MECHANICS OF FLUIDS FOR PRACTICAL MEN. It is observed of Archimedes, by his philosophical biographer Plutarch, in the Life of Marcellus, that " although we might labour long without success in endeavouring to demonstrate from our own invention, the truth of his propositions ; yet so smooth and so direct is the way by which he leads us, that when we have once travelled it, we fancy that we could readily have found it without assistance, since either his natural genius, or his indefatigable application, has given to every thing that he attempted the appearance of having been performed with ease." W^fyrf/ [.MECHANICS OF FLUIDS FOR PRACTICAL MEN, COMPHISING HYDROSTATICS, DESCRIPTIVE AND CONSTRUCTIVE: THE WHOLE ILLUSTRATED BY NUMEROUS EXAMPLES AND APPROPRIATE DIAGRAMS. BY ALEXANDER JAMIESON, LL.D. . Ml Author of" Elements of Algebra," fyc. fyc. $fe. LONDON : WILLIAM S. ORR AND CO., AMEN CORNER, PATERNOSTER ROW. 1837. or THE UNIVERSITY or PREFACE. MECHANICS is the science which inquires into the laws of equilibrium and the motion of bodies, whether solid or fluid. The term originally applied only to the doctrine of EQUILIBRIUM, and in this volume it is used in its primitive signification. The adjunct by which this work has been designated, is meant to convey the idea of a book that is self-instructing, and which, in its details, may furnish those who have not had the benefit of a regular academic education, with expeditious and practical methods of operation, in applying the principles of hydrostatic science to the general and every-day business of mechanics. The volume is therefore a manual of principles combining the twofold properties of precept and example, and exhibiting in a comprehensive view whatever is generally and particularly applicable to the mechanics of practical men. But the same construction will render it available in any course of public or private tuition, in which it may be desired to illustrate by examples those operations which, in practical science, are go- verned by the laws of fluid equilibrium, pressure, and support : for it is hoped that these laws have been demonstrated and illustrated with sufficient expansion to suit the progress of modern discoveries, and to remove some part of that uncertainty which has hitherto prevailed in the opinions of scientific men. VI PREFACE. Should the work achieve this, and contribute also to greater precision than has hitherto been attained in the arrangements, structures, and estimates required for works of public or private utility, one of the objects for which it was originally undertaken will be crowned with success. In the spring of 1838, Gcov fleXovroc, another volume of Mechanics of Fluids will appear, comprising Hydraulics, which, as the term implies, will exhibit the principles of Dynamics in the joint operation of air and water, hydraulic architecture, and the principles of construction of various machines and engines which belong to the mechanics of fluids. In conclusion, we vir tvK\tiq 0avw, I beg leave to observe, that as my own avocations did not allow me sufficient leisure to complete such an undertaking as these volumes are, in any reasonable time, I have availed myself of the services of WM. TURNBULL, the author of a treatise on Cast-iron Beams and Columns ; and it affords me unfeigned satisfaction to ac- knowledge the extent of his abilities and the accuracy of his calculations, in subjects connected with the mechanics of fluids. Quam, sit uterque, libens (censebo) exerceat artem. But having elsewhere alluded to the calculations and exam- ples which abound in this volume, I shall only here remark, that, were it necessary to plead authority for such exercises, I might quote NEWTON himself, who has thus recorded his opinion : " In scientiis ediscendis, prosunt exempla magis quam prtecepta." '' . r ' The wood-engravings, so well executed by Mr. G. VASEY, are sufficiently intelligible, and possess besides the lasting property of being destroyed only with the page in which they appear. A. J. Wyke House , Sion Hill, Isleworth, Oct. 19, 1837. I* CONTENTS. CHAPTER I. DEFINITIONS AND OBVIOUS PROPERTIES OF WATERY FLUIDS, WITH THE PRELIMINARY ELEMENTARY PRINCIPLES OF HYDRODYNAMICS, FOR ESTIMATING THE PRESSURE OF INCOMPRESSIBLE FLUIDS. THE subject introduced, art. 1, page 1. Fluid what, art. 2, page 2. Fluid presses equally in all directions, art. 3, page 2. Lateral pressure equal to the perpendicular pressure, art. 3, page 2. Fluid equally pressed in all directions, art. 4, page 2. Fluid pressure against containing surface, perpendicular to that surface, art. 5, page 2. Fluid, surface of horizontal, or perpendicular to the direction of gravity, art. 6, page 2. The common surface of two fluids which do not mix, parallel to the horizon, art. 7, page 2. The surfaces of fluids continue horizontal when subjected to the pressure of the atmosphere, art. 7, page 2. Fluid particles at the same depth equally pressed, art. 8, page 2. Fluid pressure varies as the perpendicular depth, art. 9, page 2. Fluid pressure measured by the weight of a column whose base is the surface pressed, and altitude the perpendicular depth of the centre of gravity, art. 10, page 3. Fluid pressure identified with a property of the centre of gravity, art. 11, 12, and 13, pages 3 to 7. Fluid pressure assignable, art. 13, page 8, corol. Pressure of different fluids on different plane surfaces immersed in them, as the areas of the planes, the perpendicular depths of the centres of gravity, and the specific gravities of the fluids, art. 14, page 8. Pressure of the same fluid on dif- ferent plane surfaces immersed in it, as the areas of the planes and the perpendicular depths of the centres of gravity, art. 15, page 8. Plane parallel to fluid's surface, pressure on ditto varies as the perpendicular depth, art. 16, page 8. Plane surface inclined to the surface of the fluid, pressure on ditto varies as the perpendicular depth of its centre of gravity, art. 17, page 8. Planes of equal areas, whose centres of gravity are at the same perpendicular depth, sustain equal pressures, whatever may be their form and position, art. 18, page 8. The centre of gravity remaining fixed, the pressure upon a revolving plane is the same at all points of the revolu- tion, art. 19, pages 8 and 9. A plane surface being immersed in two fluids of different densities, if the pressures are equal, the depths of the centres of gravity vary inversely as the densities of the fluids, art. 20, page 9. The same principle demonstrated, art. 21, page 9. VIII CONTENTS. CHAPTER II. OF THE PRESSURE OF NON-ELASTIC FLUIDS UPON PHYSICAL LINES, RECTANGULAR PARALLELOGRAMS CONSIDERED AS INDEPENDENT PLANES IMMERSED IN THE FLUIDS, AND UPON THE SIDES AND BOTTOMS OF CUBICAL VESSELS, WITH THE LIMIT TO THE REQUISITE THICKNESS OF FLOOD- GATES. Physical line when obliquely immersed, pressure on determined, art. 22, pages 10 and 11. Practical rule for calculating ditto, art. 23, page 11. Example to illustrate ditto, art. 24, pages 11 and 12. The same determined for a line when perpendicularly immersed, art. 25, page 12. The same determined for a line obliquely immersed, when the upper extremity is not in contact with the surface of the fluid, art. 26, pages 12 and 13. Pressure determined when the perpendicular depth of the upper extremity is given, art. 26, equation (5), page 14. Practical rule for ditto, art. 28, No. 1, page 14. The same determined v hen the perpendi- cular depth of the lower extremity is given, art. 27, equation (6), page 14. Prac- tical rule for ditto, art. 28, No. 2, page 14. Example to illustrate ditto, art. 29, pages 14 and 15. The pressure determined when the line is perpendicularly im- mersed, art. 30, page 15. Two physical lines obliquely immersed, with their upper extremities in contact with the surface of the fluid, to compare the pressures upon them, under any angle of inclination, art. 31, pages 15 and 16. The lines being unequally inclined, the pressures directly as the squares of the lengths and sines of inclination, art. 31, inf. 1, page 16. When the lines are equally inclined, the pres- sures are directly as the squares of the lengths, art. 31, inf. 2, page 16. The lines being similarly situated the pressures can be compared, art. 31, inf. 3, page 16. The lines being differently situated in the fluid, the pressure is determined, art. 32, page 17. Right angled parallelogram obliquely immersed, pressure determined, art. 33, equation (7), page 1 9. Practical rule for ditto, art. 34, No. 1, page 19. Plane perpendicularly immersed, the pressure determined, art. 33, equation (8), page 19. Practical rule for ditto, art. 34, No. 2, page 19. Example for illustration of the inclined case, art. 35, pages 19 and 20. Example for illustrating the per- pendicular case, art. 36, page 20. The parallelogram obliquely immersed and the longer side coincident with the surface of the fluid, pressure determined, art. 37, equation (9), page 20. Perpendicularly immersed, pressure determined, art. 37, equation (10), page 20. Pressures compared, art. 37,corols. 1, 2, and 3, page 21. Right angled parallelogram immersed as before and bisected by its diagonal, the pressures on the triangles determined and compared, art. 38 and 39, pages 21, 22, 23, and 24. Practical rules for the pressures, art. 40, page 25. Example for illustration, art. 41, page 25. Centre of gravity of a right angled triangle deter- mined, art. 42, pages 26 and 27. Rectangular parallelogram bisected by a line parallel to the horizon, the pressure on the two parts determined and compared, art. 43, pages 27, 28, and 29. Practical rules, art. 44, Nos. land 2, page 29. Examples for illustration, art. 45, pages 29 and 30. Rectangular parallelogram so divided by a line parallel to the horizon, that the pressures on the two parts are equal, art. 46, pages 30, 31, and 32. The same effected geometrically, art. 47, page 32. Practical rule for the calculation of ditto, art. 48, page 33. Example to illustrate ditto, art. CONTENTS. IX 49, page 33. The same determined when the point of division is estimated upwards, art. 50, page 33. Practical rule for calculating ditto, art. 51, page 34. The solution independent of the breadth of the parallelogram, art. .51, page 34, corol. Rectangu- lar parallelogram divided into two parts sustaining equal pressures, by a line drawn parallel to the diagonal, art. 52, pages 34, 35, and 36. Example for illustrating ditto, art. 53, pages 36 and 37. Centre of gravity of the upper portion determined, art. 54, pages 38, 39, and 40. Rules for the co-ordinates, art. 55, pages 40 and 41. Example for illustrating ditto, art. 56, pages 41 and 42. Construction of the figure, art. 57, page 42. The problem simplified when the line of division is parallel to the diagonal, art. 57, page 42. Example for illustrating ditto, art. 57, pages 42 and 43. Remarks on ditto, art. 57, page 43. The pressure not a neces- sary datum, compare the concluding remark, art. 57, page 43, with equations (20 and 21), art. 54, page 40. Rectangular parallelogram divided into two parts sustain- ing equal pressures, by a line drawn from one of the upper angles to a point in the lower side, art. 58, pages 43 and 44. Practical rule for ditto, art. 59, page 45. Example for illustrating ditto, art. 60, page 45. The same determined when the line of division is drawn from one of the lower angles to a point in the immersed length, art. 61, page 46. Practical rule for ditto, art. 62, page 47. Example for illustrating ditto, art. 63, page 47. Centre of gravity determined, art. 64, pages 48,49, and 50. Remark on ditto, art. 66, page 51. Rectangular parallelogram so divided, that the pressures 011 the parts shall be in any ratio, art. 67, 68, and 69, pages 51 and 52. Remarks on ditto, art. 69, page 53. Rectangular parallelogram divided by lines drawn parallel to the horizon, into any number of parts sustaining equal pressures, art. 70, pages 53, 54, 55, 56, and 57. Practical rule for ditto, art. 71, pages 57 and 58. Example for illustrating ditto, art. 72, page 58. Con- struction of the figure, art. 72, page 59. Requisite thickness of floodgates deter- mined, art. 73, page 59. Example for illustrating ditto, art. 73, pages 59 and 60. Construction of the figure, art. 74, page 60. Remarks on ditto, art. 74, page 61. Pressure on the sides and bottom of a rectangular vessel compared, art. 75, pages 61, 62, and 63. Examples for illustrating ditto, art. 76 and 77, page 64. CHAPTER III. ON THE PRESSURE EXERTED BY NON-ELASTIC FLUIDS UPON PARABOLIC PLANES IMMERSED TN THOSE FLUIDS, WITH THE METHOD OF FINDING THE CENTRE OF GRAVITY OF THE SPACE INCLUDED BETWEEN ANY RECTANGULAR PARAL- LELOGRAM AND ITS INSCRIBED PARABOLIC PLANE. Pressure on a parabolic plane, compared with that on its circumscribing rectan- gular parallelogram, art. 77, pages 65 and 66. Practical rules for ditto, art. 78, page 67. Example for illustrating ditto, art. 79, page 67. Centre of gravity of the included space determined, art. 80, pages 67 and 68. Remarks on ditto, art. 80, page 69. The same things determined when the base of the parabola is coincident with the surface of the fluid, art. 81, pages 69 and 70, Practical rules for ditto, art. 82, page 71. Example for illustrating ditto, art. 83, page 71. Remarks on ditto, art. 83, page 72. The same things determined when the base of the parabola is vertical, and just in contact with the surface of the fluid, art. 84, pages 72 and VOL. I. b X CONTENTS. 73. Corollary to ditto, art. 84, page 74. Practical rules for ditto, art. 85, page 73. Example for illustrating ditto, art. 86, page 74. Corollaries on ditto, art. 87, page 75. Pressure on a semi-parabolic plane, compared with that upon its circumscribing parallelogram, art. 88, pages 75, 76, and 77. Practical rule for ditto, art. 89, page 77. Example for illustrating ditto, art. 90, page 77. The same determined when the axis of the semi-parabola is horizontal, art. 91, pages 78 and 79. Practi- cal rules for ditto, art. 92, page 79. Example for illustrating ditto, art. 93, page 79. The centre of gravity of the included space determined, art. 94, pages 80, 81, and 82. Construction of the figure, art. 94, pages 82 and 83. Remark on ditto, art. 94, page 83. Parabolic plane with axis vertical, and vertex in contact with the surface of the fluid, divided by a horizontal line into two parts sustaining equal pressures, art. 95, pages 83, 84, and 85. Practical rule for ditto, art. 96, page 85. Example for illustrating ditto, art. 97, pages 85 and 86. The same things determined generally, art. 97, pages 86 and 87. Example for illustrating ditto, art. 97, page 87. CHAPTER IV. OF THE PRESSURE OF INCOMPRESSIBLE FLUIDS ON CIRCULAR PLANES AND ON SPHERES IMMERSED IN THOSE FLUIDS, THE EXTREMITY OF THE DIAMETER OF THE FIGURE BEING IN EACH CASE IN EXACT CONTACT WITH THE SURFACE OF ; THE FLUID. Chord of maximum pressure in a circular plane determined, art. 98, pages 88 and 89. Construction effected, corollaries 1 and 2, page 89. Practical rules for calculating ditto, art. 99, page 90. Example for illustrating ditto, art. 100, page 90. Pressures on two immersed spheres determined and compared, art. 101, pages 90 and 91. A globe being filled with fluid, the pressure on the interior surface is three times the weight of the contained fluid, art. 101, corol. page 92. Example for illustrating ditto, art. 102, page 92. The exterior surface of a sphere divided by a horizontal circle into two parts sustaining equal pressures, art. 103, pages 92, 93, and 94. Practical rule for ditto, art. 104, page 94. Example for illustrating ditto, art. 105, page 94. The solution generalized, art. 106, page 95< Practical rule for the general solution, art. 107, page 95. CHAPTER V. OF THE PRESSURE OF NON-ELASTIC OR INCOMPRESSIBLE FLUIDS AGAINST THE INTERIOR SURFACES OF VESSELS HAVING THE , FORMS OF TETRAHEDRONS, CYLINDERS, TRUNCATED CONES, &C. Pressure on the base of a tetrahedron compared with the pressure on its sides, art. 108, pages 96, 97, and 98. Comparison effected, corol. page 98. Pressure on the base of a cylinder compared with the pressure on its upright surface, art. 109, CONTENTS-. Xi pages 99 and 100. When in any vessel whatever, the sides are vertical and the base parallel to the horizon, the pressure on the base is equal to the weight of the fluid, art. 110, pages 100 and 101. The concave surface of a cylindrical vessel divided into annuli, on which the pressures are respectively equal to the pressure on the base, art. Ill, pages 101, 102, 103, and 104. The limits of possibility assigned, art. 112, page 104. The equations expressed in terms of the radius, art. 112, page 104. The equation generalized, art. 112, page 104. The practical rule for any annulus, art. 113, page 105. Example for illustrating ditto, art. 114, page 105. The pressure on the base of a truncated cone compared with that on its curved surface, and also with the weight of the contained fluid, art. 115, pages 106, 107, 108, and 109. The same principle extended to the complete cone, base downwards, art. 115, page 108, equations (77 and 80). Comparison completed, art. 115, corol. page 109. The same things determined for a truncated vessel with the sides diverging upwards, art. 116, pages 109, 110, and 111. For the case of the complete cone with the base upwards, see equation (84). Pressure on the base compared with the weight of the contained fluid, art. 117, page 112. The pressure on the base may be greater or less than the weight of the contained fluid in any proportion, art. 117, corol. 1, page 112. The pressure on the bottom of a vessel not dependent on the quantity of the contained fluid, art.117, corol. 2, page 113. Any quantity of fluid, however small, balances any other quantity, however great, art. 117, corol. 3, page 113. Pressure on the bottom of a cylindrical vessel equal to any number of times the fluid's weight, art. 118, pages 114 and 115. Practical rule for ditto, art. 119, page 115. Example for illustrating ditto, art. 120, page 115. Remarks on ditto, art. 120, corollaries 1 and 2, page 116. Concluding remarks on the Hydrostatic Press, page 116. CHAPTER VI. THE THEORY OF CONSTRUCTION AND SCIENTIFIC* DESCRIPTION OF SOME HYDROSTATIC ENGINES, VIZ. THE HYDROSTATIC PRESS, HYDROSTATIC BELLOWS, THE HYDROSTATIC WEIGHING MACHINE, AND EXPERIMENTS PROVING THE QUA QUA VERSUS PRESSURE OF FLUIDS. Principle of the Hydrostatic Press developed, art. 121, pages 117 and 118. First brought into notice by Joseph Bramah, Esq., of Pimlico, art. 122, page 118. Not a new mechanical power, ib. Known under the name of Hydrostatic Paradox ib. Principal element by which the power is calculated, art. 123, page 119. Example to illustrate ditto, art. 124, page 119. General equation for the pressure on the piston of the cylinder, art. 124, equation (89), page 119. Practical rule for reducing ditto, art. 124, page 120. Example for illustrating ditto, art. 125, page 120. General equation for the pressure on the piston of the forcing pump, art. 125, equation (90), page 120. Practical rule for ditto, art. 125, page 120. Example for illustrating ditto, art. 126, page 120. General expression for the diameter of the piston of the cylinder, equation (91), page 121. Practical rule for ditto, art. 126, page 121. Example to illustrate ditto, art. 127, page 121. General expression for the Xll CONTENTS. diameter of piston in the forcing pump, art. 127, equation (92), page 121. Practical rule for ditto, art. 127, page 121. Safety valve introduced, art. 128, pages 121 and 122. General expressions established, art. 128, equations (93 and 94), page 122. Example for illustrating the weight upon the safety valve, art. 129, page 122. General expression for the weight upon the safety valve, art. 129, equation (95), page 122. Practical rule for ditto, art. 129, page 123. Example for illustrating ditto, art. 130, page 123. General expression for the pressure on the piston of the cylinder, art. 130, equation (96), page 123. Practical rule for ditto, art. 130, page 123. Example for illustrating ditto, art. 131, page 123. General expression for the diameter of the safety valve, art. 131, equation (97), page 124. Practical rule for ditto, art. 131, page 124. Example for illustrating ditto, art. 132, page 124. General expression for the diameter of the cylinder, art. 132, equation (98), page 124. Practical rule for ditto, ib. Example for illustrating ditto, art. 133, page 124. General expression for the weight upon the safety valve, art. 133, equation (99), page 125. Practical rule for ditto, ib. Example for illustrating ditto, art. 134, page 125. General expression for the pressure on the piston of the forcing pump, art. 134, equation (100), page 125. Practical rule for ditto, ib. Example for illustrating ditto, art. 135, pages 125 and 126. General expression for the diameter of the safety valve, art. 135, equation (101), page 126. Practical rule fpr ditto, ib. Example for illustrating ditto, art. 136, page 126. General expression for the diameter of the forcing pump, art. 136, equation (102), page 126. Practical rule for ditto, art. 136, page 127. Concluding remarks, ib. The thickness of the metal in the cylinder determined, art. 137, pages 127, 128, 129, 130, and 131. Example for illustrating ditto, art. 138, page 131. Appropriate remarks, art. 139, pages 131 and 132. A determinate and uniform value assigned to the pressure on a square inch of surface, art. 140, page 132. Rules for the pressure in tons, and for the diameter of the cylinder in inches, art. 141, page 133. Remarks on the theory, art. 142, page 133. Examples for illustrating ditto, ib. and page 134. Remarks, art. 143, page 134. Observations on presses previously constructed, with rules and examples for examining them, art. 144 and 145, pages 134 and 135. The description of the Hydrostatic Press, with its several parts and appendages, both in its complete and disjointed state, arts. 146, 147, and 148, pages 136 142. The Hydrostatic Bellows and its uses introduced, art. 149, page 142. Descrip- tion and principles of ditto, art. 150, pages 142 and 143. Practical rule for calculating the weight of fluid in the tube, art. 151, page 144. Example for illustrating ditto, ib. The solidity or capacity of the fluid in the tube determined, art. 152, page 144. Altitude of the tube determined, art. 152, page 145. The solution given generally, art. 153, page 145. Practical rule for determining the height of the tube, art. 154, page 146. Illustrative example and remark, i&. Weight on the moveable board determined in the case of an equilibrium, art. 155, page 146. Practical rule and illustrative example for ditto, ib. Concluding- remark, art. 155, page 147. The diameter of the equilibrating tube determined, art. 156, page 147. Practical rule for ditto, ib.- Example for illustrating ditto, art. 156, page 148. The diameter of the Bellows Board determined, art. 157, page 148. Practical rule and illustrative example for ditto, ib. When the equilibrium obtains, if more fluid be poured into the tube, it will rise equally in the tube and in the bellows, art. 158, page 149. The ascent of the weight on the moving board determined, art. 159, page 150. Practical rule and example for ditto, art. 159, page 151. CONTENTS. Xlll The Hydrostatic Weighing Machine introduced, described, and investigated, art. 160, pages 151, 152, and 153. Practical rule and example for ditto, art. 161, page 154. Quantity of ascent above the first level determined, practical rule and example for ditto, art. 162, pages 154 and 155. Concluding remarks, art. 162, page 155. Quaquaversus pressure of incompressible fluids illustrated by experiments, pages 155-160. CHAPTER VII. OF PRESSURE AS IT UNFOLDS ITSELF IN THE ACTION OF FLUIDS OF VARIABLE DENSITY, OR SUCH AS HAVE THEIR DENSITIES REGULATED BY CERTAIN CONDITIONS DEPENDENT UPON PAR- TICULAR LAWS, WHETHER EXCITED BY MOTION, BY MIXTURE, OR BY CHANGE OF TEMPERATURE. Preliminary remarks on density, pages 161, 162, and 163. The alteration of pressure in consequence of a change of temperature determined, art. 163, pages 163, 164, and 165. The chord in a semicircular plane on which the pressure is a maximum determined, on the supposition that the diameter is in contact with the surface of the fluid, and the density increasing as the depth, art. 164, pages 165, 166, and 167. Practical rule for ditto, art. 165, page 167. Example for illus- trating ditto, art. 166, page 167. Construction of ditto, art. 167, pages 167 and 168. The section of a conical vessel parallel to the base, on which the pressure is a maximum, determined, the axis of the cone being inclined to the horizon in a given , angle, art. 168, pages 168, 169, and 170. Practical rule for ditto, art. 169, page 170. Example for illustrating ditto, art. 170, page 170. Concluding remarks on compressibility, art. 171, page 171. The diameter of a globe determined, in ascending from the bottom to the surface of the sea, on the supposition that the globe is condensible and elastic, art. 172, pages 171 and 172. Practical rule for ditto, art. 173, page 173. Example for illustrating ditto, art. 174, page 173. Remark on ditto, art. 175, page 173. The depth of the sea determined, art. 176, pages 173 and 174. Practical rule and example for ditto, art. 177, page 174. Pressure on the bottoms of vessels containing fluids of different densities deter- mined, art. 178, pages 174 and 175. Law of induction expounded, art. 179, pages 175 and 176. Example for illustration, art. 180, page 176. Another example under different conditions, art. 181, pages 176 and 177. Pressure on the inner surface determined and compared with that upon the bottom, art. 182, pages 177 and 178. Altitudes of fluids of different densities, inversely as the specific gravities, art. 183, pages 178 and 179. Practical rule for the altitudes, art. 184, page 179. Example for illustrating ditto, art. 185, page 180. A column of mercury of 2 feet and a column of water 33.995 equal to the pressure of the atmosphere, art. 186, page 180. The principle of the common or sucking pump dependent on this property, ib. Altitudes in the tubes equal when the specific gravities are equal, art. 187, page 180. Surfaces of small pools near rivers on the same level as the rivers, art. 188, page 180. Water may be conveyed from one place to another, of the same or a less eleva- tion, art. 189, page 180. When the source and point of discharge are on the same level the water is quiescent, but when the point of discharge is lower than the XIV CONTENTS. source, the water is in motion, art. 190, page 180. On this principle large towns and cities are supplied with water, art. 190, page 181. Edinburgh thus supplied, ib. The principle of the proposition art. 183 generalized, art. 191, page 181. Remark on ditto, ib. Two fluids of different densities, hut equal quantities, being poured into a circular tube of uniform diameter, their positions determined when in a state of equilibrium, art. 192, pages 181, 182, 183, and 184. The principle assumed to be similar to that of Mr. Barclay's Hydrostatic Quadrant, art.193, page 184. Practical rule for ditto, ib. Example illustrative of ditto, art. 194, pages 184 and 185. The actual position of the fluids exhibited by a construction, art. 195, pages 185 and 186. Mercury preferable to water for the tubes of philosophical instruments, art. 196, page 186. Other fluids convenient for the purpose, art. 197, page 186. The result of the investigation only applicable to mercury and water, arts. 196 and 198, page 186. The same rendered general, art. 199, pages 186 and 187. The general equation deducible from the particular one, art. 199, page 187. Practical rule for the general case, art. 200, pages 187 and 188. Example for illustrating ditto, art. 201, page 188. The positions of the fluid different in the two cases, art. 202, page 188. Inquiry into the changes produced on the general formulae, under certain conditions, art. 202, page 189. Practical rule for reducing the resulting equation, art. 203, page 189. Concluding remark, ib. CHAPTER VIII. OF THE PRESSURE OF NON-ELASTIC FLUIDS UPON DYKES, EM- BANKMENTS, OR OTHER OBSTACLES WHICH CONFINE THE FLUIDS, WHETHER THE OPPOSING MASS SLOPE, BE PERPENDI- CULAR, OR CURVED, AND THE STRUCTURE ITSELF BE MASONRY, OR OF LOOSE MATERIALS, HAVING THE SIDES ONLY FACED WITH STONE. Introductory remarks, art. 204, page 190. The manner explained in which a dyke, mound of earth, or any other obstacle, may yield to fluid pressure, art. 205, page 191. Remarks on ditto, ib. General investigation, art. 206210, pages 191 197. Reasons adduced for not giving practical rules on this subject, art. 210, page 197. Example for illustrating the reduction of the final equations, art. 211, pages 197 and 198. The same illustrated when the effect produced by the vertical pressure of the fluid is omitted, art. 212, pages 198 and 199. It is safer to calculate by omitting the vertical pressure, art. 212, corol. page 199. The breadth at the foundation determined when the slopes are equal, art. 213, page 199. Example for illustrating ditto, art. 214, pages 199 and 200. The same determined when the side on which the fluid presses is vertical, art. 215, page 200. More expensive to erect a dyke of this form, than if both sides slope, art. 215, corol. page 201. The same determined when the side of the dyke opposite to that on which the fluid presses is vertical, art. 216, page 201. The stability in this case less than when the vertical side is towards the water, art. 216, pages 201 and 202. Thickness of the dyke determined when both its sides are perpendicular to the plane of the horizon, art. 217, page 202. Practical rule and example for this case, art. 218, page 202. Thickness of the wall determined when its section is in the CONTENTS. XV form of a right angled triangle, and first when the fluid presses on the perpendi- cular, art. 219, page 203. Practical rule and example for this case, art. 220, page 203. The same determined when the fluid presses on the hypothenuse, art. 221, pages 203 and 204. Practical rule and example for this case, art. 222, page 204. The construction exhibited, art. 223, pages 204 and 205. Concluding remarks, art. 223, page 205. The thickness of the dyke determined when it yields to the pressure by sliding on its base, art. 224, pages 205 and 206. The resistances of adhesion and friction compared with the weight of the dyke, art. 225, page 206. Example for illustrating the resulting formula, art. 226, page 207. The breadth of the dyke at the top determined, art. 226, page 207. The breadth of the dyke at the bottom determined, when the side on which the water presses is perpendicular to the horizon, art. 227, page 207. The same determined when the opposite side is perpendicular, ib. The same when both sides are perpendicular, ib. The same determined when the section of the wall is triangular, having the side next the fluid perpendicular, and the remote slope equal to the breadth, art. 228, page 207. The same takes place when the remote side is perpendicular, and the slope on which the water presses is equal to the breadth, art. 228, page 208. Concluding remarks, ib. The dimensions of the dyke determined when it is constructed of loose materials, art. 229, pages 208, 209, and 210. Example for illustration of ditto, art. 230, page 210. The conditions necessary for preventing the dyke from sliding on its base determined, art. 231, page 210. When the water presses against the vertical side of a wall, the curve bounding the other side so that the strength may be every where proportional to the pressure, is a cubic parabola, ib. Introductory remarks to Chapter IX, art. 232, page 211. CHAPTER IX. OF FLOATATION, AND THE DETERMINATION OF THE SPECIFIC GRAVITIES OF BODIES IMMERSED IN FLUIDS. The buoyant force equivalent to the weight of the displaced fluid, art. 233, pages 212, 213, and 214. The pressure downwards equal to the buoyant force, art. 233, corol. page 214. The height determined to which a fluid rises in a cylindrical vessel, in consequence of the immersion of a given sphere of less specific gravity, art. 234, pages 214 and 215. Practical rule for ditto, art. 235, page 216. Example for illustrating ditto, art. 236, page 216. The same determined when the sphere and the fluid are of equal specific gravity, art. 237, page 216; Practical rule for ditto, ib. Example for illustrating ditto, art. 238, pages 216 and 217. The height determined to which the fluid rises in a paraboloidal vessel, in consequence of the immersion of a sphere of less specific gravity, art. 239, pages 217, 218, and 219. Practical rule for ditto, art. 240, page 219. Example for illustrating ditto, art. 241, pages 219 and 220. The same determined when the sphere and