State Control 8vo 7 oo 18 Mason's Examination of Water. (Chemical and Bacteriological) i2mo, i 25 Water-supply. (Considered Principally from a Sanitary Standpoint; . . 8vo, 4 oo * Merrimztn's Elements of Sanitary Engineering 8vo, oo Ogden's Sewer Design izmo, oo Parsons's Disposal of Municipal Refuse 8vo, oo Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis i2mo, 50 * Price's Handbook on Sanitation i2mo, 50 Richards's Cost of Cleanness. A Twentieth Century Problem i2mo, oo Cost of Food. A Study in Dietaries i2mo, oo Cost of Living as Modified by Sanitary Science I2mo, oo Cost of Shelter. A Study in Economics I2mo, oo * Richards and Williams's Dietary Computer 8vo, go Richards and Woodman's Air, Water, and Food from a Sinitary Stand- point 8vo, a oo Rideal's Disinfection and the Preservation of Food 8vo, 400 Sewage and Bacterial Purification of Sewage 8vo, 4 oo Soper's Air and Ventilation of Subways . Large i2mo, a 50 Turneaure and Russell's Public Water-supplies 8vo, 5 oo Venabl's Garbage Crematories in America 8vo, 2 oo Method and Devices for Bacterial Treatment of Sewage 8vo, 3 oo Ward and Whipple's Freshwater Biology i2tno, 2 50 Whipple's Microscopy of Drinking-water 8vo, 3 50 * Typhod Fever Large i2mo, 3 oo Value of Pure Water Large i2mo, i oo Winslow's Bacterial Classification ; i2mo, 2 50 Winton's Microscopy of Vegetable Foods 8vo, 7 50 MISCELLANEOUS. Emmons's Geological Guide-book of the Rocky Mountain Excursion of the International Congress of Geologists Large 8vo, i 50 Ferrel's Popular Treatise on the Winds 8vo, 4 oo Fitzgerald's Boston Machinist i8mo, i oo Gannett's Statistical Abstract of the World 24010, 75 Haines's American Railway Management i2mo, 2 50 * Hanusek's The Microscopy of Technical Products. (Winton) 8vo, 5 oo Owen's The Dyeing and Cleaning of Textile Fabrics. (Standage). (In Press.) Ricketts's History of Rensselaer Polytechnic Instituto 1824-1894. Large i2mo, 3 oo Rotherham's Emphasized New Testament , Large 8vo, 2 oo Standage's Decoration of Wood, Glass, Metal, etc i2mo, 2 oo Thome's Structural and Physiological Botany. (Bennett) i6mo, 225 Westermaier's Compendium of General Botany. (Schneider) 8vo, 2 oo Winslow's Elements of Applied Microscopy I2mo, i 50 HEBREW AND CHALDEE TEXT-BOOKS. Green's Elementary Hebrew Grammar I2mo, i 25 Qesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. (Tregelles) Smail 4 to, half mor. 5 oo 19 N| THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. APR 171933 JftN 2 1935 LD 21-50m-l,'33