the fluid are of equal specific gravity, art. 242, page 220. Practical rule for ditto, art. 243, page 220. Example for illustrating ditto, art. 244, pages 220 and 221. Remarks on ditto, art. 244, corol. page 221. A homogeneous body placed in a fluid of the same density as itself, remains at rest in all places and in all positions, art. 245, pages 221 and 222. The upward pressure against the base of a body immersed in XVI CONTENTS. a fluid, is equal to the weight of the displaced and superincumbent fluid, art. 246, page 222. The difference between the downward and upward pressures, is equal to the difference between the weight of the solid and an equal bulk of the fluid, art. 247, page 222. Absolute and relative gravity, what, art. 248, pages 222 and 223. By absolute gravity, fluids gravitate in their proper places, by relative gravity they do not, ib. A heavy heterogeneous body descending in a fluid, has the centre of gravity preceding the centre of magnitude, art. 249, page 223. The reason of this, ib. Concluding remarks, art. 250, page 223. The force with which a body ascends or descends in a fluid of greater or less specific gravity than itself, is equal to the difference between its own weight and that of the fluid, art. 251, pages 223 and 224. The force of ascent and descent is nothing when the specific gravities are equal, art. 251, page 224. When a body is suspended or immersed in a fluid, it loses the weight of an equal bulk of the fluid in which it is placed, art. 252, page 224. When a body is suspended or immersed in a fluid of equal or different density, it loses the whole or a part of its weight, and the fluid gains the weight which the body loses, art. 253, page 225. Bodies of equal magnitudes placed in the same fluid lose equal weights, and unequal bodies lose weights pro- portional to their magnitudes, art. 254, page 225. The same body placed in different fluids, loses weights proportional to the specific gravities of the fluids, art. 255, page 225. When bodies of unequal magnitudes are in equilibrio in any fluid, they lose their equilibrium when transferred to any other fluid, art. 256, page 225. When a body rises or falls in a fluid of different density, the accelerating force, what, art. 257, page 225. When the solid is heavier than the fluid it descends, when lighter it ascends, art. 258, pages 225 and 226; hence relative gravity and relative levity, ib. Practical rule for the general formula, art. 259, page 226. Example for illustrating ditto, art. 260, page 226. The distance of descent determined, when the pressive and buoyant forces are equal, art. 261, pages 226, 227, and 228. Practical rule for calculating ditto, art. 262, page 228. Example for illustrating ditto, art 263, page 2'J8. CHAPTER X. OF THE SPECIFIC GRAVITIES OF FLUIDS, AND THE THEORY OF WEIGHING SOLID BODIES BY MEANS OF NON-ELASTIC FLUIDS. Introductory remarks on specific gravity, end the principles or criteria of com- parison, page 229. The weight lost by a body, is to the whole weight, as the specific gravity of the fluid is to that of the solid, art. 264, pages 229 and 230. The weight which the body loses in the fluid is not annihilated, but only sustained, art. 264, page 230. The weight of a body after immersion determinable, art. 265, page 231. Practical rule for ditto, ib. Example for illustration, art. 266, page 231. - The weights which a body loses in different fluids, are as the specific gravities of the fluids, art. 266, corol. page 231. The real weight of a body determinable by having its weight in water and in air, art. 267, pages 231 and 232. Practical rule for ditto, art. 268, page 232. Example for illustration, art. 269, page 232. The specific gravity of a body determinable from its weight, as indicated in water and in air, art, 270, pages 233 and 234. Practical rule for ditto, art. 271, page 234. CONTENTS. XVII Example for illustration, art. 272, page 234. The magnitude of a globular body determinate from its real weight and density, art. 273, pages 234 and 235. The same determinable from its weight in air and in water, art. 274, pages 235 and 236. Practical rules for ditto, art. 275, page 236. Examples for illustration, arts. 276 and 277, pages 236 and 237. Different bodies of equal weights immersed in the same fluid, lose weights that are inversely as their densities, or directly as their magnitudes, art. 278, page 237. The difference between the absolute weight of a body and its weight in any fluid, is equal to the weight of an equal bulk of the fluid, art. 279, page 237. If two solid bodies of different magnitudes indicate equal weights in the same fluid, the larger body preponderates in a rarer medium, art. 280, page 237. Under the same circumstances the lesser preponderates in a denser medium, art. 281, page 238. If solid bodies when placed in the same fluid sustain equal diminutions of weight, their magnitudes are equal, art. 282, page 238. To determine the equipoising weight, when two bodies equally heavy in air, are placed in a fluid of greater density, the densities of the bodies being different, art. 283, pages 238 and 239. Practical rule for ditto, art. 284, page 239. Necessary remark, ib. Example for illustration, art. 285, page 239. The ratio of the quantities of matter determinable, when two bodies of different specific gravities equiponderate in a fluid, art. 286, page 240. Example for illustration, art. 287, pages 240 and 241. Problem respecting the equiponderating of the cone and its circumscribing cylinder, art. 288, pages 241, 242, and 243. Practical rule for ditto, art. 289, page 243. Example for illustration, art. 290, page 244. To compare the specific gravities of a solid body, with that of the fluid in which it is immersed, art. 291, page 245. Example for illustration, art. 292, page 245. To compare the specific gravities of two solid bodies, when weighed in vacuo and in a fluid of given density, art. 293, pages 245 and 246. Example for illustration, art. 294, page 247. The specific gravities of different fluids compared, by weighing a body of a given density, art. 295, pages 247 and 248. Example for illustration, art. 296, pages 248 and 249. Concluding remarks, ib. The specific gravity of a solid body determined by weigh- ing it in air and in water, art. 297, page 249. The principle of solution explained, ib. The practical rule for ditto, art. 298, page 250. Example for illustration, art. 299, page 250. Concluding remarks, art. 300, page 250. The specific gravity of a solid body determined from that of the fluid in which it is weighed, art. 301, page 251. Practical rule for ditto, art. 302, page 252. Example for illustration, art. 303, page 252. The specific gravity of a solid body determined, by immersing it in a vessel of water of which the weight is known, art. 304, pages 252, 253, and 254. Practical rule for ditto, art. 305, page 254. Example for illustration, art. 306, page 254. Concluding remarks on the value of the opal, ib. CHAPTER XL OF THE EQUILIBRIUM OF FLOATATION. Opening remarks on floatation, page 255. The centre of gravity of the whole body and that of the immersed part occur in the same vertical line, art. 307, pages 255 and 256. The weight of the floating body and that of the displaced fluid are equal to one another, art. 307, page 256. Corresponding remarks, ib. Homogeneous plane figures divided symmetrically remain in equilibrio with their axes vertical, art. 308, page 257. Homogeneous solid bodies generated by the revolution of a XV1I1 CONTENTS. curve, when placed upon a fluid of greater specific gravity, maintain their equili- brium with the axis vertical, art. 309, page 257. Showing under what conditions a prismatic hody remains in equilibrio, art. 310, page 257. The magnitude of a floating body to that of the immersed part, as the specific gravity of the fluid is to that of the solid, art. 311, pages 257 and 258. Examples for illustration, arts. 312 and 313, pages 258 and 259. General determination, art. 314, page 259. Practical rule for ditto, ib. Example for illustration, art. 315, page 260. General investiga- tion, ib. Practical rule and example for ditto, pages 260 and 261. Inferences arising therefrom, arts. 316, 317, 318, 319, and 320, page 261. Demonstration of a general principle, art. 321, page 262. To determine how far a paraboloidal solid will sink in a fluid, art. 322, pages 262 and 263. Practical rule for ditto, art. 323, page 264. Example for illustration, art. 324, page 264. The same determined when the vertex of the figure is downwards, ib. Practical rule for ditto, art. 325, page 265. The elevation or depression of a body determined, when the equilibrium is disturbed by the subtraction or addition of a certain given weight, art. 326, pages 265 and 266. The same thing determined independently of fluxions, art. 327, pages 267 and 268. Practical rule for ditto, art. 328, page 268. Example for illustration, art. 329, page 268. The same determined for a body in the shape of a paraboloid, art. 329, pages 26& and 269. Remark on the resulting equations, art. 330, page 269. The descent occasioned by adding a weight determined, art. 330, pages 269 and 270. Practical rule for ditto, art. 331, page 270. Example for illustration, art. 332, page 270. The ascent determined when a given weight is subtracted, art. 333, page 270. Practical rule for ditto, art. 334, page 271. Concluding remark, art. 335, page 271. The weight determined which is necessary to sink a body to the same level with the fluid, art. 336, pages 271 and 272. Practical rule for ditto, art. 337, page 272. Example for illustration, art. 338, page 272. A solid body being immersed in two fluids which do not mix, floats in equilibrio between them', when the weights of the displaced fluids are together equal to the weight of the body, art. 339, pages 272, 273, and 274. The quantity of each fluid displaced by a cubical body determined, art. 340, pages 274 and 275. Practical rule for ditto, art. 341, page 275. Example for illustration, art. 342, page 275. Another example under different conditions, art. 376, page 343. The specific gravity of a solid body determined, so that any part of it may be immersed in the lighter of two unmixable fluids, art. 344, page 276. Practical rule for ditto, art. 345, page 277. Example for illustration, art. 346, page 277. The same determined when equal parts of the body are immersed in the lighter and heavier fluids, art. 347, page 277. Practical rule for ditto, art. 347, page 278. Example for illustration, art. 348, page 278. The same determined when the lighter fluid vanishes, art. 349, page 278. A very curious property unfolded, art. 349,' page 279. The ratio of the immersed parts determined, when the body floats on water, in air, and in a vacuum, art. 350, pages 279 and 280. Practical rule for ditto, art. 351, page 280. Example for illustration, art. 352, page 280. Remark and rule for determining the same otherwise, art. 353, page 281. The Hydrometer or Aerometer introduced, art. 354, page 281. Improvements on ditto by various writers, art. 354, page 282. Description of the instrument according to Deparcieux, art. 355, pages 282 and 283. The specific gravity of a fluid determined by the aerometer, art. 356, pages 283 and 284. Practical rule for ditto, art. 357, page 284. Example for illustration, art. 358, page 284. The immersed quantity of the stem determined, art. 359, page 285. Practical rule for ditto, 360, page 285. Example for illustra- CONTENTS. XIX tion, art. 361, page 285. Transformation of the equations, art. 362, pages 285 and 286. Concluding remarks, art. 362, page 286. The change in the aerometer cor- responding to any small variation in the density of the fluid, art. 363, page 286. Practical rule for ditto, art. 364, pages 286 and 287. Example for illustration, art. 365, page 287. Remarks, ib. The sensibility of the instrument increased by decreasing the diameter of the stem, and otherwise, art. 366, pages 287 and 288. Practical rule for ditto, art. 367, page 288. Example for illustration and remarks on ditto, art. 368, pages 288 and 289. Hydrostatic Balance introduced and de- scribed, art. 369, pages 289 and 290. The specific gravity of a solid body determined by the balance, that of distilled water being given, art. 370, page 291. Practical rule for ditto, art. 371, page 291. Example for illustration, art. 372, page 291 . CHAPTER XII. OF THE POSITIONS OF EQUILIBRIUM. The positions of equilibrium of a triangular prism determined, art. 373, pages 292, 293, and 294. Remarks on the form of the resulting equation, art. 374, pages 294 and 295. Geometrical construction effected by means of two hyperbolas, art. 374, page 295. Example for illustrating the reduction of the formula, art. 375, pages 295 and 296. Construction and calculation of the figure and its parts, arts. 376, 377, and 378, pages 296, 297, and 298. More positions of equilibrium exhi- bited by the equation, and remarks on ditto, art. 379, page 299. The positions determined when the section of the prism is isosceles, art. 380, pages 299 and 300. Practical rule for ditto, art. 381, page 300. Example for illustration, art. 382, pages 300 and 301. Other positions determined, art. 383, page 301. Remarks on the resulting equation, art. 383, page 302. Practical rules for reducing ditto, art. 384, pages 302 and 303. Limits to the positions of equilibrium, art. 385, page 303. Example for illustration, art. 386, pages 303 and 304. Remarks on ditto, art. 386, page 304. The positions delineated, art. 387, page 304. The same demon- strated, pages 305 and 306. Remarks, art. 388, page 306. The positions of equi- librium dependent upon the specific gravity, art. 389, pages 307 and 308. The true positions delineated according to the conditions of the problem, art. 390, page 308. The positions of equilibrium determined when the section of the body is in the form of an equilateral triangle, art. 391, pages 308 and 309. The positions calculated, art. 392, pages 309 and 310. The positions delineated according to this determination, page 310. Other positions determined, art. 393, pages 310 and 311. The same positions delineated by construction, art. 393, page 31 1. The positions of equilibrium determined for a triangular prism, when two of its edges fall below the plane of floatation, art. 394, pages 311, 312, 313, 314, and 315.- The method of reducing the general equation described, art. 395, page 315. Example to show the method of reduction, art. 396, pages 315, 316, and 317. The position delineated and verified, art. 397, page 317. Another condition of equilibrium, what, art. 398, page 317. The same verified, page 318. Other positions of equilibrium determin- able, the determination left for exercise to the reader, art. 399, page 318. The same determined when the section of the figure is in the form of an isosceles triangle, art. 400, page 318. The equation reduced, art. 401, page 319. Practical rule for reducing the equation, art. 402, page 319. Example for illustration, art. or THE A UNIVERSITY 3 XX CONTENTS. 403, pages 319 and 320. The position delineated by construction, art. 404, page 320. Other two positions of equilibrium determinable, art. 405, pages 320, 321, and 322. Limits to the specific gravity defined when the body floats with two of its angles immersed, art. 405, page 321. Expression for the arithmetical mean between the limits, equation (247), page 321. Expressions for the extant sides of the triangular section, (equations 248 and 249), pages 321 and 322. Remarks on ditto, art. 405, page 322. Example for illustration, art. 406, page 322. Construction indicating the positions, art. 407, page 322. The construction verified by calculation, art. 407, page 323, and art. 408, page 324. The positions of equilibrium determined when the triangular section is equilateral, art. 409, pages 324 and 325. Remarks on ditto, art. 409, page 325. Example for illustration, art. 410, page 325. Construc- tion indicating the positions, art. 410, page 325. The same verified by calculation, art. 410, page 326. Two other positions assignable, ib. The limits of the specific gravity defined, art. 411, pages 326 and 327. Positions of equilibrium determined, art. 411, page 327. Construction indicating the positions, art. 412, page 327. The same verified by calculation, arts. 412 and 413, pages 327 and 328. Con- cluding remarks on ditto, art. 414, pages 328 and 329. The positions of equilibrium determined for a rectangular prism with one of its edges immersed, art. 415, pages 329 to 331. General remarks on ditto, art. 416, page 332. The positions deter- mined when the ends of the prism are squares, art. 417, page 332. General inference from ditto, art. 417, page 333. Practical rule for ditto, art. 418, page 333. Example for illustration, art. 419, page 333. Construction indicating the position, arts. 419 and 420, page 333. Condition establishing the equilibrium, art. 421, page 334. The maximum limit of the specific gravity defined, art. 422, page 334. Two other positions determined, as they depend upon the limit of the specific gravity, art. 423, page 334. Conditions that limit the specific gravity, art. 424, page 334. Positions determined according to the limits, and repre- sented by diagrams, art. 425, pages 334 and 335. Positions determined from the arithmetical mean of the limits, and represented by diagrams, pages 335 and 336. The positions determined when three edges of the prism are immersed, art. 426, page 336. The positions determined when two edges are immersed, art. 427, page 336. Remarks on the superior difficulty of this case, page 337. Pre- paratory construction, ib. The conditions necessary for a state of equilibrium, page 338. The first condition established, equation (265), page 339. The second condition determinable by a separate construction, page 339. The investigation pursued, pages 340, 341, and 342. The positions determined, art. 429, pages 342, 343, and 344. The same determined when the section is a square, art. 430, page 344. When the specific gravity of the prism is half the specific gravity of the fluid, the body sinks to half its depth, art. 431, page 344. This position repre- sented by a diagram, ib. The positions which the body would assume in the course of revolution assigned, page 345. Two other positions assignable, dependent on the limits of the specific gravity, page 345, equations (273 and 274). Construction indicating the position according to the limits, art. 432, page 346. The maximum limit of the specific gravity determined, art. 433, page 346. Remarks, accompanied by diagrams to indicate the positions, art. 434, pages 346 and 347. The arith- metical mean between the limits assigned, art. 435, page 347. The positions assigned numerically, art. 436, pages 347 and 348. The same indicated by con- struction, art. 436, page 348. The reverse positions exhibited, art. 437, page 348. CONTENTS. XXI The construction verified by calculation, art. 438, page 349. The positions of equilibrium determined for a solid, of which the transverse section is in the form of the common parabola, art. 439, pages 349-352. Practical rule for ditto, art. 440, page 352. Example to illustrate ditto, art. 441, page 352. Construction indicating the position of equilibrium, with its verification, art. 442, page 353. The same determinable for the figure in the inclined position, art. 443, pages 353 to 356. Example to illustrate ditto, art. 444, pages 356 and 357. The positions indicated by construction, page 357. The construction verified, page 358. CHAPTER XIII. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. Stability of floating bodies, the subject introduced, art. 446, page 359. General remarks concerning ditto, arts. 447, 448, and 449, pages 359 to 362. Definitions, art. 450, pages 362 and 363. A floating body displaces a quantity of fluid equal to its own weight, and in consequence, the specific gravity of the fluid is to that of the solid, as the whole magnitude is to the part immersed, art. 451, pages 363 and 364. A floating body is impelled downwards by its own weight, and upwards bj the pressure of the fluid, and these forces act in vertical lines passing through the centre of effort and the centre of buoyancy, art. 452, page 364. When these lines do not coincide, the body revolves upon an axis of motion, ib. If a floating body be deflected from the upright position, the stability is proportional to the length of the equilibrating lever, or to the horizontal distance between the vertical lines, passing through the centre of effort and the centre of buoyancy, art. 453, page 364. When this distance vanishes, the equilibrium is that of indifference, ib. When it falls on the same side of the centre of effort as the depressed parts of the solid, the equili- brium is that of stability, art. 453, pages 364 and 365. When it falls on the same side as the elevated parts, the equilibrium is that of instability, art. 453, page 365. Concluding remarks, ib. A general proposition belonging to the centre of gra- vity, art. 454, page 365. General and subsidiary remarks, ib. The stability of floating bodies determined, art. 455, pages 366 to 370. The manner of generalizing the result explained, together with the constituent elements of the equation, art. 456, page 370. Example for illustration, art. 457, pages 370 and 371. The steps of calculation illustrated by reference to a diagram, arts. 458, 459, 460, 461, 462, 463, and 464, pages 371 to 375. Concluding observations, art. 464, page 375. The principle of stability applied to ships, art. 465, page 376. The conditions of the data explained, art. 466, pages 376 and 377. The longer and shorter axes, what, art. 467, page 377. In what respects a ship may be considered as a regular body, art. 468, page 377. The principal section of the water, what, art. 468, page 377. The circumstances and conditions of calculation explained ,by reference to a diagram, arts. 469, 470, 471, and 472, pages 378 to 382. A numerical example for illustration, art. 473, page 382. Table of the measured ordinates, page 383. Construction of the example explained, pages 384 to 387. Construction continued, art. 474, pages 387 to 389. The contents of the displaced volumes, how obtained, art. 475, page 389. Approximating rules for calculating the areas and solidities, art. 476, page 390. The construction completed, art. 477, page 390. The practical delineation of the vertical and horizontal planes, arts. 478 and 479, pages 391 and XX11 CONTENTS. 392. The manner of calculation described, art. 480, page 393. The manner of calculation exemplified, art. 481, pages 39S to 396. Calculation continued, arts. 482 and 483, pages 396 and 397. Concluding remarks, art. 483, pages 397 and 398. Reflections suggested by the importance of the subject, art. 484, page 398. The principles of stability as referred to steam ships considered, art. 485, pages 398 and 399. Reference to Tredgold's work on the Steam Engine, art. 485, page 399. Tredgold's method of simplifying the investigation, art. 486, page 399. His subdivision of the inquiry, ib. The steps of investigation not necessary to be retraced, art. 487, page 399. The expression for stability when the ordinates are parallel to the depth, equation (290), art. 488, page 399. Remarks deduced from the form of the equation, art. 489, page 400. The expression for stability in the ease of a triangular section, equation (291), art. 489, page 400. The practical rule for reducing the equation, art. 490, page 400. Example for illustrating ditto, art. 491, pages 400 and 401. The expression for stability in the case when the trans- verse section is in the form of a common parabola, equation (292), art. 492, page 401. Remarks on its fitness for the purpose of steam navigation, as contrasted with the triangular section preceding, art. 492, page 401. Comparison of the results, art. 493, pages 401 and 402. Method of identifying the rule in the two cases, art. 494, page 402. When the transverse section is in the form of a cubic parabola, the stability is determined by equation (293), art. 495, page 402. Remarks on the superior form in this case, art. 496, page 402. Practical rule for ditto, ib. Example for illustration, art. 497, pages 402 and 403. The stability determined for a parabolic section of the 5th order, equation (294), art. 498, page 403. Remarks on ditto, ib. General remarks in reference to the limiting forms of steam ships, art. 499, page 403. When the ordinates of the transverse section are parallel to the breadth, the stability is expressed by equation (295), art. 500, page 404. When the transverse section is in the form of a triangle, the stability is expressed by equation (296), art. 501, page 404. When the transverse section is in the form of the common parabola, the stability is expressed by equation (297), art. 502, page 404. When the transverse section is in the form of a cubic parabola, the stability is expressed by equation (298), art. 503, page 404. This form superior for stability, ib. When the transverse section is formed by a parabola of the 5th order, the stability is expressed by equation (299), art. 504, page 405. Concluding and general remarks, i&. The stability the same at every section throughout the length, under what conditions this will obtain, art. 505, page 405. CHAPTER XIV. OF THE CENTRE OF PRESSURE. The centre of pressure, subject introduced, definition and preliminary remarks, art. 506, page 406. The centre of pressure determined for a plane surface, art. 507, pages 406, 407, 408, and 409. Formulae of condition, equation (302), page 409. The centre of pressure determined for a physical line, art. 508, pages 409 and 410. Practical rule for ditto, art. 509, page 410. Example for illustration, art. 510, pages 410 and 411. The same determined when the upper extremity of the line is in contact with the surface of the fluid, art. 511, page 411. The same principle applicable to a rectangle, art. 511, page 411. The centre of pressure determined CONTENTS. XXlll for a rectangle when its upper side is in contact with the surface of the fluid, art. 512, pages 411 and 412. The centre of pressure determined when the plane is a square, art. 513, page 412. The centre of pressure determined for a semi-paraholic plane, art. 514, pages 412 and 413. Example to illustrate ditto, art. 515, pages 413 and 414. The same determined for one side of a vessel in the form of a parallelo- pipedon, art. 517, pages 414, 415, and 416. Practical rule and example for ditto, page 416. The balancing force, the centre of pressure, and the direction of its motion determined for the side of a tetrahedron, arts. 518, 519, 520, and 521, pages 417 to 419. Example to illustrate ditto, art. 522, page 420. CHAPTER XV. OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. Capillary attraction and the cohesion of fluids, the subject introduced, with its object, and remarks thereon, art. 525, pages 421 and 422. Definition of capillary attraction, art. 526, page 422. Attraction of cohesion between glass and water, art. 527, pages 422 and 423. The particles of a fluid attract each other, art. 523, page 423. The particles of mercury have an intense attraction for each other, art. 529, page 423. The attraction between glass and water only sensible at insensible distances, art. 530, pages 423 and 424. The manner described in which the attractive influence is exerted, art. 531, page 424. The forms assumed by the summit of the elevated columns described, art. 532, pages 424 and 425. The force of attraction proposed to be determined, art. 533, page 425. The parts by which the fluid in the tube is attracted, described, art. 534, page 426. The same as respects the lower portion of the tube described, art. 535, page 426. The fluid attracted by the glass only, art. 535, pages 426 and 427, A negative force in the opposite direction, art. 536, page 427. The force of attraction determined, and expressed by equation (310), art. 536, page 427. The conditions of the rising and falling column explained, ib. The expression for the force of at- traction generalized, equation (311), page 428. The height to which the fluid ascends in the tube determined, art. 538, pages 428 and 429. The mean altitude expressed, equation (312), page 429. The general expression modified, arts. 539 and 540, pages 429 and 430. The constant determined, equation (315), page 430. Practical rule for the mean height, art. 541, page 430. Example for illustration, art. 542, page 430. The radius of the tube determinable, art. 543, pages 430 and 431. The method illustrated by a numerical example, art. 544, page 431. The height to which the fluid rises between two parallel plates determined, arts. 545, 546, and 547, pages 431, 432, and 433. The practical rule for ditto, art. 547, page 433. Example for illustration, art. 548, page 433. The distance between the plates determinable, art. 549, page 434. Concluding remarks, art. 550, page 434. When two smooth plates of glass meet in an angle, to determine the nature of the curve which the fluid forms between them, art. 551 and 552, pages 435 and 436. The curve determined to be the common or Apollonian hyperbola, art. 553, page 436. Concluding remarks, ib. Two bodies that can be wetted with water, when placed an inch asunder do not approximate or recede ; but if placed a few lines apart, they approximate with an accelerated velocity, art. 554, page 436. Two bodies that cannot be wetted, when placed a few lines apart, approximate with an accelerated velocity, art. 555, page 437. When one body can be wetted and the other XXIV CONTENTS. not, they recede from each other, art. 556, page 437. Remarks on the above, ib. General laws deducible from ditto, arts. 557, 558, and 559, pages 437 and 438. Hydrostatic pressure exemplified in springs and Artesian wells, arts. 560 to 569, pages 439 to 442. CHAPTER XVI. MISCELLANEOUS HYDROSTATIC QUESTIONS, WITH THEIR SOLUTIONS. Miscellaneous hydrostatic questions, with their solutions, arts. 570 to 581, pages 443 to 448. On a careful revision of the sheets, the following are the principal errors that have been discovered. / Art. 9, page 2, for varies in its perpendicular depth, read varies as its perpendi- cular depth. J Page 97, line 17 from top, for p= the pressure upon one of the sides, readp the pressure upon the three containing sides. 1 Page 183, line 10 from bottom,/or cos.x + sin.20 sin.0, read cos.x -J- sin.20 sin.x. _ ^ Page 215, line 16 from top, dele as. Page 230, line 6 from top,/or the specific of the solid, read the specific gravity of the solid ; and in line 10,/or respective gravities, read respective specific gravities. Page 237, line 6 from top,/ar d=\3/ Ill21ll=3.93l3, or nearly 4 inches, read V .5236X7 d= 4s/ W \t/ - 14X16 =0.3939 feet, or 4.7267 inches. V .5236S V .5236X7X1000 line 18 from bottom, far d= . 3 / (13.9975-12)16 . = /(13.9975-12) X =4 ^ 67 incheg . .5 V .52361-001262.5 . .o2o6(s-s) X 62.5 V .5236(1-0012)62. Art. 354, page 282,/or Levi read Lovi. Page 432, line 13 from bottom,/or %bd z Tr, read$d*ir. - line 11 from bottom,/or bd*%bd*ir=$bd*(2 TT), read bd z (l |TT). - line 8 from bottom,y!w m=&<#i-ffed 2 (2 ?r), read m=bdh+bd z (\\ir). - line 3 from bottom,/or {bdh-\-ibd\2 ir)}8g, read {bdh+ibd*(l-$7r)}3g. - line 1 from bottom,/or d{fc+id(l ^TT)}, read d{h+ld(l ITT)}- INTRODUCTION. THE analytical table of contents, which the reader must have perused, will have shown him that this volume is not a selection of shreds and patches, garbled from contemporary authorities : but a systematic treatise on Hydrostatic Science, containing a vast mass of valuable and interesting facts, combining indeed almost all that needs to be known on the equilibrium of fluids. But for the convenience of reference, these Mechanics of Fluids are distributed into a series of chapters, whose titles indicate the several topics that receive mathematical demonstration. The first of these contains, besides a few brief but necessary definitions, the fundamental proposition upon which all the pro- blems that are drawn up in Elementary Hydrostatics are in reality founded. The principle established in the general proposition, enables the reader to proceed in the second chapter with the pressure of incompressible fluids upon physical lines, rectangular parallelo- grams considered as independent planes immersed in the fluid, and to determine the position of the centre of gravity of the various rectangular figures which the successive problems em- brace, together with the pressures of fluids upon the sides and bottoms of cubical vessels, with the limits which theory assigns to the requisite thickness of flood-gates. One distinguishing characteristic in this inquiry is, that every problem is accompanied by a practical example ; and in order that nothing be omitted which could render the subject intelligible to VOL, i. c XXVI INTRODUCTION. the general reader, the most important formulae of a practical and general nature have been thrown into rules, in words at length, whereby all the arithmetical operations required in the solution of the examples, can be performed without any reference to the algebraical investigation, which is the surest way of uniting precept with example. After the same method, the third chapter treats of the pres- sure exerted by non-elastic fluids upon parabolic planes immersed in these fluids, with the method of finding the centre of gravity of the space included between any rectangular parallelogram and its inscribed parabolic plane. This is a valuable proposi- tion in the practice of bridge-building, and it is very satisfactory to find in prosecuting one branch of science, the means of ac- complishing another ; to discover in a subject purely hydrostatic, a method by which to find the position of the centre of gravity of the arch, with all its balancing materials, and consequently many important particulars respecting the weight and mechani- cal thrust, with the thickness of the piers necessary to resist the drift or shoot of a given arch, independently of the aid afforded by the other arches. The method laid down in Problem XII. for this purpose is presumed to be new; at any rate we have not seen it noticed by any previous writer on Mechanics. But its development belongs to Hydraulic Architecture ; the principle here established being all that is required in Hy- drostatics. Chapter IV. introduces the reader to the pressure of non- elastic fluids on circular planes, and spheres immersed in those fluids as independent bodies, the extremity of the diameter of the figure being in each case coincident with the surface of the fluid. These problems could easily have been extended to examples of elliptical planes and solids, but the investigation would not embrace any practical result : and where that is un- attainable, this work presumes not to advance. The Fifth Chapter, in which are classed the tetrahedron, cylinder, conical frustum, and indeed the frustum of any other regular pyramid, completes this branch of fluid pressure ; but the investigation is directed altogether to the pressure of the fluid upon the internal surfaces of the vessels under considera- tion. Indeed this was part of the inquiry when the sphere was INTRODUCTION. treated of in the fourth chapter; but in the fifth, the subject is purely practical, and involves some of the most important prin- ciples in the whole range of Hydrodynamics. The reader now enters upon that remarkable and important principle, That any quantity of fluid, however small, may be made to balance or hold in equilibrio any other quantity, however great ; and is enabled thence to investigate the theory and expound the construction of those mechanical contrivances known as Bramah's hydrostatic press, the hydrostatic bellows, and weigh- ing machine, which are all methods of balancing different intensities of force, by applying the simple power of non- elastic fluids to parts of an apparatus moving with different velocities : and this is all the mechanical powers can effect. The Sixth Chapter, which treats of these hydrostatic engines, their theory of construction and scientific description, com- mences with a distinct proposition ; the first having proved suf- ficient to resolve every problem connected with fluid pressure upon rectilinear and curvilinear figures considered as independent planes immersed in the fluids, together with the pressure of fluids upon the interior surfaces of vessels containing the fluids and belonging to the class of regular bodies, the second pro- position, which the reader now enters upon, involves the prin- ciple whereon depend the construction and appliancy of the hydrostatic press, an engine very generally employed in practical mechanics, and which should therefore be scientifically as it is practically known. But the same proposition extends to the investigation of the hydrostatic bellows, and furnishes the prin- ciple of a particular machine by which goods may be weighed as by the common balance. It may thence be inferred, that as yet, science has but stepped on the threshold of fluids that are heavy and liquid. How far this distinguishing property, the power of transmitting pressure equally in all directions, may yet carry mankind, it would be idle to conjecture. Enough, how- ever, is here shown to satisfy the reader, that in expounding the laws of the pressure and equilibrium of fluids, as well as those of their motion and resistance, he will encounter principles of great practical utility in the construction and use of machines, c2 XXVlll INTRODUCTION. engines, apparatus, and instruments employed, not only in the higher departments of natural philosophy, but in the every-day concerns of society, in the arts, manufactures, and domestic operations of civilized men. The occurrence of such principles seems to present the legitimate time and place for classifying the inventions to which they gave existence, and for directing genius in its attempts to elicit new applications of collateral principles : for though fortuitous circumstances and accidental hints may have led to some discoveries in Hydrodynamics, the greater part of modern improvements must be traced to patient induction, which arrives at those coincidences whereby scientific men are enabled to expound the theory of particular machines, whose construction and principles of action depend upon the equi- librium or motion of fluids. By this method, nothing is taken for granted which can be investigated from a series of mathe- matical truths ; for, as Mr. Whitehurst observes, " it is one thing to assent to truths, and another to prove them to be true : the former leaves the mind in a state of suspense, the latter in the possession of truth." * This chapter concludes with some experiments upon the quaqua versus property of non-elastic fluids ; these experiments have the lowly merit of placing that property, the power of transmitting pressure equally in all directions, in a popular point of view, " level to the capacity of ordinary minds." Our labours hitherto refer exclusively to what may be termed elementary principles in the mechanics of fluids ; we now com- mence with PRESSURE, as it unfolds itself in the action of fluids of variable density,^ or such as have their densities regulated by certain conditions, dependent upon particular laws, whether ex- cited by motion, by mixture, or by change of temperature. This is the subject of Chapter Seventh, in which it will be found that the investigation of the pressure of fluids of variable den- sity is fruitful of some remarkably curious results : among these we may notice the circumstance of a globe of condensible * " Inquiry into the Original State and Formation of the Earth." London, 1792. t The word variable is perhaps taken in a too general sense : the densities are not variable in all cases, they are only different yet they are sometimes variable also ; but there can be no more correct mode of writing upon this subject. INTRODUCTION. XXIX matter immersed in the sea to a given depth, as being likely to suggest some easy and accurate methods of determining the depth of the ocean, when it is so profound as to preclude the appli- cation of the methods npw in use. The next fact claiming our attention here, is the result we obtain by putting fluids of dif- ferent densities into bended tubes, when the perpendicular alti- tudes of these fluids above their common surface will vary inversely as their specific gravity ; for we herein settle at once the grand problem in our domestic policy what is the best method by which large towns and cities, or in fact any place, can be supplied with water from a distance. But this is not all another result is, the construction of the hydrostatic quadrant, for finding the altitude of the heavenly bodies, when from haze or atmospheric obscurity, the horizon is rendered indistinct or invisible. We trust our investigation of this beautiful principle of the pressure of fluids of variable density, will in some mea- sure facilitate the construction of the hydrostatic quadrant an instrument but as yet in its infancy. The Eighth Chapter is one of vast utility in constructive me- chanics, when it is necessary to investigate the pressure of fluids on dykes and embankments, a subject interesting and im- portant in the doctrine of Hydraulic Architecture, and peculiarly applicable to the inland navigation and the maritime accommoda- tion of a country situated like Great Britain, every where inter- sected by canals, and seamed in all the sinuosities of her coast by the tides and waves of the restless and turbulent commercial ocean. Moreover, this subject is particularly applicable to the great works now in progress, as rail-roads, docks, harbours, and basins. The brevity of this chapter is compensated by the unity it confers on separate and distinct portions of fluid pressure and support : and the exact formulae it affords to practical men in estimating expense, while their undertakings are executed with systematic regard to permanent durability. The Ninth Chapter treats of floatation, and the determina- tion of the specific gravities of bodies immersed in fluids, com- prehending therein some of the most interesting and important principles of Hydrodynamic Science. There are two general propositions embraced by this department of the philosophy of fluids : viz. XXX INTRODUCTION. 1st. That when a body floats, or when it is in a state of buoyancy on the surface of a fluid of greater specific gravity than itself, It is pressed uptvards by a force, whose intensity is equivalent to the absolute weight of a quantity of the fluid, of which the magnitude is the same as that portion of the body below the plane of floatation, or the horizontal surface of the fluid. 2dly. That if a solid homogeneous body be placed in a fluid of a greater or less specific gravity than itself, It will ascend or descend with a force which is equiva- lent to the difference between its oivn weight and that of an equal bulk of the fluid; a proposition which is almost self-evident, but which leads to a series of inferences, practically of vast importance in the me- chanics of fluids. Archimedes, the Sicilian philosopher, first established the fun- damental laws of fluid equilibrium, and the specific gravity of bodies immersed in fluids. Having determined the conditions which are requisite to produce and measure the equilibrium of a solid floating on a fluid, the philosopher readily perceived that Two bodies equal in bulk, and immersed in a fluid lighter than either of them, lose equal quantities of their weight ; or inversely, that when Two bodies lose equal quantities of their weight in a fluid, they are of equal volume ; this is the 7th Prob. of his first book De Humido Insidentibus, or Of bodies floating on a fluid. Mathematicians generally suppose Archimedes employed this proposition to solve the well- known problem proposed to him by Hiero, king of Syracuse, who having employed a goldsmith to make a crown of pure gold, and suspecting that the artist had not kept faith with him, applied to Archimedes to discover the truth without injuring the crown. The philosopher, it is said, laboured in vain at the INTRODUCTION. XXXI problem, till, going one day into the bath, he perceived that the water rose in the bath in proportion to the bulk of his immersed body ; it occurred to him at that moment that any other sub- stance of equal 'size would have raised the water just as much, though one of equal weight and of less bulk could not have produced the same effect. He immediately felt that the solu- tion of the king's question was within his reach, for taking two masses, one of gold and one of silver, each equal in weight to the crown, and, having filled a vessel very accurately with water, he first plunged the silver mass into it, and observed the quantity of water that flowed over ; he then did the same with the gold, and found that a less quantity had passed over than before. Hence he inferred that, though of equal weight, the bulk of the silver was greater than that of the gold, and that the quantity of water displaced was, in each experiment, equal to the bulk of the metal. He next made a like trial with the crown, and found it displaced more water than the gold, and less than the silver, which led him to conclude that it was neither pure gold nor pure silver. This discovery by Archimedes, which after all is but the application of the well-known axiom, that two bodies cannot occupy the same space at the same time, has been considered one of the most fortunate in the annals of science, for it has led to great advances in the arts, and become the foundation of chemical analysis ; just in the same way that his development of the properties of floating bodies has formed the rudiments of naval architecture, how much soever this branch of con- structive mechanics may boast of its modern improvements. In the Tenth Chapter, specific gravities and the methods of weighing solid bodies in fluids are treated of; and the principle here to be demonstrated is, That when a solid body is immersed in a fluid of dif- ferent specific gravity from itself, the weight which the body loses will be to its whole weight, as the specific gravity of the fluid is to the specific gravity of the solid. In this chapter we have a full developement of that fine thought, which rendered the truth of experiment an overmatch for the craft of Hiero's goldsmith ; and the examples we have pro- XXX11 INTRODUCTION. duced, though not voluminous, fully show the different ways of solving the ancient problem of Archimedes To find the respective weights of two known ingredients in a given compound. The principle enunciated above, may be popularly expounded in the following manner. Every body placed on a surface of water, has a tendency to sink by its own weight : it is, however, resisted by a force equivalent to an equal bulk of the fluid, or of as much fluid as will fill the space occupied by the body. Should the body be heavier than the fluid, bulk for bulk, its greater weight will cause it to descend, for the upward pressure of the fluid will not prevent the descent. When, on the other hand, the body is specifically, that is to say bulk for bulk, lighter than the fluid, its pressure downwards will be less than the upward pressure of the fluid at the same depth; conse- quently, as the greater force necessarily overcomes the less, and the upward pressure is the greater, the body will rise. When the body and the fluid have the same specific gravity, then equal masses of each are of the same weight, and the de- scending force being equally balanced by the ascending force, the body will float with its upper surface coincident with the surface of the fluid, or in any other position whatever in which it may be placed. It is very obvious from these laws, that if, by any contrivance or change, the specific gravity of a body can be so altered and varied, as to be at one time greater, at another time less, and then equal to the specific gravity of the fluid in which it is placed, the said body will sink, or rise, or remain at rest, accord- ing to the variations produced in its specific gravity. Lecturers amuse their audiences with glass images, which, upon the principle here adverted to, ascend or descend, or remain in mid- water, at the pleasure of these philosophers. The doctrine of the Equilibrium of Floatation, which appears in Chapter XL, is as old as the days of Archimedes, who ex- amines the conditions which are requisite to produce and pre- serve the equilibrium of a solid floating in a fluid. He shows that when a body floats in a state of equilibrium on the surface of an incompressible fluid, INTRODUCTION. XXX111 The centre of gravity of the whole body, and that of the part immersed, must occur in the same vertical line, or the line of pressure and the line of support must coincide; and, secondly, that the magnitude of the body is to that of the part immersed below the plane of floatation, as the specific gravity of the fluid is to that of the floating body. Of the truth of the doctrine which is here propounded, and, let us hope, satisfactorily demonstrated in the sequel of our work, we have a curious illustration afforded by an Arab ship- builder in Java, whose task is thus described in GEORGE EARL'S Eastern Seas: "The largest merchant vessel in Java, a ship about 1,000 tons burden, was built by an Arab merchant, in a long but shallow river, which runs into the sea near Soura- baya. As great expense is incurred by floating the timber in rafts down the river, he determined to commence the work in the forest itself, as he would thereby be enabled to select the best trees for the purpose. He accordingly ascended the river, accompanied by a sufficient number of workmen, conveying the necessary materials, and commenced the undertaking about 80 miles from the sea. When the keel and the floor timbers were laid, and a few of the bottom planks nailed on, he launched the embryo vessel, and floated her gently down the river to a place in which the water was deeper. Here the building was con- tinued, until it became necessary to seek a deeper channel, and in this manner the work proceeded, the vessel being floated further down the river, whenever the water was found to be too shallow for her to float, until at length, she was fairly launched, half finished, into the sea, and completed in the harbour." The operations of this ingenious orientalist proceeded upon the truth stated in Inference 5, page 261, that if a body float in equilibrio on the surface of a given fluid, and if the part below the plane of floatation be increased or diminished by a given quantity, the absolute weight of the body, (in order that the equilibrium might still obtain,) must be increased or dimi- nished by a weight which is equal to the weight of the portion of the fluid that is more or less displaced, in consequence of increasing or diminishing the immersed part of the body, or that which falls below the plane of floatation. XXXIV INTRODUCTION. As his work proceeded, the Arab could calmly and skilfully contemplate the effect of the antagonist forces directed to the centre of gravity and the centre of buoyancy of his ship, and survey her equilibrium as it might be permanent or instable ; even though he knew nothing of the fine theory of M. Bouguer, or the laborious calculations of the Swedish Admiral Chapman or of Mr. Atwood, on the hull of the Cuffnells. But we have other topics of equal practical importance with the floatation of vessels in this chapter, as for example : 1st. The consideration of a body floating in equilibrio between two fluids which do not mix when the weights of the fluids respec- tively displaced, are together equal to the weight of the solid body which causes the displacement: 2dly. The construction and application of the hydrometer, an instrument generally em- ployed for detecting and measuring the properties and effects of water and other fluids, such as their density, gravity, force and velocity, which depends upon the principles explained and illus- trated in the eighth proposition : 3dly. The hydrostatic balance, an instrument by which we are enabled to measure the specific gravities of bodies with great accuracy and expedition, whether the bodies be in a fluid or a solid state. A great many curious facts relating to the equilibrium of floatation could have been here brought under the reader's con- sideration; but these, as well as all popular illustrations of natural philosophy, belong essentially to Somatology, or the properties of matter, a subject which we could not amalgamate with the calculations that illustrate the Mechanics of Fluids. The Twelfth Chapter treats of the positions of equilibrium of floating bodies, to determine which, from strict theory, is one of the finest speculations in the whole range of natural philosophy : to ascertain them, as we have done, by computation, involves nothing intricate or repulsive, though the process is both laborious and irksome. To construct them geometrically, demands a know- ledge of principles higher than elementary. And although the geometrical construction may truly represent the position which the body assumes when floating in a state of equilibrium, it is the application of numbers after all, which must determine the true positions. The reason is this ; the specific gravities of the solid and fluid bodies, which are always elements of the in- INTRODUCTION. XXXV quiry, cannot be represented by lines ; but having once obtained by computation, the dimensions of the extant and immersed portions of the body, the sides of which are always given in the question, we can easily exhibit the geometrical construction. The method of proof, by calculation, which we have applied to this part of our work, seems to leave nothing to be added to an elegant branch of the Mechanics of Fluids, so highly important in the practice of naval architecture. In the Thirteenth Chapter, we have considered the stability of floating bodies and of ships. The subject of stability is the same to whatever form of floating body it may be referred, whether the body be a ship driven by wind or steam, logs of wood, or masses of ice, and it consists entirely in resolving the equation x = S sin. d>. The determination of the se- ra veral quantities of which this equation consists, depends entirely upon calculations drawn from the particular circumstances of the individual case under consideration ; and these circum- stances as referred to a ship, it is impossible to assign by esti- mation ; they must be obtained by actual measurement, and when they have been obtained in this manner, they are to be inserted in the above equation, to obtain the measure of stability. The investigation of this subject is both laborious and intricate, but from what we have done in Problems LXI. and LXIL, with their subordinate examples, it may become intelligible to the general reader. The mathematician who has consulted the writings of the Swedish Admiral CHAPMAN, and the scientific investigations of ATT WOOD, knows well that in considering the properties of a vessel, the orderly arrangement requires that we should treat, First of stability, or the power a vessel has of resisting any change of position when afloat. Secondly, the forms having stability which have the least resistance, and are therefore best adapted for speed. Thirdly, the different methods of propelling ships ; and Fourthly, the construction for strength. But our inquiries are much more limited in this Treatise, and might conveniently end with the exposition of the equation of stability. We have, however, carried the subject a little farther, and considered it in reference to steam navigation, in order to point out that the stability of a ship is greatly increased, byaug- XXXVI INTRODUCTION. meriting the lateral dimension of the water line ; for the easiest and most advantageous way of obtaining stability is by a large area of floatation, and great fulness between wind and water ; or, which is the same thing, by keeping the centre of gravity of the displacement at the least possible distance below the water's surface, in order to obtain the maximum of stability and the fastest rate of sailing : and it will not differ much from the truth to assume the cross section of the vessel, as of the form of a parabola. In this species of figure, the stability and capacity both increase as the ordinate becomes of a higher power ; but a greater breadth is necessary in proportion to the vertical height of the hull to give stability. The breadth, however, should be every where in the same ratio to the depth, to render the sta- bility equal throughout the length, or so that the vessel will undergo no strain from change of position by pitching or rolling in a boisterous sea. The distinguishing characteristic of Chapman's works on ship-building, is the application of the inductive method of philosophy to the different parts of this subject, to found a theory on experimental results, and where data failed, to con- duct his investigations on the acknowledged principles of me- chanics, and subject his conclusions to the test of observation and experiment. His w r orks have never been surpassed ; and in the treatise on ships of war, he collected and gave in detail all the data which affected the qualities of ships, calculated their effects under different circumstances, and determined on theoretical principles, deduced from his experience, the dimen- sions and forms of all ships of war, from a first-rate to the smallest armed vessel. Their calculated elements are collected in tables, and drawings of all the ships constructed agreeably to these elements complete the work, which the reader will find translated by MM. Morgan and Creuze, Naval Architects, in the Papers on Naval Architecture, published about 1830. Next to Chapman's, must be ranked the Treatise of Leonard Euler, on the Construction and Properties of Vessels. The Calculations relative to the Equipment and Displacement of Ships and Vessels of War, by John Edi/e, show by tables and plates, every element and material belonging to the British navy. INTRODUCTION. XXXVH The Fifteenth Chapter embraces cohesion and capillary attraction, subjects replete with many curious speculations, especially in our investigations of the phenomena of fluids. Whatever may be the cause of fluidity, we know that ice becomes water if a certain degree of he#t be applied to it, and steam if more heat be used. Whether therefore, caloric or motion be the cause of fluidity, we know that in the first instance of the case we have cited, the atoms are fixed in crystals in the second they are thrown into intestine motion and in the third state they are forced asunder with an amazing expansive force. Philosophers have usually assumed, that the particles of fluids, since they are so easily moved among one another, are round and smooth. This supposition will account for some circumstances belonging to fluids, as, if the particles are round, there must be vacant spaces between them, in the same manner as there are vacuities between cannon balls when piled toge- ther ; between these balls smaller shot may be placed, and be- tween these, others still smaller, or gravel, or sand, may be diffused. In a similar manner, a certain quantity of particles of sugar can be taken up in a quantity of water without in- creasing the bulk ; and when the water has dissolved the sugar, salt may be dissolved in it, and yet the bulk will not be sensibly augmented ; and admitting that the particles of water are round, this is easily accounted for. Indeed the universal law of gravitation, by which the constituent parts of all bodies mutually attract each other, will cause all such as are fluid, and do not revolve on their own axis, to assume spherical forms. Others have supposed, that the cause of fluidity is the mere want of cohesion of the particles of fluids, which in small quantities, and under peculiar circumstances, arrange themselves in a spherical manner, and form drops. Fluids are subject to the same laws with solids. The parts of a solid are so connected as to form a whole, their weight is concentrated in a single point, called the centre of gravity : but the atoms of a fluid gravitate independently of each other, and press not only like solids perpendicularly downwards, but also upwards, sideways, and in every direction. To the flexibility and cohesion of their particles, is owing the singular property which fluids possess of forming themselves into globules, and of XXXVlll INTRODUCTION. remaining heaped up above the brims of vessels ; and to their attraction of cohesion, may be referred many phenomena in evaporation and solution, their spontaneous ascent in capil- lary tubes, whether natural or artificial, the motion of the various juices through .animal bodies and vegetables, of water through layers of ashes and sand or the rocky strata of the earth and its ascent between plates of glass; to this attraction may be referred solid bodies dissolving in fluids, whose first colour or appearance is not changed, or changed without sen- sible augmentation of the volume ; the mutual action of bodies in contact with each other exhibiting this attraction, as when dry salt of tartar is exposed to the air, it becomes fluid ; the attraction of cohesion evinced in the process of evaporation, as when the warm air of a room is crystallized on the panes of glass during a cold night. We, however, are employed in con- sidering the cohesion of water, which is known to be a com- pound of hydrogen and oxygen, in the proportion of 15 parts of the former, and 85 of the latter. Now this oxygen, which exists in so large a proportion in water, makes exactly one-fourth part of the atmospheric air which all animals breathe. It is the pure part of the air, for the nitrogen or azotic gas which exists in air, in the proportion of three-fourths, is incapable of sustaining animal life or combustion for a single instant. The atmosphere contains besides various supplementary matters, but water is the most abundant, being there found in its different states of cloud, mist, rain, dew, snow answering a thousand useful purposes in the great laboratory of nature, so that upon the whole there is a perfect balancing of actions, preserving the atmospheric mass in a uniform state, constantly fit for its admirable purposes of animal and vegetable existence. The sea-water, however, contains besides hydrogen and oxygen, a solution of muriate of soda, or table salt, which probably adapts this fluid for the purposes of animal life ; at all events preserves the ocean from putrefaction. That the oxygen of the water does not by cohesion or absorption swallow up the oxygen of the atmosphere, and leave the earth to be surrounded with a covering of deadly azotic gas, is perhaps to be accounted for by the general laws of electrical attraction and repulsion, which as they respect the physical constitution of these two INTRODUCTION. XXXIX fluids, preserve a perfect equilibrium between both and to each its own due proportion of the life-giving gas, either as an elastic or a non-elastic fluid. And it is, perhaps, owing to this circumstance operating through the principle of specific gra- vities, that the barometer the prophet of the weather, indicates the changes which diversify the climate of our earth. When the atmosphere becomes surcharged with water it falls as rain, and the weight and bulk of the mass being diminished, the rising column of mercury presages serene and dry weather, as previously the falling barometer had prognosticated wind and rain. Our inquiries cease the moment we approach the limit, which separates chemical analysis from the mechanics of fluids. From the time of Archimedes till the age of Pascal,* the annals of scientific discovery present no improvement in hydro- statics. Pascal has the merit of discovering the pressure of the atmosphere, and his treatise on the Equilibrium of Liquids raised hydrostatics to the dignity of a science. The midnight of barbarism, that for a thousand years had brooded over the discoveries of the Sicilian philosopher, and had concealed the Commentary of Sextus Julius Frontinus on the Aqueducts of Rome, fled before the genius of Pascal and the powers of Newton's mind ; the former, in the most perspicuous and simple manner, demonstrating and proving by experiments the laws of fluid equilibrium ; and the latter expounding the oscillation of waves, a subject the most refined in Hydrodynamic science, which, from that time, counts among its votaries the engineers and philosophers of Italy, France, Sweden, Germany, and Britain. It is proper here to state, that we believe in the compres- sibility of water ; but we hold it true that for all general opera- tions in the mechanics of fluids this compressibility is so small as not to occasion any error in the numerous and varied formulae, from which we have drawn practical rules for the * Pascal gave proof of his skill in hydrostatics, by the celebrated well which he dug at Port Royal des Champs, about six miles from Versailles. The well still exists, in the midst of the farm yard of Les Granges ; but its machinery, by which a child of ten years old could with ease and safety draw up water, is now no more. On one side of the farm yard is a hovel, in which that good man studied during his visits to Port Royal- a place rendered famous also by the name of the devout Arnauld D'Andilli. Xl INTRODUCTION. solution of such questions as may engage the attention of our readers. LESLIE computes that air would become as dense as water at the depth of 33 j miles ; it would even acquire the density of quicksilver at a further depth of 163| miles ; and he hence concludes with the probability that the ocean may rest on a bed of compressed air. Water at the depth of 93 miles would be compressed into half its bulk ; at the depth of 362.5 miles it w r ould acquire the ordinary density of quicksilver. Even marble itself, subjected to its own pressure, would become twice as dense as before, at the enormous depth of 283.6 miles. But air, from its rapid compressibility, would sooner acquire the same density with water, than this fluid would reach the con- densation of marble. For the coincidence of air and water the depth is 35.5 miles ; for equal densities of water and marble 172.9 miles. At the depth of 395.6 miles, or one-tenth the radius of the earth, air would attain the density of 101960 billions ; while at the same depth water would acquire but the density of 4.3492, and marble only 3.8095. At the centre of the earth, the density of air would be expressed by 764 with 166 ciphers annexed ; while water would be condensed three millions nine thousand times its bulk at the surface of the ocean ; and marble would acquire the density of 119. The inference is, that if the structure of our globe were uniform, and its mass consisted of such materials as we are acquainted with, its mean density would far surpass the limits assigned by Astronomy. Now both Dr. Maskelyne and Cavendish nearly concur in representing the mean density at only about five times greater than that of water. Leslie is thence of opinion, that our planet must have a vast cavernous structure, the crust of which, for aught we know to the contrary, may cover some very diffusive medium of astonishing elasticity, as light, which when embodied constitutes elemental heat or fire.* * Elements of Natural Philosophy, vol. i. pp. 447457, second edition. MECHANICS OF FLUIDS CHAPTER I. DEFINITIONS AND OBVIOUS PROPERTIES OF WATERY FLUIDS, WITH THE PRELIMINARY ELEMENTARY PRINCIPLES OF HYDRODYNA- MICS, FOR ESTIMATING THE PRESSURE OF INCOMPRESSIBLE FLUIDS. 1 . THE phenomena of Hydrodynamics are those truths which explain the peculiarities of equilibrium and motion among- fluid bodies, espe- cially those that are heavy and liquid. As that branch of natural phi- losophy which points out and explains the properties and affections of fluids at rest, it comprehends the doctrine of pressure, specific gravity, equilibrium, together with the circumstances attending the positions, equilibrium, and stability of floating bodies, the phenomena of cohe- sion and capillary attraction. And as that other branch of natural philosophy which points out and explains the motions of such fluids as have weight and are liquids, it investigates the means by which such motions are produced, the laws by which they are regulated, the dis- charge of fluids through orifices of various dimensions, forms, and positions, the motion of fluids in pipes, rivers, and canals, and the force or effect they exert against themselves, or against solid bodies which may oppose them. Hydrodynamics, therefore, from Greek words signifying water and force, comprehend the entire science of watery fluids, whether in a state of rest or of motion ; and this science, practically considered, enables us to investigate and apply a fruitful source of maxims and principles, upon which depend the construction and efficiency of engines and machines employed in the arts, manufactures, and domestic concerns of society, together with that extensive class of mechanical combinations displayed in the more delicate and important operations of HYDRAULIC ARCHITECTURE. VOL. i. B * 2 ELEMENTARY PRINCIPLES OF FLUID PRESSURE. 2. A Fluid is a body so constituted, that its parts are all ready to yield to the action of the smallest force or pressure, in whatsoever direction it may be exerted. The following are some of the simplest and most obvious properties of fluids.* 3. Every particle of a fluid presses equally in all directions, whether it be upwards or downwards, laterally or obliquely; consequently, the lateral pressure of a fluid is equal to its perpendicular pressure. The converse of this is equally obvious, and is thus expressed. 4. Every particle of a fluid in a state of quiescence, is pressed equally in all directions. 5. When a fluid is m a state of rest, the pressure exerted against the surface of the vessel which contains it, is perpendicular to that surface. 6. When a mass of fluid is in a state of rest, its surface is horizontal, or perpendicular to the direction of gravity. 7. If two fluids which do not mix, are poured into the same vessel, and suffered to subside, their common surface is parallel to the horizon ; consequently the surfaces of fluids continue horizontal, when sub- jected to the pressure of the atmosphere. 8. The particles of a fluid, situated at the same perpendicular depth below the surface, are equally pressed. 9. When a fluid is in a state of rest, the pressure upon any of its constituent elements, wheresoever situated, is equal to the weight of a column of fluid particles, whose length is equal to the perpendicular depth of the particle or element pressed ; consequently, the pressure on any particle varies ^s its perpendicular depth, and in any vessel containing a fluid in a state of rest, the parts that are deepest sustain the greatest pressure. These principles, which flow immediately from the conditions of fluidity, are too simple and obvious to require demonstration, yet nevertheless, the writers on hydrostatical science generally accompany them with a sort of popular proof, which may be found in almost every treatise that has appeared on the subject. But our immediate object being to unfold the more important elementary principles, by the resolution of a series of examples dependent upon one general proposition, we have thought it unnecessary to exhibit the demonstra- tions here (Note A). The general proposition is as follows : * Fluids are generally divided into two sorts, compressible and incompressible, or elastic and non-elastic ; the latter of which, or incompressible and non-elastic fluids, such as water, mercury, wine, &c., form the subject of the present article ; the dis- cussion of the compressible and elastic fluids, is reserved for another place. The compressibility of water is so small, that in all practical operations in mechanics its bulk or mass may generally be considered unalterable : for at a thousand fathoms depth it can only be compressed one-twentieth of its bulk at the surface. ELEMENTARY PRINCIPLES OF FLUID PRESSURE. PROPOSITION I. 10. When an incompressible and non-elastic fluid is in a state of equilibrium, and subjected only to the action of gravity: The magnitude, or the intensity of pressure exerted by the fluid, perpendicularly to any surface immersed in it, or other- wise exposed to its influence, is measured by the weight of a column of the fluid, whose base is eqval to the area pressed, and whose altitude is the same as the depth of the centre of gravity of that area beneath the upper surface of the fluid. This is an elegant and most important proposition in the doctrine of fluid pressure, and in order that the principle may be the more readily perceived, and the demonstration the more easily comprehended, it will be proper, in the first place, to exhibit and demonstrate an analo- gous property, in reference to the common centre of gravity of a system of bodies, or of the particles of matter of which the system is composed. The property which we have alluded to above, is noticed by almost every writer on the principles of mechanical science, and it has at various times received most beautiful and rigorous demonstrations ; it may therefore, at first sight, appear superfluous to introduce it here; but in order to bring the subject more immediately before the atten- tion of our readers, we do not hesitate to repeat the process. PROPOSITION (A). 11. If there be any system of bodies and a plane given in position with respect to them : The distance of that plane from the common centre of gravity of the system, is equal to the aggregate of the pro- ducts, arising from multiplying each body into its distance from the given plane, divided by the sum of the bodies. The proposition just enunciated, being of the greatest use in many departments of philosophical inquiry, and of essential importance in establishing the truth of the hydrodynamic principle above specified, we shall therefore bestow some attention on its illustration for the purpose of rendering it as clear as possible, by connecting the steps with separate diagrams, -and pursuing the reasoning, until we shall have proceeded so far that the law of induction becomes manifest, and from thence, the truth of the principle announced in the proposition. To accomplish this purpose, let a and b, be two very small bodies E 2 V 4 ELEMENTARY PRINCIPLES OF FLUID PRESSURE. or particles of matter, supposed to be col- lected into their respective centres of gravity, and let A B c D be a smooth rectangular plane or surface, placed in any position with respect to the bodies a and b. Connect a and b by the straight line ab, and let m be the place of their common centre of gravity; draw the straight lines a p, m q and br respectively perpendicular to the plane A B c D, and consequently parallel to one another ; join pr, then because the points a, m, b are situated in a straight line, the points p, q, r are also in a straight line, and therefore p r will pass through the point q. Through m t the common centre of gravity of the two bodies a and b, draw st parallel to pr, meeting br in s, and pa produced in t ; then the triangles ami and bms, are similar to one another; but by the property of the lever, we have a : b :: bm : am, and by similar triangles, it is bm : am : : bs : at; therefore, by the equality of ratios, we obtain a : b : : bs : at ; from which, by equating the products of the extreme and mean terms, we get a X at bx bs. Now, it is manifest by the construction, that at ptpa, and b s r b r s; therefore, by substitution, we obtain a (p tpa)=b (rb rs)', but by reason of the parallels p r and t s, the lines p t and r s are respectively equal to m q ; hence we have a (m q p a) m: b (r b m q), and from this, by collecting the terms and transposing, we get (a + b) mq a X p a ' -f- b X r b, and finally, by division, we obtain a xa b X rb - - a -f- b COROL. Here then, the truth of the proposition is manifest with respect to a system composed of only two bodies ; that is, The distance of the common centre of gravity from the plane to which the bodies are referred, is equal to the sum of the products, arising by multiplying each body into its dis- tance from the given plane, divided by the sum of the bodies. ELEMENTARY PRINCIPLES OF FLUID PRESSURE. 5 12. Again, let a, b and c, be a system of three very small bodies or particles of matter, any how situated with respect to the plane A B c D, and connected together by the straight lines a b, be and a c ; and suppose the two bodies a and b to be collected into their common centre of gravity at the point m. Join the points m and c by the straight line me, and let n be the place of the common centre of gravity of the three bodies a, b and c ; draw the lines mq,nu and cv parallel to each other, and respectively perpendicular to the plane A B c D; join qv, and because the points m, n and c are situated in the straight line m c ; it follows, that the points q, u and v must also occur in a straight line ; consequently, q v will pass through the point u. Through n, the common centre of gravity of the three bodies o, b and c, draw st parallel to qv, meeting mq in t and vc produced in s ; then are the triangles m n t and ens similar to one another ; but by the property of the lever, and because the body at m is equal to a 4- 6, we obtain a ~\- b : c : : c n : m w, and by similar triangles, we have c n : m n : : c s : m t ; therefore, by the equality of ratios, we get a -|- b : c : : c s : m t ; consequently, by equating the products of the extreme and mean terms, we shall obtain (a -f b) X m t zz c X c s ; now mt mq t q, and c sv s vc; hence we get (a -f- b) (mq tq) c(ysvc). But it is manifest by the construction, that t q and v s are each of them equal to nu ; therefore, by substitution we have (a -|- b (mq n ) zr c (nu v c) ; therefore, by collecting the terms and transposing, we get (a + b + c) Xnu=(a+b) Xmq + cXvc; now, it has already been shown in the case of two bodies, that a X^a-f b X rb n <*= ^+T~ therefore, by substituting this value of m q in the step immediately preceding, we shall obtain ELEMENTARY PRINCIPLES OF FLUID PRESSURE. (a 4-6-f- c) Xnu aXp and finally by division, we have _aXpa-\-bXrb-\-cXvc (a + b + c) Now, n u is the distance of the common centre of gravity of the three bodies a, b and c, from A B c D the plane to which they are referred ; hence again, the truth of the proposition is manifest, and if another body were added to the system, a similar investigation would exhibit the same law, and thus we might proceed to any extent at pleasure, the nature of the induction being fully disclosed. COROL. If therefore, we suppose the system to be constituted of an indefinite number of small bodies or particles of matter, it will become assimilated to a fluid mass, and consequently, the proposition which we have just demonstrated in reference to the centre o'f gravity, is identified with the well-known theorem for estimating the pressure of fluids ; to which subject we must now return. 13. Resuming therefore, the conditions specified in Proposition I preceding, let us suppose that ABCD, de- notes a vertical section of a reservoir full of water, E and F representing the corresponding sections of the walls or embankments by which it is contained ; then, since the fluid is supposed to be quiescent or in a state of equilibrium, it follows, that the surface AB is parallel to the horizon. Let b df hkm\>e the portion of the containing section or boundary, on which the pressure exerted by the water is required to be investi- gated, and conceive it to be constituted of an indefinite number of minute bodies or particles of matter, placed at infinitely small dis- tances from one another, or so near, that their aggregate or sum shall make up the entire area which forms the subject of our investigation. Suppose the points b, d, f, A, k and m, to be so many individual particles of the surface pressed, and through the points thus assumed draw the vertical lines b a, dc,fe, hg, ki and m /, which lines are severally in the direction of gravity, and consequently perpendicular to the surface of the fluid, indicating by their lengths, the respective depths of the several bodies of which our immediate system is corn- But according to Proposition I, the pressures exerted by the fluid on the particles 6, d,f, h, k and m, are respectively represented by the products b X ba, dXe?c,/X fe, h X h g, k X k i and m X m I, ELEMENTARY PRINCIPLES OF FLUID PRESSURE. 7 and the aggregate or sum of these products becomes p b X ba + d X dc+fxfe + h X hg -\- k X ki + mX ml, where p denotes the sum of the computed pressures. Now, it is manifest, from what we have demonstrated in Proposition (A), respecting the centre of gravity of a system of bodies, that The sum of the products, arising from multiplying each body into its distance from a certain plane given in position, is equal to the sum of the bodies, drawn into the distance of their common centre of gravity from that plane. Let therefore, the particles b, d,f, h, k and m be considered as a system of very minute bodies, and let the surface of the fluid denote the plane given in position, to which the system is referred ; then, if G be the place of the common centre of gravity of that system, put 71 G == 3, and we shall obtain 8 (b+d+f+h+k+m) = b.ba+d.dc+f.fe+h . hg+k . ki+m.ml. But we have seen above, that the sum of the products on the right hand side of the equation, expresses the aggregate pressure on the several points of the containing surface, to which the present step of the inquiry refers, and that pressure we have briefly represented by the symbol p ; therefore we have and this expression implies, that the pressure exerted by a fluid, on any number of points of the surface that contains it, Is equal to the sum of the points, drawn into, the perpendi- cular distance of their common centre of gravity below the upper surface of the fluid. Now, it is evident, that the same law would obtain if another point were added to the system, and even if the number of points were to become indefinite, or such that their aggregate or sum shall be essentially equal to the area pressed, the law of induction would remain the same ; consequently, if a denote the sum of the material points, or particles of space in the surface on which the fluid presses ; then we shall have p = Za. (1). This equation supposes, that the specific gravity of the fluid by which the pressure is propagated, is represented by unity, which cir- cumstance only holds in the case of water; therefore, let s denote the specific gravity of any incompressible fluid whatever, and the general form of the equation becomes p=$as. (2). 8 ELEMENTARY PRINCIPLES OF FLUID PRESSURE. Now, it is obvious, that the expression 5 a s indicates the weight of a column of the fluid, the area of whose base is a, perpendicular alti- tude S, and the specific gravity s ; hence the truth of the proposition is manifest. COROL. From what has been demonstrated above, it appears, that whatever may be the form of the surface on which the fluid presses, if its area, and the position of its centre of gravity can be ascertained, the intensity of pressure which it sustains, is from thence assignable. The truth of the proposition being thus established, we shall proceed to deduce from it a few of the most useful and obvious inferences. 14. INF. 1. If different planes be immersed perpendicularly, hori- zontally, or obliquely, in fluids of different specific gravities : The pressures upon those planes perpendicularly to their surfaces, are as their areas, the perpendicular depths of their centres of gravity, and the specific gravities of the fluids jointly. 15. INF. 2. If different planes be immersed perpendicularly, hori- zontally, or obliquely in the same fluid : The pressures upon those planes perpendicularly to their surfaces, are as their areas, and the perpendicular depths of their centres of gravity. 16. INF. 3. If a plane surface of given dimensions be parallel to the surface of the fluid in which it is immersed : The pressure sustained by the plane, in a direction perpen- dicular to its surface, varies directly as its vertical depth below the upper surface of the fluid. 17. INF. 4. If a plane surface of given dimensions be any how inclined to the surface of the fluid in which it is immersed : The pressure sustained by the plane, in a direction perpen- dicular to its surface, varies directly as the vertical depth of its centre of gravity, below the upper surface of the fluid. 18. INF. 5. If any number of planes of equal areas be immersed in the same fluid, and have their centres of gravity at the same vertical depth below the surface : The pressures which they sustain are equal to one another, whatever be their form, and whatever be their position with respect to the surface of the fluid. 19. INF. 6. If any plane surface revolve about its centre of gravity, which remains fixed in position : ELEMENTARY PRINCIPLES OF FLUID PRESSURE. 9 The pressure which it sustains in a direction perpendicular to its surface, will be the same at every point of the revolution .-, as if it remained constantly horizontal. 20. INF. 7. If the perpendicular pressures upon a given surface be equal, when it is immersed in two fluids of different densities : The perpendicular depths of the centres of gravity below the surface, will vary inversely * as the densities or specific gravities of the fluids. 21. The above inferences are immediately deducible from the general proposition, but it is probable that the last may require a little illustration ; for which purpose Put p =. the pressure sustained by the plane in both the fluids, a~ the area of the plane or the surface pressed, s the density or specific gravity of one of the fluids, d = the depth at which the given surface is immersed in it, / the density or specific gravity of the other fluid, and 5 = the depth of immersion. Then, according to the principle indicated by the general equation (2), we have, in the case of the first fluid, pdas, and in the case of the second fluid, it is p :zr 5 a s f ; but according to the conditions of the question, these expressions are equal to one another, for the pressure is the same in both cases ; con- sequently by comparison, we have d a s ~ S a /, and this, by suppressing the common factor, becomes ds = W; therefore, by converting this equation into an analogy or proportion, we shall exhibit the precise conditions of the inference ; hence, we have d : a : : / : s. * One quantity is said to vary inversely as another, when of two quantities the one increases as the other decreases. CHAPTER II. OF THE PRESSURE OF NON-ELASTIC FLUIDS UPON PHYSICAL LINES, RECTANGULAR PARALLELOGRAMS CONSIDERED AS INDE- PENDENT PLANES IMMERSED IN THE FLUIDS, AND UPON THE SIDES AND BOTTOMS OF CUBICAL VESSELS, WITH THE LIMIT TO THE REQUISITE THICKNESS OF FLOODGATES. 1. OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. THE principle established in the general proposition enables us now to proceed with the resolution of a numerous class of curious and important problems, which will be found of the greatest practical utility, in all cases in which the pressure of watery fluids is concerned. These problems we shall accompany by examples, which will unfold their geometrical and analytical character, and leave no truth in the phenomena of this branch of hydrodynamics unrevealed. PROBLEM I. 22. A physical line,* of a given length, is obliquely immersed in an incompressible fluid in a state of equilibrium, in such a manner that its upper extremity is just in contact with the surface ; It is required to determine what pressure it sustains, the angle of obliquity being a given quantity. Let ABC, represent a vertical or upright section of a lake or pool of stagnant water, confined by the walls or embankments of which E E is a vertical section, and let A B be the | surface of the water, supposed by the problem to be in a state of equilibrium. In A B take any point a, and at the point a thus assumed, immerse the line a b of the given length, and tending downwards at the given inclination or angle b a A. * A physical line is that which belongs to, or exists in nature, and is so called to distinguish it from a mathematical line, which exists only in the imagination. OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. 11 Bisect a 6 in m, and through the point m draw mn perpendicular to AB, the surface of the fluid; then, because the centre of gravity of, a straight line is at the middle of its length, m is the place of the centre of gravity, and nm its perpendicular depth below the surface AE; through b draw the straight line be parallel to mn, and cb is the perpendicular depth of the lower extremity at b. Put Z zr a 6, the length of the line whose upper extremity is at a, d~=.nm, the perpendicular depth of the centre of gravity, $ ~ b a c, the angle of inclination, or the given obliquity. Then, because m is the centre of gravity of the straight line a b, we have a m zz \ I, and by the principles of Plane Trigonometry ,we obtain rad. : sin. (j> : : \ I : d, and since the tabular radius is expressed by unity, we get d \ I sin. . Now, the whole pressure which the line sustains in a direction perpendicular to its length, according to the second inference pre- ceding, Is proportional to its area, drawn into the perpendicular depth of its centre of gravity below the upper surface of the fluid. But the area of a physical line is simply equal to its length ; therefore, if the symbol p denote the pressure, and s the specific gravity of the fluid by which it is propagated, we shall have p = isl*sm.. (3). and this, in the case of water, where the specific gravity is expressed by unity, becomes p~ JZ*sin. 0. 23. This equation, as well as the more general one from which it is derived, is sufficiently simple in its form for practical application; but in order that nothing may be omitted, which tends to render the subject intelligible to our readers, we shall in this, and in all the succeeding formulae of a practical or general nature, draw up a rule, describing the manner in which the several steps of the process are to be performed ; pursuant to this plan, therefore, the rule for the present case will be as follows : RULE. Multiply the square of the length by half the specific gravity of the fluid, and again by the natural sine of the angle of inclination, and the product will express the required pressure on the line in the oblique position. 24. EXAMPLE 1. A physical line whose length is 36 feet, is im- mersed in a cistern of water, in such a manner that the upper extremity 12 OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. is just in contact with the surface, and the other inclining downwards in an angle of 67 35'; what pressure does the line sustain, supposing the fluid in which it is placed to be in a state of equilibrium ? Here by the question, the fluid in which the line is supposed to be immersed is water, of which the specific is unity; consequently, according to the rule, we have p zz 36 X 36 X \ X sin. 67 35' ; but by the Trigonometrical Tables, the natural sine of 67 35' is .92444 ; hence we get p=\296 X J X .92444 = 599.03712. In this case, however, the resulting pressure is only relative, the absolute pressure being indeterminable, upon a line where length merely is indicated and no breadth assigned ; the existence of surface being indispensable for the expression of a determinate measure. 25. If the line were immersed perpendicularly in the fluid, or so as to make a right angle with its surface, the equation (3) would become transformed into p=%sl*sin. 90; but by the principles of Trigonometry, we have sin. 90 = 1 ; hence, by substitution, we obtain j.= JP; (4). and this, in the case of water, where the specific gravity is unity, becomes P =ii*. Therefore, the relative pressure for a perpendicular immersion, on the line, as given in the above example, is p = 36*36* 1=648. 26. If the upper extremity of the line be not in contact with the surface of the fluid, but placed as in the annexed diagram, then the method of solu- tion, and consequently the form of the re- sulting equation, will be somewhat different. Let A B be the surface of the water or fluid in which the line is immersed, and A B c D a vertical section, in whose plane the line a b is situated, E E being the corresponding section of the walls or embankments by which the fluid is contained. Bisect the given line ab in m, and through the point m thus deter- mined, draw m n perpendicular to A B, the surface of the fluid ; and through a and b the extremities of the given line, and parallel to mn, draw ad and be, and produce ba to meet AB in A, or in any other OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. 13 point, according to circumstances ; then is mn the depth of the centre of gravity of the line a b, below the surface of the quiescent fluid, and ad, be are respectively the depths of its extremities, b AC being the angle which the direction of the given submerged line makes with the horizontal line A B. Put d=ad, the depth of the upper extremity of the given line, ^zrmTi, the depth of the centre of gravity, D in b c, the depth of the lower extremity, I zz a b y the length of the proposed line, p zz the relative pressure upon it as propagated by the fluid, and ^ zz b A c, the angle which the given line makes with the horizon. Through a the upper extremity of the given line, draw ae parallel to AB the surface of the fluid; then is the angle bae equal to the angle b A c, and by the principles of Plane Trigonometry, we have a b : b e : : rad. : sin. ; but be is manifestly equal to be ad; that is, 6ezzD c?, and according to our notation, ab l; hence, the above analogy becomes I : (D c?) : : rad. : sin. -\- d; and the relative pressure becomes />zz J I* sin. p Id. 14 OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. But the equation, in its present form, supposes the specific gravity of the fluid to be expressed by unity, which only takes place in the case of water ; in order, therefore, to generalize the formula, we must introduce the symbol which denotes the specific gravity ; hence, we obtain pi= |/ 2 s sin. -\- Isd; or by collecting the terms, we get p ls(\ Zsin.^>-|-c?). (5). 27. This is the general form of the equation, on the supposition that the perpendicular depth of the upper extremity of the line is given; it however assumes a different form, when the depth of the lower extremity is known ; for by Plane Trigonometry, we have as above b erz:sin.0, and by subtraction, we obtain ec bc be; that is, dznp / sin. ; therefore, the perpendicular depth of the centre of gravity is 3=iD \ I sin. cf), and consequently, the general expression for the pressure becomes p=ls(o \ I sin. 0). (6). 28. Therefore, the practical rule for each of these cases, when expressed in words at length, is as follows : 1. When the perpendicular depth of the upper end is given (5). RULE. To half the length of the given line drawn into the natural sine of the angle of inclination, add the depth of the upper extremity ; then., multiply the sum by the length of the line, drawn into the specific gravity of the fluid, and the pro- duct will give the pressure sought. 2. When the perpendicular depth of the lower end is given (6). RULE. From the perpendicular depth of the lower extre- mity, subtract half the length of the given line drawn into the natural sine of the angle of inclination ; then, multiply the remainder by the length of the line, drawn into the specific gravity of the fluid, for the pressure sought. 29. EXAMPLE 2. A physical line, whose length is 56 feet, is immersed in a cistern of water, in such a manner that its upper extremity is at the distance of 9 feet below the surface, and its direction making with the horizon an angle of 58 degrees ; required the relative pressure on the line, the water being in a state of quiescence ? The natural sine of 58 degrees, according to the Trigonometrical Tables, is .84805 ; therefore by the rule, we have 28 X .84805 + 9 = 32.7454, the perpendicular OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. 15 depth of the centre of gravity ; then, finally, because the specific gravity of water is unity, we have p 32.7454 X 56 = 1833.7424. Let the length of the line and its inclination to the horizon remain as above, and suppose the depth of the lower extremity to be 56.4908 feet ; then, by the rule for the second case, we have 56.4908 - 28 x .84805= 32.7454, the depth of the centre of gravity, the same as above, from which the relative pressure is found to be 1833.7424, as it ought to be. 30. If the line were immersed perpendicularly, or at right angles to the horizon, then sin. is equal to unity, and the formulse for the pressure become p=l8($l + d), and^ = Z 5 (D J I), where it is manifest, that the parenthetical expressions are equal to one another, each of them expressing the perpendicular depth of the centre of gravity, or the middle point of the given line. PROBLEM II. 31. Two physical lines of different given lengths, have their upper extremities in contact with the surface of an incompressible and non-elastic fluid in a state of equilibrium : It is required to compare the pressures which they sustain at right angles to their lengths, supposing them to be immersed at given inclinations to the horizon. Let A B c D, represent a vertical section of a vessel filled with water, or some other incompressible and non- elastic fluid, and suppose the lines a b and cdto be situated in the plane of the section, in such a manner that the upper extremities a and c are respectively in contact with A B the surface of the fluid, while their directions make with the horizon the angles ba A and dc B respectively. Through the points b and d, the lower extre- mities of the lines ab and cd, draw be and df respectively perpen- dicular to A B the surface of the fluid ; and through m and r, the middle points of a b and cd, draw the lines mn and rs respectively parallel to the perpendiculars b e and df; then are mn and rs the perpendicular depths of the centres of gravity. Put I zz a b, the length of the line whose upper extremity is a, I' cd, the length of that whose upper extremity is c; d nm, the perpendicular depth of the centre of gravity of the line a b ; 16 OF THE PRESSURE OF FLUIDS OK PHYSICAL LINES. 3rz s r, the perpendicular depth of the centre of gravity of the line e d ; ^>:n b a A, the inclination of the line a b to the horizon, 0'= d c B, the inclination of c d to the horizon, or to the line A B ; j^nrthe relative pressure upon a b, j/zzthe relative pressure upon cd, and 5 the specific gravity of the fluid. Now, because the centre of gravity of a physical straight line is at the middle of its length, we have a m \ I, and c r rz J /'; therefore, by the principles of Plane Trigonometry, we obtain from the right-angled triangle a m n d=% lsiu.(j>, and from the right-angled triangle c r s we get = | V sin.0. consequently, the general expressions for the relative pressures on the lines a b and c d, according to equation (5) are p~ \ r s sin.0, and;/ J l'*s sin. 0', from which, by comparison, we get p :p' : : I* sin. : Z' a sin. ^'. INF. 1. Hence it appears, that the pressures on the lines, when their directions make different angles of inclination with the horizon, Are directly as the squares of the lengths, and the sines of the inclinations jointly. 2. Where $=<', that is, when the lines are equally inclined to the horizon, whatever may be the magnitude of the inclination, then p:p'::l'-.l"; therefore, when the lines are perpendicularly immersed, or when they are equally inclined to the surface of the fluid, with which their upper extremities are supposed to be in contact, The pressures which they sustain perpendicular to their lengths, are directly proportional to the squares of those lengths. 3. Consequently, if two or more lines are similarly situated in the same fluid, the relative pressures can easily be compared ; thus, for example : Suppose two physical lines, whose lengths are respectively 36 and 56 feet, to be perpendicularly immersed in the same fluid, and having their upper extremities in contact with the surface, or equally depressed below it; then, the pressures sustained by these lines, are to one ano- ther as the numbers 1296 and 3136 ; that is, pip':: 36 2 : 56' :: 1296 : 3136. OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. 17 But when the lines are differently situated in the fluid, the compa- rison of their relative pressures requires a more particular exemplifi- cation; for which purpose take the following example. 32. EXAMPLE 3. Two physical straight lines, whose lengths are respectively 18 and 27 feet, are immersed in the same fluid, in such a manner that their upper extremities are just in contact with its surface, and the angles which they make with the horizon are respec- tively equal to 42 and 29 degrees ; what is the pressure on the longer line, supposing that on the shorter to be expressed by the number 78.54?. If we convert the preceding analogy for the oblique lines of different inclinations into an equation, by making the product of the mean terms equal to the product of the extremes, we shall obtain Now, by assimilating the several quantities in this equation to the lines in the foregoing diagram, and according to the conditions of the question, it appears that p' is the required quantity, all the rest being given ; therefore, let both sides of the equation be divided by / 2 sin. 0, and we shall obtain t , _ p /* sin. ' But it is a well-known principle in the arithmetic of sines, that to divide by the sine of any arc, is equivalent to multiplying by the cosecant of that arc ; hence we have , p'=(p sin. 0' cosec.). Let therefore the numerical values, as proposed in the example, be substituted for the respective symbols in the above equation, and we shall obtain .972 ' = (78.54 sin. 29 cosec.42) ; 18 2 now, the natural sine of 29, according to the Trigonometrical Tables, is .48481, and the natural cosecant of 42 is 1.49447 ; therefore, by substitution, we get if ?Z!x 78.54 X .48481 X 1.49447 128 nearly ; 18 2 consequently the pressures on the inclined lines, are to one another as the numbers 78.54 and 128; but had the inclinations been equal, the comparative pressures would have been as 78.54 to 176.72 very nearly. VOL. i. c 18 2. OF THE PRESSURE OF FLUIDS THAT ARE NON - ELASTIC UPON RIGHT ANGLED PARALLELOGRAMS CONSIDERED AS INDEPENDENT PLANES IMMERSED IN FLUIDS. PROBLEM III. 33. A right angled parallelogram is immersed in a quiescent fluid, in such a manner, that one of its sides is coincident with the surface, and its plane inclined to the horizon in a given angle : It is required to determine the pressure perpendicular to the plane, both when it is inclined to the surface of the fluid, and when it is perpendicular to it, the nature of the fluid, and consequently its specific gravity, being known* Let A B c D represent a vertical section of a volume of incompressible fluid in a state of equilibrium, of which A B E F is the surface, and consequently parallel to the horizon ; let a b c d be a rectangular plane immersed in the fluid, in such a manner that the upper side a b coincides with the surface, and the plane abed is inclined to the horizon in a given angle. Draw the diagonal a c, which bisect in m, and through m the centre of gra- vity of the parallelogram, draw mn parallel to ad or be, meeting ab the line of common section perpendicularly in the point n. In the horizontal plane A BE F, and through the point n, draw nr also at right angles to ab, and from m the centre of gravity of the immersed plane abed, let fall the perpendicular m r ; then is the angle m n r the inclination of the plane to the horizon, and rm the perpendicular depth of its centre of gravity below the upper surface of the quiescent fluid. Put b = ab, the horizontal breadth of the immersed parallelogram, / zn a d or b c, the immersed length, d-=rm, the perpendicular depth of the centre of gravity, mn r, the inclination of the plane to the horizon, p = the pressure on the plane perpendicularly to its surface, and 5 the specific gravity of the fluid. * By the pressure upon any plane or curvilineal surface, is always understood the aggregate of all the pressures upon every point of those surfaces, estimated in directions perpendicular to them at each point, no part heing lost by obliquity of direction. OF THE PRESSURE OF FLUIDS ON RIGHT ANGLED PARALLELOGRAMS. 19 Then, because the point m is at the middle of e, and mn parallel to ad, it follows, that m win \ I; and by reason of the right-angled triangle m r n, we have, from the principles of Plane Trigonometry, rm~d^l sin.^ ; consequently, the entire pressure upon the plane perpendicularly to its surface, is expressed by p ~\ b I* s sin. (f>. This is manifest from Problem I. (art. 22), for \l sin. expresses the perpendicular depth of the centre of gravity, and b I the area of the surface pressed ; therefore, the solidity of the fluid column is \ I sin. X bl%bl* sin. 0, and since s denotes the specific gravity of the fluid, the weight of the column is | I sin. XblXs %bl*s sin. ; but the perpendicular pressure upon the plane, is equal to the weight of the fluid column ; therefore, we obtain p~ J6Z s ssin. 0. (7). When the plane of the immersed rectangle is perpendicular to the surface of the fluid, we have in 90, and sin.^> zz: 1 ; consequently, by substitution, the above equation becomes P ibl z s. (8). These equations are sufficiently simple in their form for practical application, and we shall show hereafter, that they are extremely useful in many important cases of hydrostatical construction. 34. The practical rules derived from these equations, for determining the pressure in the particular cases, may be expressed as follows. 1. When the plane is oblique to the horizon. (Eq. 7). RULE. Multiply the square of the immersed length of the plane, by the horizontal breadth drawn into the specific gravity of the fluid, and again by the natural sine of the angle of inclination,, and half the product will give the pressure sought. 2. When the plane is perpendicular to the horizon. (Eq. 8). RULE. Multiply the square of the immersed length of the plane, by the horizontal breadth drawn into the specific gravity of the fluid, and half the product will give the pressure sought. 35. EXAMPLE 4. A rectangular parallelogram, whose sides are re- spectively 1 8 and 3 feet, is immersed in a quiescent body of water, in such a manner, that its shorter side is in contact with the surface, c 2 20 OF THE PRESSURE OF FLUIDS and its plane inclined to the horizon in an angle of 68 degrees; required the pressure which it sustains, both in the inclined and the perpendicular position ? In this example the area of the parallelogram is 1 8 X 3 == 54 square feet, and the longer side is that which is immersed downwards in the fluid; therefore, according to the rule for the oblique position, the solidity of the column by which the pressure is propagated, becomes 3 X 18 2 X s X 1 sin. 68 = 486 X s sin. 68. Now, in the case of water, the specific gravity is represented by unity, and by the Trigonometrical Tables, the natural sine of 68 degrees, is 0.92718 ; consequently, by substitution, the pressure becomes ^ = 486x .92718 = 450.60948; the pressure here obtained, however, is estimated in cubic feet of water ; but in order to have it expressed in a more appropriate and definite measure, it becomes necessary to compare it with some weight ; now, it has been found by experiment, that the weight of a cubic foot of water is very nearly equal to 62 1 Ibs. avoirdupois; therefore, the absolute pressure upon the plane, is p 450.60948 X 62.5 = 28163.0925 Ibs. 36. Let the dimensions of the plane remain as in the preceding case, which condition is supposed in the example ; then, the pressure on its surface, when perpendicular to the horizon, is p 3 x 18 X 18 X 1 X J =486 cubic feet of water ; but we have stated above, that the weight of one cubic foot is equal to 62 \ Ibs. ; therefore, we have p 486 X 62^ zz 30375 Ibs. ; consequently, the pressures on the plane in the two positions, are to one another as the numbers 450.60948 and 486, when expressed in cubic feet of water ; but when expressed in pounds avoirdupois, they are as the numbers 28163.0925 and 30375. 37. If the longer side of the rectangular parallelogram were coin- cident with the surface of the fluid, while its plane is obliquely inclined to the horizon ; then, the formula for the pressure perpendicular to its surface becomes p=. b*ls sin. . (9). But if the plane of the parallelogram, instead of being inclined to the horizon, or which is the same thing, to the surface of the fluid, were immersed perpendicularly to it ; then, zz 90, and sin. zz 1 ; hence, the formula for the pressure becomes p =*#**. (10). ON RECTANGULAR PARALLELOGRAMS. 2l Therefore, by retaining the data of the preceding example, the absolute pressure on the plane in the oblique position, is p=3* X 18 x 62.5 X \ X .92718 = 4693.84875 Ibs. But when the plane is perpendicularly immersed, the absolute pressure on its surface is p=3* X 18 X 62.5 X J = 5062Jlbs. COROL. 1 . Hence, the pressures on the plane in the oblique and per- pendicular positions, are to one another as the numbers 4693.84875 and 5062 \ ; but in order to compare the pressures under the same conditions, when the shorter and longer sides of the parallelogram are respectively in contact with the surface of the fluid, we have as follows, viz. 2. When the shorter side of the parallelogram is horizontal, the absolute pressure in the inclined position is 28163.0925 Ibs. ; but when the longer side is horizontal, the absolute pressure is 4693. 84875 Ibs. ; consequently, the absolute pressures in the two cases are to one another as 6 to 1 . 3. Again, when the shorter side of the parallelogram is horizontal, the pressure in the perpendicular position is 30375 Ibs. ; and when the longer side is horizontal, the pressure is 5062 1 Ibs. ; therefore, the pressures in these two cases are to one another as 6 to 1 , the same as before ; from which we infer, that the quantity of inclination affects only the magnitude of the pressures, and that in so far as it changes the position of the centre of gravity, but it has no effect upon the ratio ; therefore, if the plane were to vibrate round its shorter and longer sides respectively as axes, the pressures on its surface, in the two cases, would be to one another in a constant ratio. 3. OF THE AGGREGATE PRESSURE EXERTED BY THE FLUID ON THE IMMERSED PARALLELOGRAM, AND ON EACH THE CONSTITUENT TRIANGLES FORMED BY ITS DIAGONAL. PROBLEM IV. 38. Suppose the parallelogram to be placed under the same circumstances as in the preceding problem, and let it be bisected by one of its diagonals : It is required to determine the aggregate pressure exerted by the fluid, in a direction perpendicular to the surface of each triangle into which the diagonal divides the parallelo- gram, and to compare the pressures on the two triangles. 22 OF THE AGGREGATE PRESSURE OF FLUIDS Let A B c D represent a vertical section of a mass or collection of quiescent fluid, contained by the walls or embankments indicated by the shaded boundary ; and let A B E F be the horizontal surface of the fluid, with which one side of the immersed rectangle is supposed to be coincident. Now, suppose abdc, to be the immersed rectangle, and draw the diagonal b d; then are a b d and bdc the triangles, into which the parallelogram abed is divided by the diagonal b d, and for which the pressures are required to be investigated. Draw the diagonal a c, which divide into three equal portions in the points m and n > then are m and n respectively the centres of gravity of the constituent triangles a b d and bdc. Through the points m and n, and parallel to a d or b c, the immersed sides of the figure, draw me and nf meeting a b perpendicularly in the points e and/; then, through the points e and /thus determined, and in the plane of the fluid surface, draw er and/s respectively perpendicular to ab; then are the angles mer and nfs equal to one another, and each of them is equal to the angle which the plane of the immersed parallelogram makes with the surface of the fluid. From m and n, the centres of gravity of the triangles a b d and b d c, demit the lines m r and n s respectively perpendicular to e r and fs; then are rm and sn the perpendicular depths of the centres of gravity. Put b ab, the horizontal breadth of the immersed parallelogram, / = ad or be, the immersed or downward length, d = rm, the perpendicular depth of the centre of gravity of the triangle abd, S ~ sn, the perpendicular depth of the centre of gravity of the triangle bdc, D ac or ba, the diagonal of the parallelogram, ty~mer, or nfs, the inclination of the plane to the surface of the fluid, Pzz the whole pressure on the parallelogram abed, p m the pressure on the triangle abd, ythe pressure on the triangle bdc, and s the specific gravity of the fluid. Then, because the parallelogram abed is rectangular, the triangle ON IMMERSED RIGHT ANGLED PARALLELOGRAMS. 23 udc is right angled at d; therefore, by the property of the right- angled triangle, we have aczz -v/ ad* or by employing the appropriate symbols, we have Dizrv/r-M 2 . But, according to the construction, and by the nature of the centre of gravity, we have am"=z -i-ac, and an='%ac, or symbolically, we obtain am -ly Z 2 -f 6 2 , and an zr f^/^-j-fe 2 . Now, by reason of the parallel lines em, fn, and be, the triangles a em, afn, and a be, are similar among themselves ; consequently, by the property of similar triangles, we have ac : be : : an :fn : : am : em; therefore, by separating the analogies, and employing the symbols, it is D:Z::|V~F+& 2 :/ w , and again, we have D : I :: i\/^ + & : em '> from these analogies, therefore, we obtain fn n: |7, and em~\l\ which is otherwise manifest by drawing the dotted lines mt and nu. Now, in the right angled triangles erm and fsn, there are given the hypothenuses em and fn, and the equal angles mer and nfs, to find rm and sn, the perpendicular depths of the centres of gravity; consequently, by Plane Trigonometry, we have, from the triangle mer, rad. : sin. $ : : $1 : d, and from the triangle nfs, it is rad. : sin. : : %l : , and since radius is equal to unity, these analogies become d-=.\l sin. 0, and zz f I sin, 0. But according to Inf. 2, Proposition (A), the pressure sustained by each triangle, in a direction perpendicular to its surface, Is expressed by the product of its area, drawn into the perpendicular depth of the centre of gravity. Now, the area of each triangle is manifestly equal to half the area of the given parallelogram, and by the principles of mensuration, the area of the rectangular parallelogram is equal to the product of its two dimensions ; that is, of the length drawn into the breadth ; there- fore, we have for the pressure on the triangle bd, 24 OF THE AGGREGATE PRESSURE OF FLUIDS sn. 0, and in like manner, the pressure on the triangle bdc is p' ~\bl* sin. . These equations, however, express the pressures simply by the magnitude of a fluid column, whose base is the area pressed, and whose altitude is equal to the depth of the centre of gravity below the upper surface of the fluid. In order, therefore, to have the pressures expressed in general terms, the specific gravity of the fluid must be taken into the account; in which case, the pressure on the triangle abd becomes p=bl*ssm., (11). and the pressure on the triangle bdc is p'i=i&Z 2 ssin.;>. (12). COROL. Hence it appears, that the pressure perpendicular to the plane of a triangle, when its vertex is upwards and coincident with the surface of the fluid, is double the pressure on the same triangle, when its base is upwards, and placed under the same circumstances. 39. If the immersed plane be perpendicular to the surface of the fluid, then =:mgr, the inclination of the plane to the surface of the fluid, Pn= the pressure on the whole parallelogram abed, p the pressure on the lower portion efcd, and p the pressure on the upper portion abfe. Then, because the straight line gh is bisected in 0, and each of the portions g and h respectively bisected in the points n and m ; it follows that g n zz |, and gm % of gh; that is g n nz J I, and g m j / ; consequently, by the principles of Plane Trigonometry, we have swrzrSzz \l sin. , and rm d=%l sin. 0; therefore, since the area of each portion of the parallelogram is expressed by \ bl, the pressure on each portion is as below, viz. The pressure perpendicular to the surface a bfe, is p'zz: bl z sin. 0, and the pressure perpendicular to the surface efcd, is p m %bl* sin.0 ; consequently, by comparison, the pressures on the upper and lower portions of the parallelogram, are to each other as the numbers 1 and 3 ; that is pT:p::l:3. But according to the third problem, the aggregate pressure sustained by the plane, in a direction perpendicular to its surface, is consequently, the pressures on the two portions and on the whole plane, are to one another as the numbers 1, 3 and 4. ON DIFFERENT SECTIONS OF PARALLELOGRAMS. 29 In the preceding values of the pressure, it is supposed, that the specific gravity of the fluid in which the plane is immersed, is repre- sented by unity, which is true only in the case of water ; therefore, in order to render the formulae general, we must introduce the symbol for the specific gravity, and then the above equations become, 1. For the upper half of the parallelogram, p'=zbl 2 s sin. 0. (13). 2. For the lower half of the parallelogram, pz=f bl*s sin. 0. (14). When the plane is perpendicularly immersed in the fluid, or when 90, then sin. 1, and the equations (13) and (14) become p' \bl^s, andprzi&/ 2 5. In which equations the co-efficients or constant quantities remain ; therefore, the ratio of the pressure is not varied in consequence of a change in the angle of inclination, the variation takes place in the magnitude of the pressures only, and not in the ratio, the magnitude increasing from zero, where the plane is horizontal, to its maximum where the plane is perpendicular. 44. The practical rules for calculating the pressures, as derived by the equations (13) and (14) are as follows. 1. For the pressure on the first, or upper half of the paral- lelogram. RULE. Multiply the square of the immersed length, by the breadth drawn into the specific gravity of the fluid, and again by the natural sine of the angle of elevation ; then, one eighth part of the product will be the pressure sought. (Eq. 13). 2. For the pressure on the second, or lower half of the paral- lelogram. RULE. Multiply the square of the immersed length, by the breadth drawn into the specific gravity of the fluid, and again by the natural sine of the angle of elevation; then, three eighths of the product will be the pressure sought. (Eq. 14). 45. EXAMPLE 6. A rectangular parallelogram, whose sides are respectively 20 and 30 feet, is immersed in a cistern of water, in such a manner, that its breadth or shorter side is just coincident with the surface ; required the pressures on the upper and lower portions of the plane, supposing it to be bisected by a line drawn parallel to the horizon, the inclination of the plane being 59 38' ? Here, by operating according to the rule, we have p 30* X 20 X sin. 59 38' X i ; 30 OF RECTANGULAR PARALLELOGRAMS DIVIDED INTO but by the Trigonometrical Tables, the natural sine of 59 38' is .86281 ; therefore, we have p'=i 30 2 X 20 x .86281 X i= 1941.3225, and again, by a similar process we have p= 30 2 X 20 X .86281 x 1 = 5823.9675. Now, these results are obviously expressed in cubic feet of water, for they are respectively equal to the solidity of a fluid column, whose base is equal to one half the given parallelogram, and whose altitude, in the one case, is expressed by \l sin. $ rz 7.5 x .86281, and in the other by |J sin. =: 22.5 X .86281 ; but the weight of one cubic foot of water is .equal to 62J Ibs. ; consequently, the pressures expressed in Ibs. avoirdupois, are p f = 1941.3225 X 62.5 = 121332.65625 Ibs. and;? =5823.9675 X 62.5 = 363997.96875 Ibs. When the plane is perpendicular to the surface of the fluid, the pressure is a maximum, and in that case, the respective pressures on the two portions of the parallelogram, are p' = 30 2 x 20 X 62.5 x I = 140625 Ibs. and jo rz30 2 X 20 x 62.5 x |zr 421875 Ibs. and the sum of these, is obviously equal to the whole pressure on the plane ; hence we get P = 140625 + 421875 == 562500 Ibs. COROL. If the plane, instead of being immersed in the fluid, as we have hitherto supposed it to be, should only be in contact with it, as we may conceive the surface of a vessel to be in contact with the fluid which it contains ; then, the pressure will be the same ; for the quan- tity of pressure at any given depth upon a given surface, is always the same, whether the surface pressed be immersed in the fluid or just in contact with it, and whether it be parallel to the horizon, or placed in a position perpendicular or oblique to it. 5. OF RECTANGULAR PARALLELOGRAMS IMMERSED IN NON -ELASTIC FLUIDS, AND DIVIDED INTO TWO PARTS SUCH THAT THE PRESSURES OF THE FLUID UPON THEM SHALL BE EQUAL BETWEEN THEMSELVES. PROBLEM VI. 46. A rectangular parallelogram is obliquely immersed in an incompressible and non-elastic fluid, in such a manner, that one side is just coincident with the surface : It is required to divide the parallelogram into two parts by a line drawn parallel to the horizon, so that the pressures on the two parts shall be equal to one another. TWO PARTS SUSTAINING EQUAL PRESSURES. 31 Let A E D represent a rectangular vessel filled with water, or some other incompressible and non-elastic fluid, of which ABEF is the surface, and ABCD the fluid as exhibited in the vessel, on the supposition that one of its upright sides is removed. Let abed be the immersed parallelo- gram, having its upper side a b coincident with the surface of the fluid, and its plane tending obliquely downwards in the given angle of inclination. Bisect a b in g, and through g draw the v straight line gh parallel to ad or be, the side of the given immersed rectangle, and let ef parallel to ab or cd, denote the line of division; then, by the problem, the pressure on the rectangle abfe, is equal to the pressure on the rectangle efcd. Draw the diagonals dfo.ud.fa, cutting the bisecting line gh in the points m and n ; then are m and n respectively, the places of the centres of gravity of the spaces efcd and abfe. Through the point g and in the plane of the fluid surface, draw g r at right angles to a b, and from m and n demit the straight lines mr and ns, respectively perpendicular to the horizontal line gr ; then are sn and rm the perpendicular depths of the centres of gravity of the rectangles a bfe and efcd on which the pressures are equal. Put b rz ab, the horizontal breadth of the proposed rectangular plane, Z=n ad or be, the immersed length of ditto, or that which tends downwards, d = rm, the vertical depth of the centre of gravity of the lower portion efcd, $i=sn, the vertical depth of the centre of gravity of the upper portion abfe, 0z= mgr, the inclination of the plane to the surface of the fluid, P=r the pressure on the entire parallelogram, p =z the pressure on each of the portions into which the paral- lelogram is divided, s rr the specific gravity of the fluid, and x ae, the immersed length of the upper portion abfe. Then is edl x; gn \x, and gm # -|- \ (I x) ; con- sequently, by the principles of Plane Trigonometry, we have 57i^rS= | a sin. 0, and rm d [x -\- %(l #)J sin.0, and moreover, by the principles of mensuration, the area of the upper portion is expressed by b x, and that of the lower portion by b (I x) ; 32 OF RECTANGULAR PARALLELOGRAMS DIVIDED INTO consequently, the absolute pressures as referred to the respective portions, are but according to the conditions of the problem, these pressures are equal to one another ; hence by comparison, we have and this by a little farther reduction, becomes 2* 2 = Z 2 . (15). 47. The equation in its present form, suggests a very simple geometri- cal construction ; for since Z 2 is equal to twice z 2 , it is manifest, that Z is the diagonal of a square of svhich the side is x ; hence the following process. Draw the straight line AB of the same length as the side of the given parallelogram, and bisect AB per- pendicularly in c by the straight line CD; on AB as a diameter, and about the centre c describe the semi- circle ADB, cutting the straight line CD in the point D; join AD, and about the point A as a centre, with the distance A D, describe the arc DE meeting AB in E ; then is F the point of division sought. Upon A B and with the given horizontal breadth, describe the paral- lelogram ABHG, and through the point E, draw the straight line EF parallel to AG or BH ; then will EF divide the parallelogram, exactly after the manner required in the problem. The truth of the above construction is manifest ; for by the property of the right angled triangle, we have A D 2 A c 2 -J- c D ? ; but AC is equal toco, these being radii of the same circle, hence we get AD 2 =2 AC 2 ; but by the construction, we have AE m AD ; consequently, by substitution, it is AE 2 Z= 2 AC 2 , and doubling both sides of the equation, we get now AC is equal to one half of AB, and it is demonstrated by the writers on geometry, that the square of any quantity is equal to four times the square of its half; consequently, we have 4AC 2 =IAB 2 ; therefore, by substitution, we obtain 2AE 2 ZZZ AB 2 , TWO PARTS SUSTAINING EQUAL PRESSURES. 33 being the very same expression as that which we obtained by the foregoing analytical process, a coincidence which verifies the pre- ceding construction. Returning to the equation numbered (15), and extracting the square root of both sides, we obtain xy2=li and by division, we have x = kW2. (16), 48. The practical rule for determining the point of division, as supplied by the above equation, is extremely simple ; it may be thus expressed : RULE. Multiply half the length of the immersed side of the parallelogram by the square root of 2, or by the constant number 1.4142, and the product will express the distance downward from the surface of the Jluid. 49. EXAMPLE 7. A rectangular parallelogram, whose sides are respectively 14 and 28 feet, is immersed in a cistern of water, in such a manner, that its shorter side is just coincident with the surface; through what point in the longer side must a line be drawn parallel to the horizon, so that the pressures on the two parts, into which the parallelogram is divided, may be equal to one another? Here, by operating according to the rule, we have x i (28 X 1.4142)= 19.7988 feet. 50. If the point through which the line of division passes, were estimated in the contrary direction ; that is, upwards from the lower extremity of the immersed side of the parallelogram; then, the ex- pression for the place of the point will be very different from that which we have given above, as will become manifest from the follow- ing process. Recurring to the original diagram of Problem 5, and putting x ed, the rest of the notation remaining, we shall have by sub- traction, ae=il x; consequently, sn the depth of the centre of gravity of the rectangle abfe, is 3 J (/ x) sin. 0, and in like manner, it may be shown, that rm, the depth of the centre of gravity of the rectangle efcd, is c?= (/ !#) sin. (f>. Now, according to the writers on mensuration, the area of the rectangle abfe is expressed by b (i x}, and that of the rectangle efcd by bx; consequently, the respective pressures are VOL, i. i> 34 SECTIONS OF RECTANGULAR PARALLELOGRAMS p \ b(l a;) 8 s sin.0, andprrr bx(l \ x} s sin.0, but by the conditions of the problem, these pressures are equal ; hence we get J(Z *) 2 = a? (/ I *), and this, by reduction, becomes ** 2/3? = i/ 2 ; consequently, the root of this equation is and more elegantly, by collecting the terms, it becomes x = l(l \ V~)' 51. This is manifestly the same result as would arise, by subtract- ing the value of x in equation (16), from the whole length of the parallelogram ; and the rule for performing the operation is simply as follows : RULE. From unity subtract one half the square root o/2 ; then multiply the remainder by the length of the parallelo- gram, and the product will be the distance of the point required from the lower extremity of the immersed dimension. Therefore, by taking the length of the parallelogram, as proposed in the preceding example, we shall have for the distance from its lower extremity, through which the line of division passes, x = 28 (1 i ^/2) = 8.2012 feet. COROL. It is manifest from the equations (16) and (17), that the solution is wholly independent of the breadth of the parallelogram, its inclination to the horizon, and the specific gravity of the fluid ; these elements, therefore, might have been omitted in the investigation ; but since it became necessary to express the pressure either absolutely or relatively, we thought it better to exhibit the several quantities, of which the measure of the pressure is constituted. PROBLEM VII. 52. A given rectangular parallelogram is immersed in a fluid, in such a manner, that one side is coincident with the surface, and its plane tending obliquely downwards at a given inclination to the horizon : It is required to draw a straight line parallel to one of the diagonals, so that the pressures on the parts into which the parallelogram is divided, may be equal to one another. SUSTAINING EQUAL PRESSURES Let A ED represent a cistern filled with fluid, of which ABEF is the surface, sup- posed to be perfectly quiescent, and con- sequently, parallel to the horizon ; and let ABCD be a vertical section of the cistern, exhibiting the fluid with the immersed rectangle abed. Draw the diagonal ac, and in a d take any point e ; through the point e thus assumed, draw the straight line ef parallel to a c the diagonal of the parallelogram; then is edf the triangle, on which the pressure is equal to that upon the polygonal figure e a b cf. Take dn and dt respectively equal to one third of de and df, and through the points n and t, draw nm and tm respectively parallel to ab and ad, the sides of the parallelogram, and meeting one another in the point m ; then, according to problem B, m is the place of the centre of gravity of the triangle e df. Produce tm directly forward, meeting ab the upper side of the parallelogram perpendicularly in s ; then, through the point s, and in the plane of the fluid surface, draw the straight line sr also at right angles to a b, and from m, the centre of gravity of the triangle edf, demit the line mr perpendicularly on sr ; then is rm the perpendicular depth of the centre of gravity of the triangle edf, and msr is the angle of inclination of the plane to the horizon. Put b=^ab, the horizontal breadth of the given parallelogram, I = ad, the length of the immersed plane tending downwards, d rm, the perpendicular depth of the centre of gravity of the triangle efd, p ~ the whole pressure perpendicular to its surface, msr, the angle which the immersed plane makes with the horizon, s =the specific gravity of the fluid, and x ed, the perpendicular of the triangle edf, of which the base isrf/. Then, by reason of the parallel lines ac and ef, the triangles adc and ec?/are similar to one another, and consequently, by the property of similar triangles, we have ad : dc : : ed : df, which, by restoring the symbols, becomes I : b : : x : df, and from this analogy we have D2 36 SECTIONS OF RECTANGULAR PARALLELOGRAMS therefore, by the principles of mensuration, the area of the triangle efd is bx bx* . 4* X T = 2T Now, according to the construction, dn is equal to one third of ed, and an is equal to ad minus dn; but sn is obviously equal to an\ hence we have sninl \x, and by the principles of Plane Trigonometry, it is rm zz c?~ (I ^ x) sin. ; consequently, the pressure on the triangle edf becomes _ bx*s (31 x} sin.

'z= the pressure on the irregular figure ABCFE, OF IMMERSED RECTILINEAR FIGURES. 39 P= the pressure on the entire parallelogram A B c D, xrn, the perpendicular depth of the centre of gravity of the figure ABCFE, when the side AB is horizontal, and y sn, the perpendicular depth, when the side BC is horizontal. Then, because the sides AE and CF are given quantities, it follows, that DE and DF are also given, and consequently, AC or cm, and cb or dm are given ; therefore, the perpendicular pressure on the triangle EDF can easily be ascertained. Now, A a is manifestly equal to the difference between A D and a D, and by the construction an is equal to one third of D E ; therefore, by restoring the analytical representatives, we have cm = d = l il'. Again c b is equal to the difference between c D and D b ; but D b by the construction, is equal to one third of DF ; hence, by restoring the analytical symbols, we shall obtain dm $=:b ft. But, according to the writers on mensuration, the area of the triangle EDF is equal to half the product of the base DF by the per- pendicular DE ; that is JZ'X/3 = jr|3; consequently, if we suppose the plane to be perpendicularly immersed in the fluid, while the side AB is coincident with its surface ; then, the pressure on the triangle EDF becomes p = %pl's(3l-l'). Now, the pressure on the irregular figure ABCFE, is obviously equal to the difference between the pressures on the entire paral- lelogram A B c D, and the triangle EDF; but the pressure on the entire parallelogram, according to equation (8), is consequently, by subtraction, the pressure on the figure ABCFE, becomes but its area is also equal to the difference between that of the parallelogram and triangle; therefore, we obtain J(2Z (31') for the area of the irregular figure ABCFE; consequently, by division, the perpendicular depth of the centre of gravity below the line AB, becomes and if we suppose the fluid in which the plane is immersed to be water, the specific gravity of which i* unity, we finally obtain 40 CENTRE OF GRAVITY OF MIXED SPACE zbi* fii'(3i r) 3(2ft/-00 Again, if we suppose the side BC to be horizontal, the area of the triangle remains the same, and the pressure which it sustains in a direction perpendicular to its surface, becomes p = tfll's(3b 0). But the pressure on the whole parallelogram A BCD, on the suppo- sition that the side BC is horizontal, according to what has been proved in Problem 3, is consequently, the pressure on the irregular figure ABCFE, becomes Now, the area of the figure corresponding to the above pressure, is obviously the same as we have previously determined it to be ; that is, the difference between the areas of the triangle and the entire paral- lelogram ; consequently, by division, we shall obtain _ The equations (20) and (21) are manifestly symmetrical ; if there- fore, we carefully attend to the conditions of the problem, from which they are respectively derived, the position of the centre of gravity of the figure ABCFE can easily be ascertained byresolvng the equations. 55. The practical rules for determining the co-ordinates which fix the position of the centre of gravity, may be expressed in the follow- ing manner : 1. When the side AE is horizontal, as indicated by equation (20). RULE. From three times the vertical length of the given rectangular parallelogram, subtract the perpendicular of the triangle, and multiply the remainder by twice its area ; then^ subtract the product from three times the square of the length of the parallelogram drawn into its breadth, and the remain- der will be the dividend. Divide the dividend above determined, by three times the difference between twice the area of the parallelogram^ and twice that of the triangle, and the quotient will give the co-ordinate of the line A B. 2. When the side BC is horizontal, as indicated by equation (21). RULE. From three times the vertical breadth of the paral- lelogram, subtract the base of the triangle, and multiply the OF IMMERSED RECTILINEAR FIGURES, 41 remainder by twice its area ; then, subtract the product from three times the square of the breadth of the parallelogram drawn into its length, and the remainder will be the dividend. Divide the dividend above determined, by three times the difference between twice the area of the parallelogram, and twice that of the triangle, and the quotient will give the co-ordinate of the line BC. 56. EXAMPLE 9. The sides of a rectangular parallelogram are respectively 28 and 50 feet, and from one of the lower corners, is separated a right angled triangle, by means of a straight line ter- minating in the adjacent sides; it is required to determine the position of the centre of gravity of the remaining part, the base and perpendicular of the separated triangle, being respectively equal to 20 and 42 feet ? Here then, by operating as directed in the first rule, we have 3 x 50 42 = 150 42 = 108, and by the principles of mensuration, twice the area of the triangle, is 42 X 20 = 840 square feet ; therefore, by multiplication, we obtain 108 x 840 = 90720. Again, three times the square of the length of the parallelogram, is 3x 50 2 = 7500, which being multiplied by its breadth, gives 7500 x 28 = 210000; consequently, by subtraction, the dividend is 210000 90720=119280. Now, twice the area of the parallelogram, is 2 x 50 x 28 = 2800 square feet, and twice the area of the triangle, is 42 x 20 = 840 square feet ; therefore, by the second clause of the rule, we obtain a = H928 = 20.286 feet nearly. 3(2800 840) Hence it appears, that the co-ordinate of the line AB, according to the proposed data, is very nearly 20.286 feet; and by operating as directed in the second rule, we shall have 3 x 28 20 = 84 20 = 64, and by the principles of mensuration, twice the area of the triangle, is 42 X 20 = 840 square feet ; therefore, by multiplication, we obtain 64 x 840 = 53760. Again, three times the square of the breadth of the parallelogram, is 3 x 28* = 2352, 42 CENTRE OF GRAVITY OF MIXED SPACE which being multiplied by its length, gives 2352 X 50=117600; consequently, by subtraction, the dividend becomes 117600 53760 = 63840. Now, the second clause of the second rule, being the same as the second clause of the first rule, it follows, that the divisor must here be the same, as we have found it to be in the preceding case ; conse- quently, by division, we obtain therefore, from the numerical values of the co-ordinates as we have just determined them, the position of the centre of gravity of the proposed figure can easily be found, in the following manner. 57. Let ABCD represent the rectangular parallelo- gram, of which the side AB is 28 feet, and the side EC 50 feet; and let EDC be the right angled triangle, whose perpendicular E D is 42 feet, and its base D F 20 feet, all taken from the same scale of equal parts. From the angle B, and on the sides BC and BA, set off BS and Br respectively equal to 20.286 and 10.857 feet; then, through the points s and r, draw the lines sn and rn, respectively parallel to AB and BC, and the point n is the Centre of gravity of the figure ABCFE, which remains after the right angled triangle EDF is separated from the parallelogram ABCD. If the line of division, or hypothenuse of the triangle EF, were parallel to AC the diagonal of the parallelogram, as is distinctly speci- fied in the foregoing problem, the solution would become much more simple ; for then, in order to determine the position of the centre of gravity, it is only necessary to reduce one of the equations, and it is altogether a matter of indifference which of them it is, provided that the conditions of the equation be strictly attended to. Supposing E D the perpendicular of the triangle, to remain as above ; then the base, when the hypothenuse is parallel to the diagonal of the rectangle, will be found by the following analogy, viz. 50 : 28 : : 42 : 23.52. Then, by calculating according to rule first, or equation (20), our dividend and divisor are 103313.28 and 5436.48 respectively; con- sequently, we get 103313.28 *= 5436.48 = 19fe OF IMMERSED RECTILINEAR FIGURES. 43 therefore, by analogy, we obtain 50 : 19 :: 28 : y = 10.64 feet. Here, the whole process of calculating the second co-ordinate, is replaced by the simple analogy above exhibited. The example now before us, affords a striking instance of the advantages to be derived from this mode of considering the centre of gravity; in the case of the triangle illustrated under Problem (B), its immediate utility was not so conspicuously displayed ; but we are convinced, that in figures of more difficult and complicated forms, its usefulness will become still more evident. In the investigation of the formulae, we have thought it necessary to consider the pressure on the surface whose centre of gravity is sought ; but in the actual application of the resulting equations, the consideration of pressure does not enter; for it is manifest, that besides the dimensions of the figure and constant numbers, no other element is found in the equations, and consequently, the reduction depends upon them alone. 7. OF EQUAL FLUID PRESSURES ON THE SECTIONS OF A RECTANGULAR PARALLELOGRAM AND THE PERPENDICULAR DEPTHS OF THE CENTRE OF GRAVITY. PROBLEM VIII. 58. A given rectangular parallelogram is immersed in an incompressible and non-elastic fluid, in such a manner, that one of its sides is coincident with the surface, and its plane tending downwards at a given inclination to the horizon : It is required to draw a straight line from one of the upper angles to the lower side, so that the pressures on the two parts into which the parallelogram is divided, may be equal to one another. Let AED represent a rectangular cistern filled with water, or some other incompressible and non-elastic fluid, of which ABEF is the horizontal surface, and suppose one of the upright sides, as ABCD to be removed, exhibiting the fluid together with the immersed rectangle abed. In dc the lower side of the immersed parallelogram, take any point/, and draw af to represent the line of division ; then the triangle adf, and the trapezoid abcf, T> are the figures into which the parallelogram is divided, and on which the pressures are equal. 44 PARALLELOGRAM DIVIDED TO SUSTAIN From the angle d, set off dn and dt respectively equal to one third of da and df, and through the points n and t, draw nm and tm parallel to df and da the sides of the triangle, and meeting each other in the point m ; then, according to what has been demonstrated in Problem (B), m is the centre of gravity of the triangle adf. Produce tm directly forward, meeting a b at right angles in the point r, and through the point r and in the plane of the fluid surface, draw rs also at right angles to a b, and demit ms meeting rs perpen- dicularly in s; then is mrs the inclination of the plane to the horizon, and sm the perpendicular depth of the centre of gravity of the triangle ad/* below the upper surface of the fluid. Put b ab, the horizontal breadth of the parallelogram abed, I ad, the immersed length tending downwards, p zn the pressure perpendicular to the surface of the triangle a df. P= the pressure on the entire parallelogram abed, = mrs, the angle which the immersed plane makes with the horizon , d^ism, the perpendicular depth of the centre of gravity of the triangle adf, s =z the specific gravity of the fluid, and x zz df, the distance between d and the point through which the line of division passes. Then, according to the principles of Plane Trigonometry, the per- pendicular depth of the centre of gravity of the triangle adf, becomes 6?rr-|Zsin.0; consequently, the pressure on its surface, is p %l*xs sin.0 ; J see equation (10) J . But according to equation (7) under the 3rd problem, (art. 33), the pressure on the entire rectangle, is and by the conditions of the present problem, the pressure on the triangle, is equal to one half the pressure on the entire parallelogram ; therefore, we have /?== !P; that is from which, by expunging the common terms, we get 4x = 3b; consequently, by division, we obtain _3b ~ 4 ' EQUAL FLUID PRESSURES. 45 59. This equation is too simple in its arrangement to require any formal directions for its resolution ; nevertheless, the following- rule may be useful to many of our readers. RULE. {Take three fourths of that side of the given rec- tangular parallelogram, in which the line of division terminates , and the point thus discovered, is that through which the line of division passes. 60. EXAMPLE 10. A rectangular parallelogram, whose sides are respectively equal to 24 and 42 feet, is immersed in a cistern full of water, in such a manner, that its shorter side is coincident with the surface of the fluid, and its plane inclined to the horizon in an angle of 52 degrees ; it is required to determine a point in its lower side, to which, if a straight line be drawn from one of the upper angles, the parallelogram shall be divided into two parts sustaining equal pres- sures ; and moreover, if a straight line be drawn from the same point in the lower side, to the other upper angle, it is required to assign the pressure on the triangle thus cut off? Here, by operating according to the rule, the point of division is x = % X 24= 18 feet. In the next place, to determine the pressure sustained by the triangle bcf t cut off from the parallelogram abed, by means of the line/6 drawn from the point f to the angle at b, we have according to equation (12), (Problem 4), p z= i (b x) I 9 s sin.0, where (b #) in this equation, takes place of b in the one re- ferred to. The natural sine of 52 degrees according to the Trigonometrical Tables, is .78801 ; hence, by substituting the respective data in the above equation, we shall have pl (2418) X 42 2 X .78801 2746.09928 cub. ft. of water ; consequently, the pressure expressed in Ibs. avoirdupois, is p 1 = 2746.09928 X 62.5 = 181631.205 Ibs. This seems to be an immense pressure, on a triangle whose surface is only 126 square feet ; it is however but one sixth part of the pres- sure on the entire parallelogram ; this is manifest, for the pressure on the triangle adf, is three times the pressure on the triangle bcf, since the base df is equal to three times the base cf, and the altitudes of the triangles, as well as the perpendicular depths of the centres of gravity, are the same; but the pressure on the parallelogram abed, 46 PARALLELOGRAM DIVIDED TO SUSTAIN according to the problem, is double the pressure on the triangle adf; hence we have P 2p i= 6p' = 1089787.23 Ibs. 61. If the line of division were drawn from one of the lower angles to a point in the immersed length, after the manner represented in the annexed diagram ; then, the equation (22), would assume a different form, as will become manifest from the following investigation. From the angle d on da and dc the sides of the parallelogram, set off dn and dt, respectively equal to one third of df and dc, and through the points n and t thus found, draw the straight lines nm and tm \ parallel to dc and df, the base and perpendicular of the triangle fdc, and meeting one another in m, the place of its centre of gravity. Produce tm directly forward, meeting a b, the hori- zontal side of the given parallelogram perpendicularly in the point r ; at the point r in the straight line mr, make the angle mrs equal to the angle of the plane's inclination, and draw ms perpendicularly to rs ; then is sm the perpendicular depth of the centre of gravity of the triangle /We. Let therefore, the notation of the preceding case be retained, and put x == df; then we have am^irmm. I' ^x, and consequently sm di=: (I ^x) sin .0 ; but the area of the triangle fdc is expressed by \b x; therefore, the pressure perpendicular to its surface, is now, according to the conditions of the problem, the pressure on the separated triangle is equal to half the pressure on the entire paral- lelogram ; consequently, we obtain and this, by expunging the common quantities, becomes 2 x (3 / x) = 3 l\ or dividing by 2 we get x(3l x} = l.5l\ and from this, by separating and transposing the terms, we have x* 3lx 1.5 Z 2 . (23). If the equations (18) and (23) be compared with one another, it will readily appear, that they are precisely similar in form, but dif- ferent in degree ; the former being an incomplete cubic, wanting the first power of the unknown quantity, and the latter an adfected quadratic, having all its terms. Indeed, the diagrams from which the EQUAL FLUID PRESSURES. 47 two equations are derived, as well as the specified conditions of the problems, are nearly similar, the difference consisting simply in the position of the dividing line, it being parallel to the diagonal of the parallelogram in the one case, and oblique to it in the other. 62. Let the quantity 2JZ 2 be added to both sides of the preceding equation, and we shall obtain from which, by extracting the square root, we get x i$i = + \i V3"; therefore, by transposition, we have (24). The practical rule by which the point of division is to be determined, may be expressed as follows : RULE. Multiply the difference between 3 and the square root of 3, by half the length of that side of the parallelogram in which the line of division terminates, and the product will 'be the distance of the required point from the lower extremity of the given length. 63. EXAMPLE 11. Let the numerical data remain precisely as in the preceding case; from what point in the length of the paral- lelogram, must a straight line be drawn to the opposite lower angle, so that the parallelogram may be divided into two parts sustaining equal pressures ; and moreover, if a straight line be drawn from the same point, to the opposite upper angle, what will be the pressure on the triangle thus cut off? Here, by proceeding as directed in the above rule, we have x 21 (3 V 3) = 26.628 feet. In order to find the pressure on the triangle a bf cut off by the line bf, we have afl x, and ae rp \(l x)\ conse- quently, vp^n^(l x) sin.0, where it must be observed, that ep and vp are respectively parallel to a b and sm. Now, the pressure perpendicular to the surface of the triangle a bf, is found by multiplying its area into vp, the perpendicular depth of its centre of gravity ; hence, we have p' ^bs (I #) 2 sin.^. ; but the value of x, according to equation (24), is x = .634 I ; consequently, by substitution, we have p' = %b Z 2 s (1 .634)* sin.0, from which, by substituting the several numerical values, we obtain >' = 4 X 42 s X 62.5 X .366* X .78801 = 46551. 35 Ibs. 48 S. PERPENDICULAR DEPTH OF THE CENTRE OF GRAVITY Of A PARAL- LELOGRAM DIVIDED INTO TWO PARTS SUSTAINING EQUAL PRESSURES. 64. With respect to the centre of gravity of the figure a b cf, which remains after the triangle adf, or fdc has been separated from the parallelogram, it is in this particular instance very easily determined ; for, since the area of trapezoids, whose parallel sides and perpendi- cular breadths are equal each to each, are also equal ; it follows, that the centre of gravity must occur in the straight line which bisects the parallel sides ; it is therefore, only necessary to investigate the theorem for calculating one of the co-ordinates, the other being determinable from the circumstance just stated. Let ABCF be the trapezoid, having the angles at B and c respec- tively right angles, and of which the position of the centre of gravity is required. Produce the side CF directly forward to any convenient length at pleasure, and through the point A, draw the straight line AD parallel to BC, the longer side of the trapezoidal figure, and meeting c F produced perpendicularly in the point D. Then, the pressure upon the trapezoid ABCF, is manifestly equal to the difference between the pressure on the parallelogram A BCD, and that upon the triangle ADF, and its area, is also equal to the difference be- tween their areas. Bisect the parallel sides AB and CF in the points a and b, and join a b ; then, according to what lias been demonstrated by the writers on mechanics, the centre of gravity of the trapezoid ABCF occurs in the straight line ai. Suppose it to occur at m, and through the point m draw mr and ms respectively parallel to BC and BA, meeting AB and BC perpen- dicularly in the points a and b ; then are rm and sm the co-ordinates, whose intersection determines the position of the point m. Put b =z AB, the breadth of the parallelogram A BCD, / AD, or BC, its corresponding length, = m r the depth of the point m as referred to the line A B con- sidered to be horizontal, cTzu ms, the depth of the point m as referred to the line BC under similar circumstances, and /3~ DF, the base of the triangle ADF. Then, by conceiving the plane to be immersed perpendicularly in a fluid whose specific gravity is expressed by unity, the pressure upon CENTRE OF GRAVITY OF PARALLELOGRAMS, &C. 49 the entire surface ABCD, according to equation (8) under the third problem, becomes P = J&/ 2 ; and moreover, by equation (12) under the fourth problem, the general expression for the pressure on the triangle ADF, is p 4/3 P s sin.0 ; but according to the particular case now under consideration, the above expression becomes the terms s and sin.0, being each equal to unity, they disappear in the equation. Now, according to what we have stated above, the pressure on the trapezoid ABCF, is equal to the difference between the pressure on the entire parallelogram ABCD, and that on the triangle ADF ; that is or, by reducing the fractions to a common denominator and collecting the terms, we obtain ^ j/ = ^(36 2/3). By the principles of mensuration, the area of the trapezoid ABCF, is equal to the product that arises, when half the sum of the parallel sides AB andcF, is multiplied by BC the perpendicular distance between them ; that is, BC X |(AB + CF)Z= (^ /?), and the perpendicular depth of the centre of gravity, is equal to the pressure on the surface, divided by the area of the figure ; conse- quently, we obtain jS) ' The form of this equation is extremely simple, but it may be arrived at independently of the preceding investigation, by having recourse to equation (20) under Problem 6 ; for according to the conditions of the question, the line of division AF originates at the angle A, and consequently, the perpendicular of the triangle and the length of the parallelogram are equal ; therefore, by putting / instead of /' in equa- tion (20), the above expression immediately obtains. Now, by taking the length and breadth of the parallelogram, as given in the preceding example, and the base of the triangle as com- puted by equation (22), we shall obtain, 3(2 x24 18) 65. Having thus determined the magnitude of the co-ordinate BS or rm from the equation (25), the magnitude of the corresponding VOL. I. E 50 CENTRE OF GRAVITY OF PARALLELOGRAMS co-ordinate sr or sm, can very easily be found; for through the point b the bisection of FC, draw bn parallel to BC and meeting AB perpendicularly in n ; then, the triangles ban and mar are similar to one another, and the sides bn, mr and an, are given to find ar, ajid from thence the rectangular co-ordinate sr or sm\ consequently, we have b a : na : : mr : r a; therefore, by subtraction, we get BrorsrazzaB r a. Now, CZZBW., is obviously equal to half the difference between D c, the breadth of the parallelogram, and DF, the base of the triangle ADF ; therefore, we have B = j(6.-0); but an zr a B BW ; that is, a n zz J/3, and 3(24-0)- therefore, by reducing the analogy, we get _0(36 20). ' 6(26-0)' hence, by subtraction, we obtain 36(6-0)4-0* 3(26 0) (26). After the same manner that equation (25) is deducible from equa- tion (20), by putting Z'zzZ; so also, is equation (26) deducible from equation (21), by means of the same equality ; we might therefore have dispensed with the preceding investigation, and derived the expression from principles already established ; we however preferred obtaining it as above, for the purpose of exhibiting that agreeable variety which gives additional embellishment to scientific investigations. The method of establishing the formulee, on the supposition that the side BC is horizontal, is sufficiently obvious from what has been done in the sixth problem preceding, and therefore, it need not be repeated here. COROL. By substituting the numerical values of b and 0, as given in the preceding example, we shall have from equation (26) Therefore, from the point B, set off BS and sr respectively equal to 16.8 and 8.4 feet; and through the points s and r, draw sm and rm parallel to AB and BC, the perpendicular sides of the given trapezoid, SUSTAINING EQUAL PRESSURES. 51 and meeting one another in the point m ; then is m the required place of the centre of gravity. 66. In computing numerically the values of the rectangular co- ordinates mr and ms, we have supposed, that DF the base of the applied triangle, is determiriable by the application of equation (20) ; this supposition however is perfectly unnecessary, for the base of the triangle is always equal to the difference between the parallel sides of the given trapezoid ; and moreover, the equation (20), applies only to the particular case for which it has been deduced, viz. when the pressure on the applied triangle and that on the trapezoid to which it is applied are equal to one another. 9. WHEN THE PARALLELOGRAM IS SO DIVIDED, THAT THE PRESSURES ON THE TWO PARTS ARE TO ONE ANOTHER IN ANY RATIO WHATEVER. 67. In the sixth, seventh and eighth problems preceding, we have supposed the given rectangular parallelogram to be divided into two parts, such, that the pressures upon them shall be equal between themselves, and the investigation has accordingly been limited to that particular case ; but in order to render the solution general, we shall consider the division to be so effected, that the pressures on the two parts may be to one another in any ratio whatever, such as that of m to n, For which purpose then, by referring to the fifth problem, where the given parallelogram is divided horizontally, we find, that the pressure on the upper portion is expressed by J6a; 2 ssin.^, and that on the lower portion, by { J (I x)*-\-x(l ,z) } 6ssin.0 ; but these ex- pressions in their present state are equal to one another, and they are now required to be reduced in the ratio of m to n ; consequently, we have 1*': \ k(l x )* + x(l x}} ::m:n, and this, by expanding the second term, becomes x* : I* a 8 : : m : n; or by equating the products of the extremes and means, we obtain n a; 2 nz m / 2 ma? \ therefore, by transposition, we get (m -\- n} o^zr mZ% and finally, by division and evolution, we have E2 52 SECTIONS SUSTAINING PRESSURES 68. Again, in the case of the sixth problem, where the given paral- lelogram is divided by a line drawn parallel to the diagonal ; we find, that the pressure on the triangle cut off by the line of division, is expressed by bx*s(3l x) sin.0 -H 6 /, and consequently, by sub- traction, that on the remaining portion is expressed by bssm . { 3 l a x* (3 I #) } x ~ 6 / ; now, these expressions, by the condi- tions of the problem, are equal to one another ; but in the present case, they are to be reduced in the ratio of m to n ; for which purpose we have a* (3 / x) : 3 / 8 x*(3 I x) : : m : n ; therefore, by equating the products of the extreme and mean terms, we get nx*(3l x) 3ml 3 mx*(3l a?) ; and from this, by transposition, we shall obtain (m + ) (3 lx* x s ) = 3 m I s ; therefore, by dividing and transposing the terms, we have 3ml 9 (28). Now, in order to reduce the above equation, there must be substi- tuted the numbers which express the given ratio, together with the length of the parallelogram, and then, the value of a; will be obtained by any of the rules for resolving cubic equations. 69. In like manner as above, by referring to the eighth problem, where the given parallelogram is divided by a line drawn from one of the upper angles, and terminating in the lower side ; we find, that the pressure on the triangle cut off by the line of division, is expressed by ^/ 8 #ssin.0, and consequently, by subtraction, the pressure on the remaining portion is expressed by Fssm.(f>(3b 2#); and these expressions, according to the conditions of the problem, are equal to one another ; but in the present instance, they are to be reduced in the ratio of m to n ; hence, we have consequently, by equating the products of the extreme and mean terms, we get 2nx nr 3 bm 2 war, from which, by transposition, we obtain 2(m + n)x = 3bm, and finally, by division, we have 36m r -2(m-fn)' (29). ANY RATIO TO ONE ANOTHER. 53 Hence then, the equations (27,) (28,) and (29,) express generally the relation between the parts of division, which in the several pro- blems is restricted to a ratio of equality ; and it is presumed, that by paying a due attention to the examples that have been proposed and illustrated, the diligent reader will find no difficulty in resolving any example that may present itself under one or other of the general forms above investigated. In all the above cases, we have supposed the breadth, or that side of the parallelogram which is denoted by b to be horizontal, and coincident with the surface of the fluid ; but it is manifest, that equations of the same form would be obtained from the other side, having b in place of I, and / in place of b. 10. OF RECTANGULAR PARALLELOGRAMS DIVIDED INTO SECTIONS SUSTAINING EQUAL PRESSURES; WITH THE METHOD OF DETER- MINING A LIMIT TO THE NECESSARY THICKNESS OF FLOOD-GATES, AND OTHER CONSTRUCTIONS OF A SIMILAR NATURE. PROBLEM IX. 70. A given rectangular parallelogram, is immersed in an incompressible and non-elastic fluid, in such a manner, that one of its sides is coincident with the surface, and its plane inclined at a given angle to the horizon : It is required to divide the rectangle by lines drawn parallel to the horizon, into any number of parts, such, that the pressures on the several parts of division shall be equal to one another. Let A ED represent a rectangular cistern filled with water, or some other transparent and incompressible fluid in a state of rest ; one side of the vessel being removed, for the purpose of exhibit- ing the fluid and the immersed parallelo- gram, together with the several subordinate lines on which the investigation depends. Suppose e, I, g and i to be the several points of division, and through these points draw the lines em, If, gk and ih, respec- tively parallel to a b or dc, the horizontal sides of the figure. Bisect the sides a b and dc in the points G and H; join GH, and draw the zigzag diagonals am, ml, Ik, ki and ic, cutting the bisecting line 54 HORIZONTAL SECTIONS CONTAINING EQUAL FLUID PRESSURES. GH in the points z, q, p, o and n, which points are the respective centres of gravity of the several parts into which the given parallelo- gram is divided, and on which the pressures are supposed to be equal among themselves. Through the point G, and in the plane of the horizon, draw the straight line or at right angles to ab, making the angle ron equal to the given angle of the plane's inclination, and from the points n, o,p, q and z, let fall the perpendiculars zv, qu, pt, os and nr, meeting the line or respectively in the points v, u, t, s and r; then are the lines zv, qu, pt, os and nr, the perpendicular depths of the centres of gravity of the several portions into which the proposed rectangle is divided, the points of division being estimated from the surface down- wards. Put =iab or dc, the horizontal breadth of the given parallelogram abed, I ad or be, the entire immersed length, or that tending down- wards, ^ zzrGH, the given angle of inclination, d zn vz, the vertical depth of the centre of gravity of the part abme, d ' zz w , the vertical depth of - e mfl, d" tp, the vertical depth of S so, the vertical depth of -- ykhi, I' rn, the vertical depth of -- iked', n zzthe number of parts into which the parallelogram is divided, P zzrthe entire pressure on the parallelogram abed, p nz the pressure, common to each of the parts into which the given parallelogram is divided, v =n ae, the required length of the upper portion abme, w ~el, the length of the second portion emfl, x m Ig, the length of the third portion If kg, y gi, the length of the fourth portion gkhi, z zn id, the length of the fifth portion i hcd. Then, according to equation (7) under the third problem, the entire pressure on the parallelogram abed, is therefore, the pressure on each part of the divided figure, is bl 2 But because the value of $, the angle of inclination, and s the specific gravity of the fluid, are the same for all the parts ; those HORIZONTAL SECTIONS SUSTAINING EQUAL FLUID PRESSURES. 55 quantities may be omitted in the equation, and then the element of comparison, or the n th part of the total pressure becomes _bl* p -^~n (30). Now, according to the principles of Plane Trigonometry, the lines vz, uq, tp, so and rn, are respectively as below, viz. $ nz G o sin.^>, and 3' zz G n sin.0 ; but the lines GZ, G^, Gp, GO and GW, when expressed in terms of the respective lengths, are as follows, viz. G2 = |v; Gq=:v-\-%w; Gp v -f- w -f- J#; Gozzv-j-w-|- a; 4" iy> an d ftwss:* +'IIP + * + y + J; therefore, by substitution, the above values of the vertical depths of the respective centres of gravity, become d \v sin.< ; d' zz: (v 4- Jw;) sin.^ ; d w zz (w + w 4- Jar) sin.0 ; 5 zz (v 4- w 4- a? + Jy) sin.^, and ^ zz: (v 4- w -f x 4- 2/ + |z) sin.0. Consequently, by throwing out the common factor sin.0 and neglecting the specific gravity of the fluid, the value of jo, or the pressure sustained by each of the parts, may be expressed as follows, viz. The pressure on the part abme, is p~%bv*, (1). - em fl y isp~bw (v -\- Jw), (2). - If kg, isp=ibx(v-{-w+^, (3). - gkhi, isp = by(v-\-w + x+ jy), (4). - ihcd, isp = bz (v + w -{- x+y+z).(5). Now, according to the conditions of the problem, all these expres- sions for the value of p, are equal to one another, and each of them is equal to the element of comparison, as given in the equation (30) ; hence, from the first of the above equations, or values of p, we have or by expunging the common factor, |6, we obtain *< = -?!; n therefore, by extracting the square root, we have 56 HORIZONTAL SECTIONS SUSTAINING EQUAL FLUID PRESSURES. Again, by comparing equation (30), with the second of the fore- going expressions for the value of p, we shall have bl* bw(v + \w}= - and substituting the value of v in terms of / and n, we obtain n w* -f- 2/-V//T7 w P, from which, by dividing by n, we get w -- . w =. > n n therefore, if this be reduced by the rule which applies to the resolution of adfected quadratic equations, we shall obtain I _ ___ w :rr -( ^/2w V n )' n Proceed as above, by comparing the equation (30), with the third of the preceding expressions for the value of p, and we shall have = and if the above values of v and w, as expressed in terms of I and w, be substituted instead of them in this equation, it will become and dividing by n, we get n n therefore, by completing the square, evolving and transposing, we obtain By pursuing a similar mode of comparison, and reasoning in the same manner, with respect to the fourth value of p foregoing, we shall have let the values of v, w and x, as determined above, be respectively substituted in this equation, and it becomes complete the square, and we obtain PARALLELOGRAM DIVIDED TO SUSTAIN EQUAL FLUID PRESSURES. 57 and from this, by evolution and transposition, we get Pursuing still the same mode of induction for the fifth value of p, and substituting the respective values of v, w, x and y, as we have determined them above in terms of I and n ; then we shall have z { |/5 \/4n~}. And in like manner we may proceed for any number of divisions at pleasure ; but what we have now done is sufficient to exhibit the law of induction. The formulae which we have investigated, for determining the several sections of the given parallelogram, may now be advantage- ously collected into one place ; for it is manifest, that by exhibiting them in juxta-position, the law of their formation is more easily detected, and the difference which obtains between the co-efficients of the successive terms becomes at once assignable. The several equations therefore, when arranged according to the order of the corresponding sections, will stand as under, viz. 3 - x = (V^- &C.&C.ZZ &C. &C. 71. The practical rule for determining the points of section, in reference to their respective distances from the upper extremity of the parallelogram, may be expressed in words, as follows, viz. RULE. Multiply the number of parts into which the paral- lelogram is proposed to be divided, by the number that indicates the place of any particular section; then, multiply the square 58 PARALLELOGRAM DIVIDED TO SUSTAIN EQUAL FLUID PRESSURES. root of the product by the length of the parallelogram, and divide by the whole number of sections, and the quotient will express the distance of any particular point from the upper extremity of the divided length. If the length of any particular section, or the distance between any two contiguous points should be required, which is the condition expressed by each of the above equations ; then, Calculate for each of the points according to the preceding rule, and the difference of the results will give the length of the required section. 72. EXAMPLE 12. A rectangular parallelogram whose length is 25 feet, is perpendicularly immersed in a fluid, in such a manner, that its breadth or upper side is just in contact with the surface ; now, if it be proposed to divide the parallelogram by lines drawn parallel to the horizon, into five parts sustaining equal pressures ; it is required to determine the distance of each point of section from the surface of the fluid, and the respective distances between the several points ? Here then, we have given I 25 feet, and n =r 5 an abstract number ; therefore, by proceeding according to the rule, we shall have, for the distance of the first point, 25V5--- 5 = 11. 18034 feet. For the distance of the second point, we get 25V2 x 5 + 5 = 15.81139 feet. For the distance of the third point, we obtain 25 /3~x^5 H- 5 = 19.36492 feet, and for the distance of the fourth point, it is 25 V 4~X~5 ~ 5 = 22.36068 feet. The preceding is all that is necessary to be calculated, for the distance of the fifth point is manifestly equal to the whole length of the parallelogram, and consequently, by the question, it is a given quantity. Having thus determined the distances of the several points of section below the upper surface of the fluid, the respective distances between them, or the breadths of the several sections can easily be ascertained, since they are merely the consecutive differences of the quantities above calculated ; hence, we have Distances 11.18034, 15.81139, 19.36492, 22.36068, 25. Differences 4.63105, 3.55353, 2.99576, 2.63932 ; LIMIT TO THE REQUISITE THICKNESS OF FLOOD-GATES. 59 therefore, the breadths of the respective sections, estimated in order from the surface of the fluid, are 11.18034, 4.63155, 3.55352, 2.99576, 2.63932 feet. Let A BCD be the rectangular parallelogram, whose length AD or B c is equal to 25 feet, and the breadth A B or DC of any convenient magnitude at pleasure. Upon the length A D, set off the distances AE, A a, A b and AC, respectively equal to the preceding num- bers, taken in the order of their arrangement, and through the points E, a, b and c, draw the straight lines EF, af, be and cd, respectively parallel to AB or DC the horizontal sides of the parallelogram ; then are the rectangular spaces AF, ae, bd and cc, the respective portions into which the given parallelogram ABCD is divided, and on which, according to the conditions of the problem, the perpendicular pres- sures are equal among themselves. OF THE REQUISITE THICKNESS OF FLOOD-GATES, &C. 73. The problem which we have just resolved is a veiy important one; by it we can determine a limit to the requisite thickness of flood-gates and other constructions of a similar nature, and also the form which the section ought to assume, in order that the strength in every part may be proportional to the pressure sustained. For according to the preceding notation, and by equation (7) under the third problem, the pressure on the rectangle ABFE, is p J6v 2 ssin.^, and the pressure on the whole rectangle ABCD, is Pzz: ^bFssin.ty; consequently, by analogy and comparison, we get P:p:i?i v\ And in like manner it may be shown, that the same relation obtains in respect of any other rectangle AB/a, when compared with the entire figure ABCD; consequently, the pressures are uni- versally as the squares of the depths, the breadth being constant ; therefore, the thickness should be as the square of the depth, being greatest at the bottom and decreasing upwards to the surface of the fluid. Thus for example, let the flood-gate be of the same depth as the rectangle in the foregoing question, and let the thickness at the bottom be equal to one foot or twelve inches ; then, the corresponding thicknesses for the several feet of ascent estimated upwards, will be as follows. 60 LIMIT TO THE REQUISITE THICKNESS OF FLOODGATES. At 25 feet, we have 25 s : 25 2 OC2 . 0x14 : 12 : 12 inches. l n 11 060 058 . 032 12 : 10.156 - ro , OC2 . 00 1 9 9 O1 OAJ2 . 012 1 8 4fi8 on 052 . OQ2 1 7 680 in . 5 2 19 2 12 6 93 1 9 , 052 . 1 g2 . JO . Q OOQ V7 .. OA2 . 172 1 5 'i/IR Ifi OC2 . Ifi2 . 10 . xi qcf? 1 ^ 052 . ^52 1 1 30 \*\ OC2 . 1/12 1 S 7fi^l 11 . 1 02 . 10 . 0^^ , ., .. l n , . 102 . 10 . o 74 2 . 112 . 10 . o qo^ , 10 052 . 1Q2 1 1 90 o 052 . os 1 1 556 Q . C2 . Q . 10 . 1 000 7 o . 72 . 10 . C]A() , fi . . f!2 . 10 . o fiQ2 * 052 . y, 1 160 /I OC2 . /|2 1 308 '} j 052 . 02 10 . 017 o OC2 . 02 . 10 n 07fi - 1 9/5* : I 8 19 0.090 .. - 74. Having calculated the numerical values of the thicknesses or ordinates, corresponding to each foot in length, estimated from the bottom where the pressure is a maximum, upwards to the summit where it vanishes ; we shall now proceed to construct the section, in order to exhibit the particular form which the preceding theory assigns. Let the straight line A B represent the perpen- dicular depth of the flood-gate supporting the fluid F, and whose vertical section is denoted by the figure ABC, the exterior boundary of which is the curve line ABC, and the greatest thickness, or that at the bottom, equal to BC. Divide the depth AB into twenty five equal parts, having an interval of one foot for each ; then, through the several points of division, and parallel to the horizon, draw straight lines, beginning at the bottom and proceeding upwards, making these lines respectively equal to 12, 11.06, 10.156, 9.292, &c. OF FLUID PRESSURE ON THE SIDES AND BASE OF CUBICAL VESSELS. 61 inches, according to the numbers in the foregoing tablet, and through the remote extremities of the several ordinates, trace the curve line ADC, which will mark the exterior boundary of the section. The intelligent reader will readily perceive, that in the actual con- struction of the above figure, it has been found impossible to preserve the proper proportion between the several abscissse and their corre- sponding ordinates ; if this had been attempted, the figure must either have been enlarged to an inconvenient size, or the ordinates would have been so small as to render the general appearance of the section very indistinct. We have therefore thought it preferable to preserve the line of the abscissee within moderate bounds, and to enlarge the ordinates in a given constant ratio ; by this means the form of the curve is correct, and the whole diagram is sufficiently distinct for practical illustration. In all that has hitherto been done respecting the rectangular parallelogram, we have constantly considered it as being an inde* pendent plane immersed in the fluid, and having its upper side coincident with the surface ; but we must now observe, that whatever relations have been shown to exist on such a supposition, the same will hold, if the plane be considered as the side of a vessel filled with the fluid by which the pressure is propagated. We have already alluded to this principle, at the conclusion of our illustration of the fourth problem ; it therefore only remains to deter- mine by it, the pressure on the bottom and sides of a vessel filled with a fluid of uniform density, on the supposition that the bottom and the sides are respectively rectangular planes. 11. METHOD OF COMPARING THE PRESSURE ON THE PERPENDICULAR SIDES AND ON THE BOTTOM OF ANY RECTANGULAR CISTERN, BASIN, OR CANAL LOCK. PROBLEM X. 75. Suppose that a vessel in form of a rectangular paral- lelopipedon, is filled with fluid of uniform density, and placed with its sides perpendicular to the horizon : It is required to compare the pressure on the upright sides with that upon the bottom, both when the sides are all equal, and when the opposite sides only are equal. 62 OF FLUID PRESSURE ON THE SIDES AND BASE OF CUBICAL VESSELS. In the solution of the present problem, it will be unnecessary to exhibit the construction of a separate diagram ; because, the bound- aries when considered individually, being rectangular parallelograms, the investigation for each would be similar to that required in Pro- blem 4, and the resulting formulae would coincide in form with that exhibited in equation (8). Therefore, put b = the horizontal breadth of the greater opposite sides, /3 n: the horizontal breadth of the lesser ditto ditto, I z= the perpendicular depth of the fluid, whose den- sity is uniform, P= the aggregate, or total pressure on the upright surface, and p zz the pressure on the bottom. Then, according to equation (8) under Problem 3, the pressure on each of the narrower sides is expressed by |/3/ 2 s, and that on each of the broader sides by \bl*s\ consequently, the entire pressure on the upright surface, is P=l'04..&)< and by the third inference to Proposition (A), the pressure sustained by the bottom of the vessel, is consequently, by analogy, we obtain P:p::l(P + b):pb. Therefore, when the opposite sides of the rectangular vessel only, are equal to one another, The total pressure on the upright surface, is to the pressure on the bottom, as half the area of the former is to the area of the latter. If b in /3, or if all the sides of the vessel are equal to one another ; then, the entire pressure on the upright sides, is and that on the bottom, is p=b*ls; therefore, by analogy, we obtain P :p :: 2/ : 6. OF FLUID PRESSURE ON THE SIDES AND BASE OF CUBICAL VESSELS. 63 Consequently, when all the four sides of the rectangular vessel are equal to one another, The total pressure on the upright surface, is to the pressure on the bottom, as twice the length of the side is to its breadth. Again, when all the sides of the vessel have the same breadth, and the length I equal to the breadth b ; then, the vessel becomes a cube, and the total pressure on the upright surface, is P=26 3 s, and that on the bottom, is p = b 9 s ; therefore, by analogy, we obtain P :p :: 2 : 1. Hence it appears, that when the vessel is a cube, that is, when the bottom and the four upright sides are equal to one another, The total pressure upon the four sides, is to the pressure on the bottom, in the ratio of 2 to 1 . Since the pressure on the upright surface of a cubical vessel, is double the pressure on the base ; it follows, that the entire pressure which the vessel sustains, is equal to three times the pressure upon its bottom ; that is, P+p=:'3b*s. (32). But the expression b*s is manifestly equal to the weight of the fluid ; consequently, the total pressure upon the sides and base of the vessel, Is equal to three times the weight of the fluid which it contains. Now, in the case of water, where the specific gravity is represented by unity, the equation marked (32) becomes P+p = 3b*; but when the dimensions of the vessel are estimated in feet, and the pressure expressed in pounds avoirdupois, of which 62| are equal to the weight of one cubic foot of water ; then, the above equation is transformed into p' = 187.56 s . (33). COROL. This equation in its present form implies, that if the solid content of the vessel in cubic feet, be multiplied by the constant 64 OF FLUID PRESSURE ON THE SIDES AND BASE OF CUBICAL VESSELS. co-efficient 187.5, the product will express the number of Ibs. to which the pressure on the bottom and the four upright sides is equivalent. 76. EXAMPLE 13. Suppose the length of the side of a cubical cistern to be 35 feet ; what is the pressure sustained by it, when it is completely filled with water ? Here we have given, b zz 35 feet ; therefore, by proceeding accord- ing to the composition of the foregoing equation, we shall obtain p' = 35 X 35 x 35 X 187.5= 8039062.5 Ibs. Hence it appears, that the aggregate pressure upon the bottom and the upright surface of a cubical vessel whose side is 35 feet, is 8039062.5 Ibs. or 8039062.5 -f- 2240 = 3588.867 tons, while the absolute weight of the contained fluid, is only one third of that quantity, or 35 X 35 x 35 X62.5 -; 2240 = 1196.289 tons. 77. EXAMPLE 14. Let the dimensions of the vessel remain as in the preceding example, and suppose it to be filled with wine, of which the specific gravity is .96, instead of water, whose specific gravity is unity, what pressure does it then sustain ? For the weight of a cubic foot of wine, we have 1 : 62.5 : : .96 : 60 Ibs. and the pressure on the bottom and sides of a vessel containing a cubic foot is 60 X 3 zz: 180 Ibs. ; consequently, the pressure on the bottom and sides of a vessel whose side is 35 feet, is p' = 35 X 35 X 35 X 180 = 7717500 Ibs., and this, by reducing it to tons, is = 3445 - 3125 tons ' ; :" while the absolute weight of the contained fluid, is only one-third of that quantity, or 35 X 35 X 35 x 60 4- 2240 = 1 148.4375 tons. CHAPTER III. t)N THE PRESSURE EXERTED BY NON-ELASTIC FLUIDS PARABOLIC PLANES IMMERSED IN THOSE FLUIDS, WITH THE METHOD OF FINDING THE CENTRE OF GRAVITY OF THE SPACE INCLUDED BETWEEN ANY RECTANGULAR PARALLELOGRAM AND ITS INSCRIBED PARABOLIC PLANE. 1. WHEN THE AXIS OF THE PARABOLIC PLANE IS PERPENDICULAR TO THE HORIZON, AND ITS VERTEX COINCIDENT WITH THE SURFACE OF THE FLUID. PROBLEM XL 77. If a parabolic plane be just perpendicularly immersed beneath the surface of an incompressible fluid : It is required to compare the pressure upon it, with that upon its circumscribing rectangular parallelogram, and to determine the intensity of pressure, according as the vertex or the base of the parabola is in contact with the surface of the fluid. First, when the vertex of the parabola is uppermost, and just in contact with the surface of the fluid ; let AC B be the parabolic plane, of which AB is the base or double ordiriate parallel to the hori- zon, and CD the vertical axis just covered by the fluid whose surface is EF, and let ABFE be a rectangular parallelogram circumscribing the parabola. Now, it is demonstrated by the writers on mechanics, that the centre of gravity of a parabolic plane is situated in the vertical axis, and the point where it occurs, is at the distance of three fifths of that axis from the summit of the figure. Therefore, if the axis CD be divided at m, into two parts such, that cm is to Dm as 3 is to 2,* then is m the centre of gravity of the VOL. I. F 66 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. parabolic space ACB ; and if the axis CD be bisected in the point n ; n is the centre of gravity of the circumscribing rectangular parallelo- gram ABEF. Put b zz A B, the base of the parabola, or the horizontal breadth of its circumscribing rectangular parallelogram, /CD, the vertical axis of the parabola, or the depth of its sur- rounding rectangle, dm en, the depth of the centre of gravity of the parallelogram ABFE below the upper surface of the fluid, S zz cm, the, depth of the centre of gravity of the parabola ACB, Pzz the pressure on the circumscribing rectangle, p zz the pressure on the parabolic plane, A zz the area of the parallelogram, a zz the area of the parabola, and s the specific gravity of the fluid. Then, according to the writers on mensuration, the area of the cir- cumscribing rectangular parallelogram, is A = bl', but the area of a parabola, is equal to two thirds of the area of its circumscribing rectangle ; therefore, we have a zz %bl. Now, 5 zz %l by the construction, and we have shown in Proposition (A), that the pressure upon any surface, Is expressed by the area of that surface, drawn into the perpendicular depth of its centre of gravity, and also into the specific gravity of the fluid. Consequently, the pressure perpendicular to the surface of the parabolic plane, is />==#/ X#X * = lbP*. (34). But in order to compare the pressure on the parabola, as repre- sented by, or implied in the above equation, with that upon its cir- cumscribing parallelogram, we have only to recollect, thatc?zz|/, and consequently, the pressure on the rectangle, is consequently, by omitting the common factors, and rendering the fractions similar, we have p : P : : 4 : 5. 78. Now, the practical rule for determining the pressure on the parabolic plane, as deduced from the equation (34), or from the pre- ceding analogy, may be expressed in words, as follows. OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 67 RULE. Multiply two fifths of the base of the parabola, by the square of the length of its axis drawn into the specific gravity of the fluid, and the product will express the pressure sustained by the parabolic plane, in a direction perpendicular to its surface. Or thus : Find the pressure on the circumscribing rectangular paral- lelogram, according to the second case of the rule under the third problem, and four fifths of the pressure so determined, will express the pressure perpendicular to the parabolic surface. 79. EXAMPLE 14. A parabolic plane, whose base and vertical axis are respectively equal to 28 and 42 feet, is perpendicularly immersed in a reservoir of water, so that its vertex is just in contact with the surface ; what weight is equivalent to the pressure on the plane, the weight of a cubic foot of water being equal to 62 J Ibs. ? Here, according to the rule, we have p = f x 28 X 42 2 X 62J = 1234800 Ibs., or by the second clause of the rule, it is p \ X 28 x 42* X 62J X * = 1234800 Ibs. Either of these methods is sufficiently simple for every practical purpose ; but it will be found of essential advantage, to bear in mind the relation between the pressure on the parabola and that on its circumscribing rectangle ; for which reason, the latter method may probably claim the preference. 2. METHOD OF FINDING THE CENTRE OF GRAVITY OF THE SPACE INCLUDED BETWEEN ANY RECTANGULAR PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 80. It is a principle almost self-evident, that the centre of gravity, and the centre of magnitude of a rectangular parallelogram, exist in one and the same point ; consequently, admitting the position of the centre of gravity of the rectangle to be known or determinable a priori, the position of the centre of gravity of the inscribed parabola can very readily be found. For by knowing the position of the centre of gravity of a rectangu- lar surface, the pressure upon it can easily be ascertained, and we have shown above, that the pressure upon a parabolic plane, and that upon the surface of its circumscribing parallelogram, are to one another in the ratio of 4 to 5 ; hence, when the pressure on the F2 68 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. rectangular parallelogram is known, the pressure on the inscribed parabola is also known, being equal to four fifths of that upon the parallelogram. Now, it has already been demonstrated, that the pressure upon any surface, whatever may be its form, is always equal to its area, drawn into the perpendicular depth of the centre of gravity, below the upper surface of the fluid ; therefore, conversely, the perpendicular depth of the centre of gravity of any surface, is equal to the pressure which it sustains, divided by the area. But from what, has been demonstrated above, it is manifest that the area of the parabola and the pressure upon it, are respectively expressed by f&Z, and f&/ 2 s; consequently, by division, we obtain and when s is expressed by unity, as in the case of water, we get a=# Now, because the parabola ACB is symmetrically divided by the axis CD, it follows, that the centre of gravity occurs in that line, and we have just shown, that it occurs at the distance of three fifths of its length from the vertex ; hence, the position is determined, and that independently of computing the corresponding horizontal rectangular co-ordinate, whose intersection with the axis fixes the place of the centre sought. The aggregate pressure upon the two equal and similar spaces A E c and BFC, is obviously equal to the difference between the pressures on the rectangular parallelogram ABFE, and that on the inscribed parabola A c D ; that is, where p' denotes the pressure on the spaces A EC and BFC. Again, the area of the spaces A EC and BFC, is equal to the dif- ference between the area of the parallelogram ABFE, and that of the inscribed parabola ACB; therefore, if a 1 denote the area of the trian- gular spaces, we have a' A a=zbl %bl = %bl. But the depth of the centre of gravity of any surface, is equal to the pressure upon that surface divided by its area ; consequently, the depth of the centre of gravity of the figure A EC FB, which is composed of the two triangular spaces AEC and BFC, is OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. hence, if the specific gravity of the fluid be expressed by unity, we get V = Thl. (35). COROL. It therefore appears, that the centre of gravity of the space, included between any rectangular parallelogram and its inscribed parabola, is situated in the axis, at the distance of three tenths of its length from the vertex. The preceding investigation determines the place of the centre of gravity to be at three tenths of the axis below a tangent line passing through the vertex of the parabola, and that it is situated in the axis is manifest; for the spaces A EC and BFC are equal, and they are similarly and symmetrically placed with respect to the axis c D ; there can therefore be no reason, why the centre of gravity should occur at a point, which is nearer to the one than it is to the other ; it conse- quently occurs at a point which is equally distant from both, and that point must obviously be found in the axis of the figure. The above is a valuable proposition in the practice of bridge build- ing ; for by it, we can readily assign the position of the centre of gravity of the arch with all its balancing materials, and consequently, many im- portant particulars respecting the weight and mechanical thrust, may be determined and examined with the greatest facility : all of which will be investigated and applied in our treatise on Hydraulic Architecture. 3. WHEN THE PARABOLIC PLANE PERPENDICULARLY IMMERSED HAS ITS BASE COINCIDENT WITH THE SURFACE OF THE FLUID. 81 . What has hitherto been done under the present problem, applies only to the case, in which the parabola is perpendicularly immersed in the fluid, and having its vertex coincident with the surface ; but when the parabolic plane is perpendicularly immersed, and having its base coincident with the fluid surface, the circumstances will be something different, as will become manifest from the following investigation. Let ABFE be a rectangular parallelogram immersed in a fluid, with its plane perpendicular to the plane of the horizon, and having its upper side coincident with the surface of the fluid ; and let A c B be a parabola described upon the rectangular plane, in such a manner, that its vertex may be downwards, its axis vertical, and its base in contact with the surface of the fluid in which it is placed. JV C 70 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. Draw the diagonal AF intersecting CD, the vertical axis of the parabola in the point n ; then is n the centre of gravity of the rectan- gular parallelogram ABFE ; and because, as we have stated in the construction of the preceding case, the centre of gravity of a parabolic plane is situated in the axis, at the distance of three fifths of its length from the vertex ; it follows, that if the axis DC be divided in the point m, into two parts such, that Dm is to me in the ratio of 2 to 3 ; then shall m be the centre of gravity of the parabola ACB. Put 6 = AB, the horizontal breadth of the rectangular parallelogram ABF,E, or the base of its inscribed parabola ACB, / DC or AE, the vertical axis of the parabola, or the depth of its circumscribing rectangle, d D n , the depth of the point n, below A B the surface of the fluid , nr DW, the depth of the point m as referred to AB, P the pressure on the surface of the rectangle ABFE, j9=:the pressure on the parabolic surface ACB, Az=the area of the rectangular parallelogram, a the area of its inscribed parabola, and s zn the specific gravity of the fluid. Now, it is manifest from the principles of mensuration, that the area of a rectangular parallelogram, is equal to the product that arises when the two dimensions of length and breadth are multiplied into one another ; that is, A=bl, and according to the writers on conic sections, the area of a parabola is equal to two thirds of the area of its circumscribing rectangle ; therefore, we have oz=f6Z. Referring to the construction of the figure, we find that the axis DC is divided at m, into the two parts Dm and me, having to one another the ratio of 2 to 3 ; it therefore follows, that Dmzi: -:/; consequently, the pressure on the parabolic surface ACB, is p ^blX^lXs ^b l*s. (36). Again, since AF the diagonal of the parallelogram, bisects DC the axis of the parabola in the point n ; it follows, that Dw = d|/; therefore, the pressure perpendicular to the surface of the rectangular parallelogram ABFE, becomes P blX i/X s= OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 71 hence, by analogy, we obtain p : P :: T \bl*s : \bl*s, and this, by omitting the common quantities, b, I 1 and s, becomes P : P : : rV : 1, and finally, by reducing the fractions in this analogy to a common denominator, we shall obtain p : P : : 8 : 15. 82. Having thus established the formula for determining the pres- sure on the parabolic plane, and also the ratio which compares the said pressure with that upon the circumscribing rectangular paral- lelogram ; we shall in the next place deduce the rules, by which the numerical process is to be performed, when the principle is applied to the actual determination of the pressure in reference to cases of prac- tice. The rule, as derived from the equation numbered (36), may be expressed in the following manner. RULE. Multiply the base of the parabola by the square of its vertical axis, and again by the specific gravity of the fluid ; then, four fifteenths of the product will express the pressure perpendicular to the surface of the parabolic plane. The rule for determining the pressure on the surface of a parabola, as deduced from the analogy of comparison investigated above, may be expressed in words in the following manner. Find the pressure perpendicular to the surface of the circumscribed rectangular parallelogram, after the manner described in the second case of the rule under the third problem; then, eight fifteenths of the pressure so determined, will express the pressure on the parabolic plane. 83. EXAMPLE 15. The data remaining as in the preceding case ; what will be the pressure on the parabolic plane, its axis being verti- cal, and its base in contact with the surface of the fluid ? 28 X 42 X 42 X 62J = 3087000, and four fifteenths of this, is 3087000 X 4 -r 15 zz 823200 Ibs. This result is derived from the first of the above rules, or that which corresponds to the equation (36) ; and the process as performed by the second rule, or that obtained from the ratio of comparison, will stand as below. } X 28 X 42 X 42 X 62 X T V = 823200 Ibs. 72 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. Hence, the pressure on the plane in this case, is only two thirds of what we found it to be in the foregoing case, where the vertex is in contact with the surface of the fluid. With respect to the position of the centre of gravity in this case, it is manifest, that the mode of discovering it, is similar to that which we employed in the case immediately preceding, where the axis of the parabola was supposed lo be vertical, and its s-ummit in contact with the surface of the fluid ; it is therefore unnecessary to repeat the investigation, but there is another condition of the figure remaining to be considered, in which a knowledge of the position of the centre of gravity becomes of more importance, as will readily appear from the circumstances which present themselves in the solution of the following problem. 4. WHEN THE BASE OF THE PARABOLIC PLANE IS PERPENDICULAR TO THE HORIZON, ITS AXIS HORIZONTAL, AND THE PRESSURE UPON IT IS TO BE DETERMINED AS COMPARED WITH THAT UPON ITS CIRCUMSCRIBING RECTANGULAR PARALLELOGRAM. PROBLEM XII. 84. If a parabolic plane be perpendicularly immersed in an incompressible fluid, in such a manner, that its base may be vertical, and just in contact with the surface : It is required to determine the pressure upon it, and to compare it with that upon its circumscribing rectangular parallelogram. Let ABEF be a rectangular parallelogram immersed in a fluid, with its plane perpendicular to the plane of the horizon, and its upper side AB coincident with the surface of the fluid in which it is immersed. Bisect AF and BE in the points D and c; join DC, and upon AF as a base, with the corresponding axis DC, describe the parabola ACF, touching AB the surface of the fluid in the point A ; then is AC F the surface for which the pressure is required to be investigated. Join BF, intersecting DC the axis of the parabola in the point n- r then is n the centre of gravity of the rectangular parallelogram ABEF. OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 73 Divide the axis DC into two parts Dm and cm such, that Dm is to cm, in the ratio of 2 to 3; and the point m thus determined, is the place of the centre of gravity of the parabola ACF. Through the points m and n, draw the straight lines mr and ns respectively per- pendicular to the axis DC; then are mr and ns 7 the perpendicular depths of the points m and n below AB, the surface of the fluid, and which in the present case are equal to one another. Put &ZZIAB, the horizontal breadth of the rectangular parallelogram ABEF, or the axis of its inscribed parabola ACF, I AF, the length of the circumscribing rectangle, or the base of the inscribed parabola, d rm or sn, the vertical depths of the centres of gravity, below AB the surface of the fluid, Pmthe pressure on the rectangle ABEF, /> = that on the inscribed parabola ACF, A = the area of the circumscribing rectangular parallelogram, a the area of the parabola, and s the specific gravity of the fluid in which they are immersed. Then, according to the principles of mensuration, the area of the rectangular parallelogram ABEF, is equal to the product of the breadth AB drawn into the depth AF ; that is, A = bl; and by the property of the parabola, its area is a = \b I. But the pressure perpendicular to the surface of the rectangular parallelogram, is, as we have already frequently stated, expressed by the area drawn into the perpendicular depth of the centre of gravity ; and this being the case, whatever may be the form of the surface pressed, it follows, that the pressure on the rectangle ABEF, is P bl X d X s bdls' y and that on the parabola ACF is Now, it is manifest from the nature of the figure, and from the principles upon which it is constructed, that rm and sn are each of them equal to JAF; that is, d=L\l\ therefore, let \l be substituted for d in each of the above equations, and we shall obtain For the rectangle ABEF, and for the parabola ACF, it is (37). 74 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. Consequently, by analogy, the comparative pressures on the para- bola and its circumscribing rectangle, are as follows : p: P ::ibl*s:%bl*s; and from this, by casting out the common quantities and assimilating the fractions, we get p : P : : 2 : 3. COROL. Hence it appears, that when the axis of the parabola is horizontal, and its base perpendicular to the horizon ; the pressure perpendicular to its surface, when compared with that on its circum- scribing parallelogram, bears precisely the same relation, that its area bears to the area of the rectangle by which it is circumscribed. 85. The practical rule for determining the pressure on the parabolic plane, when placed in the position specified in the problem, may be expressed in words at length in the following manner. RULE. Multiply the horizontal axis, by the square of the vertical base or double ordinate, and again by the specific gravity of the fluid ; then, take one third of the product for the pressure perpendicular to the surface of the parabolic plane. Or thus, Calculate the pressure on the circumscribing rectangle, and take two thirds of the result for the pressure on the parabola. 86. EXAMPLE 16. The data remaining as in the example to the foregoing problem, it is required to determine the pressure on the parabolic plane, supposing its axis to be horizontal, its base or double ordinate vertical, and the upper extremity of the base in contact with the surface of the fluid, which, according to the conditions of the previous question, is water, whose specific gravity is expressed by unity, and the weight of one cubic foot of which is equal to 62 Ibs. avoirdupois ? Referring the numerical data to the same parts of the figure, as in the preceding cases, we have given 6 = 42 feet; /iz:28 feet, and s zz: 62 j Ibs. ; therefore, by proceeding according to the rule, we have 42 X 28 X 28 X 62 J = 2058000, which being divided by 3, gives p = 2058000 -r 3 = 686000 Ibs. Hence it appears, that the total pressures perpendicular to the parabolic surface, according to the several positions in which we have placed it, are to one another respectively as the numbers 3087, 2058 and 1715; OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 75 the first two of which, by reason of the parts of the figure being the same in each, are obviously dependent upon one another; but the third, in which the parts of the figure are reversed, is wholly inde- pendent and distinct from the other two. 87. COROL. 1. Admitting therefore, that the pressure upon the parabolic surface, under the three circumstances of position in which we have considered it, is represented by the equations (35), 36) and (37) ; it follows, that the situation of the centre of gravity can easily be ascertained ; for the pressure in each case, as we have elsewhere shown, is represented by the area of the figure, drawn into the per- pendicular depth of its centre of gravity ; consequently by reversing the process, the depth of the centre of gravity will become known, if the pressure be divided by the area of the surface on which the fluid presses. COROL. 2. Since the parabola is a figure symmetrically situated with respect to its axis, it is obvious, that the centre of gravity of its surface must occur at the same point, in whatsoever position it may be placed; but when the place of its centre is referred to the surface of the fluid in which it is immersed, the distance varies for each particular case: thus, In the first instance, the perpendicular distance, is in f ths of the axis, second, izrfths third, m ^ the base. But as we have just stated, the centre of gravity of the parabolic sur- face as referred to its vertex, or any other fixed point, in all these cases, remains unaltered, in whatever position the figure itself may be placed. 5. THE METHOD OF DETERMINING THE PRESSURE OF THE FLUID UPON A SEMI-PARABOLIC PLANE AS COMPARED WITH THAT ON THE CIRCUMSCRIBING RECTANGULAR PARALLELOGRAM. 88. When the semi-parabola only is considered, the determina- tion of its centre of gravity, and consequently, of the pressure on its surface becomes more difficult ; for, since the figure is not sym- metrical with respect to its axis, we are under the necessity of com- puting the two rectangular co-ordinates, whose in- tersection determines the place of the required centre. Let CBD be a semi-parabola, perpendicularly immersed in a fluid, so that its axis CD is vertical, and the vertex in contact with CF the surface of the fluid, and let CFBD be the circumscribing rectan- gular parallelogram. 76 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. Now, since by Problem XI, it has been proved that the pressure on the entire parabola with its axis vertical, is equal to four fifths of that upon its circumscribing parallelogram ; it follows, that the pressure on the semi-parabola in the same position, is also equal to four fifths of that upon its circumscribing parallelogram, it being manifestly equal to half the pressure on the whole parabola. Divide the axis CD into two parts, such, that Dm and cm shall be to one another in the ratio of 2 to 3 ; and in like manner, let the ordinate or base BD be divided into two parts, such, that DW and vn shall be to one another in the ratio of 3 to 5* ; then, through the points m and n, and respectively parallel to DB and DC, draw the straight lines WIG and no, meeting each other in G, the centre of gravity of the semi-parabola DC B. Put &IZICF or DB, the horizontal breadth of the rectangle CFED, or the base of the semi-parabola c B D, I =: CD or FB, the vertical depth of the rectangle, or axis of its inscribed semi-parabola, cZzz cm or EG, the perpendicular depth of the centre of gravity, Pzzthe pressure on the rectangular parallelogram CFBD, p zz the pressure on its inscribed semi-parabola, A zz the area of the parallelogram, a zz the area of the semi-parabola, and s zz the specific gravity of the fluid in which they are immersed. Then, according to the principles of mensuration, the area of a rectangle is expressed by the product of its two dimensions ; that is, by its length drawn into its breadth ; therefore, we have A = bl, and by Proposition (A), the pressure exerted by a fluid, perpendicu- larly to any surface immersed in it, or otherwise exposed to its influence, Is equal to the area of the surface pressed, drawn into the perpendicular depth of its centre of gravity, and again into the specific gravity of the fluid. Consequently, the pressure on the circumscribing rectangular paral- lelogram CFBD, becomes * It is demonstrated by the writers on mechanics, that the centre of gravity of a semi-parabola is situated in its plane, at the distance of three eighths of the ordinate from the axis, and two fifths of the axis from the ordinate, or three fifths of the axis from its vertex. OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 77 Now, since the pressure on the semi-parabola is equal to four fifths of the pressure on its circumscribing parallelogram, we shall obtain f P = $ X Ibl^s ^b^sp. (38). The expression which we have here determined for the pressure on the surface of the semi-parabola D c B, is precisely the same as that which we have given in equation (35) for the entire figure ; only in the pre* sent instance, the value of 6, the horizontal breadth of the parallelogram, is but one half the value as applied to the parabola, when placed under the conditions specified in the eleventh problem foregoing. 89. If the symbol b retain its former value, that is, if it be referred to the base of the entire parabola, or to the breadth of the parallelo- gram circumscribing the entire parabola, then, the pressure on the semi-parabola, becomes ' p \bl^s. (39). Consequently, the practical rule for determining the pressure on the semi-parabola as deduced from this equation, may be expressed as follows. RULE. Multiply one fifth of the base, or double ordinate of the whole parabola, by the square of the length of its axis, and again by the specific gravity of the fluid, and the product will express the pressure on the semi-parabola in a direction perpendicular to its surface. But if the symbol b refer to the ordinate, or base of the semi- parabola, then, the rule as deduced from the equation (38), will be precisely the same as that which we have given under the equation numbered (35) in Problem XI, to which place the reader is referred for the purpose of avoiding a direct repetition. 90. EXAMPLE 17. A plane in the form of a semi-parabola whose base or ordinate is 16 feet, and its axis 40 feet, is perpendicularly immersed in a cistern of water, in such a manner, that its axis is vertical, and its vertex in contact with the surface of the fluid ; what pressure does it sustain, the weight of a cubic foot of water being equal to 62 Jibs.? The equation in its present state, supposes the ordinate, or base of the semi-parabola to be given, and therefore, the pressure is deter- mined by the rule to the equation (35) or (38), in the following manner : pl x 16 X40 2 X 621= 640000 Ibs. But in order to determine the pressure by the rule immediately preceding, we must suppose the breadth or base of the figure to be 78 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. doubled, in which case it will have reference to the whole parabola, and the pressure will be reduced to that upon its half, by employing the constant \ instead of f- according to the rule, thus, p = i x 32 x 40 2 X 62J = 640000 Ibs. 91. If the axis of the semi-parabola were horizontal and its ordinate vertical, as in the annexed diagram ; then, the area of the semi-parabolic figure, as well as that of its circumscribing parallelogram, will remain the same, but the pressures perpendicular to their respective surfaces will be very different. Divide BD in n, in such a manner, that En and DW may be to one another in the ratio of 5 to 3 ; and in like manner, divide CD in m, so that cm and Dm shall be to each other in the ratio of 3 to 2. Through the points n and m, and parallel respectively to B F and BD, draw the straight lines no, and ma, intersecting one another in the point G, arid produce m G to E ; then , by the note to the pre- ceding case, the point G is the centre of gravity of the semi-parabola DCB, and EG is its perpendicular depth below BF, the horizontal surface of the fluid. Therefore, let the preceding notation remain, and let the several symbols refer to the same parts of the figure as in the preceding case, disregarding the change of position which has taken place ; then, as before, the area of the parallelogram BFCD, is A = 6/, and the pressure perpendicular to its surface, is For draw the diagonals B c and F D intersecting each other in the point r, and through r draw rs parallel to BD or Em; then, r is the centre of gravity of the rectangle BFCD, and sr its perpendicular depth below BF the surface of the fluid; but according to our notation srzr \b, and we have seen above, that A zz: bl; now, the pressure on any surface, whatever may be its form, Is equal to the product that arises, when the area of the surface pressed, is drawn into the perpendicular depth of its centre of gravity, and again, into the specific gravity of the fluid. Consequently, the pressure perpendicular to the surface of the rectangular parallelogram BFCD, is OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 79 Again, the area of the semi-parabola BCD, is equal to two thirds of its circumscribing rectangular parallelogram BFCD ; therefore we have a = l X bl=$bl, and the pressure perpendicular to its surface, is p = &Vl8. This is manifest, for according to the construction and the nature of the figure of the parabola, BW or EG is equal to five eighths of BD ; therefore, we have p ^blXibX s = -frb*ls', (40). consequently, by analogy, we obtain p : P : : &&ls : b*ls. Therefore, by suppressing the common factors, and rendering the fractions T \ and | similar, we shall get p:P::5: 6; hence it appears, that when the ordinate of the semi-parabola is vertical, and its upper extremity in contact with the surface of the fluid : The pressure upon the semi-parabola, is to that upon its circumscribing rectangular parallelogram, as 5 is to 6, or as 1 is to 1.2. 92. Consequently, the practical rule for determining the pressure in the present instance, as deduced from the equation marked (40), or from the above analogy, may be expressed as fdllows. RULE. Multiply the square of the given ordinate by the axis of the semi-parabola, and again by the specific gravity of the fluid ; then, Jive twelfths of the result will give the pressure sought. Or thus, Find the pressure on the circumscribing parallelogram, and take five sixths of the pressure thus found, for the pressure on the semi-parabola. 93. EXAMPLE 18. Let the numerical values of the axis and ordi- nate remain as in the preceding example ; what will be the pressure on the surface of the semi-parabola, supposing the axis to be hori- zontal, the ordinate vertical, and its remote extremity in contact with the surface of the fluid ? If the operation be performed according to the rule deduced from equation (40), we shall obtain p= 16 2 X 40 X 62 X T V = 2666661- Ibs. 80 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. but if the operation be performed according to the rule derived from the analogy of comparison, we shall have p = f P ; that is, p \ x 16* X 40 X 62 1 X = 266666f Ibs. 6. THE METHOD OF DETERMINING THE POSITION OF THE CENTRE OF GRAVITY OF THE SPACE COMPREHENDED BETWEEN THE PARABOLIC CURVE AND ITS CIRCUMSCRIBING PARALLELOGRAM. 94. By means of the pressure on the semi-parabola, as we have investigated it in the two foregoing cases, we are enabled to determine the pressure on the space CFB, and from thence, the position of its centre of gravity. This is an important inquiry in the practice of bridge building, for, in determining the thickness of piers necessary to resist the drift or shoot of a given arch, independently of the aid afforded by the other arches, it becomes requisite to find the centre of gravity of the span- drel or space BFC, which is used for the purpose of balancing the arch and filling up the haunches or flanks. Now, the method which has generally been employed for the deter- mination of this centre is extremely operose, and in many cases it involves considerable difficulty, requiring the calculations of solids and planes, which are by no means easy ; but the method which we are about to employ, requires no such tedious and prolix operations, as will become manifest from the following investigation, which refers to the space comprehended between a semi-parabola and its circum- scribing rectangle. Let BCD be a semi-parabola, having the axis DC vertical while the corresponding ordinate DB is horizontal, and let B DC F be the circumscribing rectangular parallelogram. Suppose the point D to remain fixed, and conceive the semi-parabola BCD to revolve about the point D until it comes into the position A ED, where the axis DE is horizon- tal, and the corresponding ordinate DA vertical; then it is manifest, that the circumscribing rectangular parallelogram ADEH in this latter position, is equal to BDCF in the former, and consequently, the space AHE comprehended between the sides of the rectangle AH, HE and the curve AE, is equal to the space BFC similarly con- stituted. OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 81 It is further manifest, that while the semi-parabola revolves about the point D, from the position BCD to that of A ED, the points B and c, the extremities of the ordinate and axis, describe respectively, the circular quadrants BA and CE, while the point F describes another quadrant, whose containing radii are the diagonals DF and DH. Put 6:zr BD or AD, the ordinate of the semi-parabola in either posi- tion, I n= CD or ED, the corresponding axis, d nG, the depth of the centre of gravity of the space BFC, when the axis is vertical, 3 rz m G, or A ft, the depth of the centre of gravity of the space AHE or BFC, when the axis is horizontal, Pzz: the pressure on the circumscribing rectangular parallelogram BDCF, Or AH ED, p the pressure on the inscribed semi-parabola, and p the pressure on the space comprehended between the semi- parabola and its circumscribing rectangular parallelo- gram. Then, according to equation (8) under the third problem, the pressure on the circumscribing rectangular parallelogram when the length is vertical, is and agreeably to equation (38) under the eleventh problem, the pressure on the inscribed semi-parabola with the axis vertical, is consequently, by subtraction, the pressure upon the space BFC, com- prehended between the sides of the parallelogram BF, FC and the curve of the parabola BC, is hence, by suppressing the symbol for the specific gravity, we get p' bl^(l f)rr T V^ 2 . (41). Now, according to the writers on mensuration, the area of the semi-parabola is equal to two thirds of the area of the circumscribing parallelogram ; it therefore follows, that the area of the space BFC, is equal to one third of the rectangle BDCF; that is, bi $bl=;$bl. But it has been demonstrated, that the pressure upon any surface, is equal to the area of that surface, drawn into the perpendicular depth of the centre of gravity ; consequently, we have VOL. i. G 82 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. and we have shown above in equation (41), that when the axis of the semi-parabola is vertical, the pressure on the space BFC is consequently, by comparison, we obtain #*!=&*+ and finally, dividing by ^bl, we shall have Again, when the axis of the semi-parabola is horizontal, as indicated by A ED, the pressure on the circumscribing rectangle, according to equation (10) under the third problem, is and the pressure upon the inscribed parabola, according to equation (40) under the eleventh problem, is therefore, by subtraction, the pressure upon the space comprehended between the rectangular parallelogram and its inscribed semi-parabola, p' P p = J6* /* T M> 2 ' , and by suppressing the symbol for the specific gravity, we have Now, the area of the inscribed semi-parabola is, as we have seen above, equal to two thirds of its bounding rectangle, and the area of the space comprehended between the rectangle and the semi-parabola, is therefore, equal to one third of the same quantity ; that is, bllbl = *bl; consequently, the pressure on the irregular space A HE, is p'=ibtl; hence, by comparison, we shall have ^= T V^; therefore, by division, we obtain Having thus determined the values of the rectangular co-ordinates, as represented by the equations (42) and (43), the position of the centre of gravity can easily be found ; for, from the point F, set off rm equal to three tenths of the axis CD, and jfn equal to one fourth of the ordinate BD, or its equal FC ; then, through the points m and n, and parallel respectively to the ordinate BD and axis CD, draw the OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 813 straight lines ma and HG, intersecting each other in the point G ; and the point G thus determined, is the position of the centre of gravity of the space comprehended between the semi-parabola and its circum- scribing rectangular parallelogram. This method of determining the position of the centre of gravity of the space comprehended between the curve and its circumscribing parallelogram, will be illustrated and applied in all its generality, when we come to treat on the subject of Hydraulic Architecture, to which it more properly belongs ; and for this reason, we shall take no further notice of it in this place, but proceed with our inquiry respecting pressure, which is more immediately the object of our research. 7. METHOD OF DIVIDING A PARABOLIC PLANE PARALLEL TO ITS BASE, SO THAT THE FLUID PRESSURES ON EACH PART MAY BE EQUAL TO ONE ANOTHER. PROBLEM XIII. 95. If a parabolic plane be immersed perpendicularly in an incompressible fluid, in such a manner, that its vertex is j ust in contact with the surface : It is required to determine at what distance from the vertex, a straight line must be drawn parallel to the base, so that the figure may be divided into two parts, on which the pressures are equal to one another. Let ACB be the given parabola, immersed in the fluid after the manner specified in the problem, and let aAEb be the circumscribing rectangular parallelogram. Take cm for the distance from the vertex through which the line of division passes, and draw EF parallel to the base AB; then are the spaces ABFE and ECF, the parts into which the parabola is divided, and on which, by the conditions of the problem, the pres- sures are equal. Through the points E and F, the extremities of the line of division, draw EC and FG? respectively parallel to CD the axis of the figure; then is CEFC?, the rectangular parallelogram circumscribing the para- bola ECF. G2 84 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. Put 26 = AB or ab, the base of the parabola ACB, or the horizontal breadth of the circumscribing parallelogram OAB&, I zz CD or a A, the vertical axis of the parabola ACB, or the depth of the rectangle by which it is encompassed, 2y zz EF or cd, the base of the parabola ECF, or the breadth of the rectangle c E F d, x zz cm or CE, the axis of the parabola ECF, or the depth of its circumscribing rectangle CEFC?, P zzthe pressure on the rectangular parallelogram CEAB&, cir- cumscribing the parabola ACB, p zz the pressure on the inscribed parabola, P' zz the pressure on the rectangular parallelogram CEF^, cir- cumscribing the parabola ECF, p' zz the pressure on the inscribed parabola, and * zz the specific gravity of the fluid. Then since ABzz26 and EFZz2y, it follows, that AD 6 and EWI zz y ; therefore, by the property of the parabola, we have and consequently, by equating the products of the extreme and mean terms, we shall obtain y=p*, and by division, it is b*x y =T> and this, by extracting the square root, becomes and finally, multiplying by 2, we obtain EFZZ ^ ^ Now, the pressure perpendicular to the surface of the rectangular parallelogram aAB#, according to equation (8) under the third problem, is but we have seen elsewhere, that the pressure on the surface of a parabola, is equal to four fifths of that upon its circumscribing paral- lelogram ; consequently, the pressure on the parabola ACB, is (44). OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 85 Again, the pressure perpendicular to the surface of the rectangular parallelogram CEFS = the surface of the lesser sphere a b d, p the pressure perpendicular to its surface, and TT nr 3.1416, the circumference of a circle whose diameter is expressed by unity. Then, according to the principles of mensuration, the surface of a sphere or globe Is equal to four times the area of one of its great circles, or that whose plane passes through the centre of the sphere. Consequently, the convex surface of the greater sphere ABD, is expressed as follows. and that of the lesser sphere abd, is S =: 3.1416 eP. But the pressure perpendicular to any surface, is equal to the area of that surface multiplied by the perpendicular depth of the centre of gravity, and again by the specific gravity of the fluid ; consequently, when the specific gravity of the fluid is denoted by unity, we have for the pressure on the surface of the greater sphere, P=r3.1416D a X iD=1.5708D 8 . (50). and for the pressure on the lesser sphere, it is hence, by comparison, we shall have P : p : : D 3 : d 3 . Consequently, if two spheres of different diameters be placed in a fluid under similar circumstances, the pressures perpendicular to their surfaces, are to one another as the cubes of their diameters. By the principles of mensuration, the solid content of a sphere or globe, is equal to the cube of the diameter multiplied by the constant number .5236 ; therefore, if c denote the solid content, we have cin.5236D 3 , 92 OF FLUID PRESSURE ON CIRCULAR PLANES AND ON SPHERES. or multiplying both sides of the equation by 3, we get 3czzl.5708D 3 ; consequently, by equation (50), we have or if s denote the specific gravity of the fluid, we shall obtain COROL. Hence we infer, that if a hollow sphere or globe be filled with an incompressible and non-elastic fluid : The whole pressure sustained by the internal surface of the sphere is equal to three times the weight of the fluid which it contains. 102. EXAMPLE 21. A hollow spherical shell or vessel, whose inte- rior diameter is equal to 30 feet, is completely filled with water ; what weight is equivalent to the pressure sustained by its internal surface ? Here, by operating according to the process indicated in equation (50), we have P=: 1.5708 X 30 3 zr42411.6cub.ft. Now, since the fluid with which the vessel is filled, is water, giving a weight of 62 \ Ibs. to a cubic foot, we have P =. 4241 1.6 X 62 J =: 2650725 Ibs. ; but 2240 Ibs. are equal to one ton; therefore, the pressure on the internal surface of a hollow spherical vessel whose diameter is 30 feet, when completely filled with water, is P = 2650725 H- 2240 1 183 Hi tons. PROBLEM XVI. 103. Suppose a sphere or globe to be immersed in an incom- pressible and non-elastic fluid, in such a manner, that the upper extremity of the vertical diameter is just in contact with its surface : It is required to determine through what point of the axis a horizontal plane must pass, so to divide the sphere, that the pressure on the convex surface of the lower segment, may be equal to the pressure on the convex surface of the upper. Let AD BE represent the sphere in question, so placed, that A the upper extremity of the vertical diameter, is just in contact with FG the surface of the fluid. OF FLUID PRESSURE ON CIRCULAR PLANES AND ON SPHERES. 93 Suppose that P is the point in the vertical diameter through which the plane of division passes, separating the sphere into the seg- ments DAE and DBE, sustaining equal pres- sures on their convex surfaces. Bisect A p and B P in the points m and n ; then are m and n y the points thus determined, respectively the centres of gravity of the surfaces of the spheric seg- ments DAE and DBE,* and Am, A.n are their perpendicular depths below FG the horizontal surface of the fluid. Put D AB, the vertical diameter of the sphere or globe A DBE, d=:Am, the depth of the centre of gravity of the surface of the upper segment DAE, zz A n, the depth of the centre of gravity of the surface of the lower segment DBE, S zz the surface of the upper segment, P zn the pressure upon it, &'zz the surface of the lower segment, p the pressure upon it, s zz the specific gravity of the fluid, x zz AP, the perpendicular depth of the point through which the plane of division passes, and 7TZZ3.1416, the circumference of the circle whose diameter is unity. Then, according to the principles of mensuration, the convex sur- face of a spheric segment : Is equal to the circumference of the sphere, drawn into the versed sine or height of the segment ivhose surface is sought. And moreover, the circumference of a sphere, or the circumference of any of its great circles : Is equal to the diameter multiplied by the constant quantity TT, or the number 3.1416. consequently, the convex surface of the upper segment DAE, is and that of the lower segment DBE, is S zz D TT (D x) zz 3.1416 D (D x}. * It is demonstrated by the writers on mechanics, that the centre of gravity of the surface of a spheric segment, is at the middle of its versed sine or height. 94 OF FLUID PRESSURE ON CIRCULAR PLANES AND ON SPHERES. But the pressure perpendicular to any surface, whatever may be its form, as we have already sufficiently demonstrated : Is equal to the area of the surface multiplied by the perpendicular depth of the centre of gravity, and again by the specific gravity of the fluid. Therefore, the pressure perpendicular to the convex surface of the upper segment DAE, is P = 3.1416Da: X i* X * = 1.5708 DSX\ (51). and the pressure perpendicular to the convex surface of the lower segment DBE, is (52). Now these two expressions, according to the conditions of the problem, are equal to one another; consequently, by comparison, we get 1 .5708 D s x* 3.1416 D s {(D x) x -f g (D a;) 8 }, and from this, by suppressing the common quantities, we have * a :=:2{(D-aO*+i(D-.af}; therefore, by expanding the terms, we obtain 2x* D S ; consequently, by division and evolution, we get ar=zJDV2. (53). 104. The ultimate form of this equation is extremely simple, and the practical rule which it supplies, may be expressed as follows. RULE. Multiply the radius, or half the diameter of the sphere by the square root of 2, and the product will give the point in the vertical diameter through which the plane of division passes, estimated downwards from the surface of the fluid. 105. EXAMPLE 22. A sphere or globe, whose diameter is 18 inches, is immersed in a fluid, in such a manner, that the upper extremity is just in contact with the surface ; through what point of the diameter must a horizontal plane be made to pass, so to divide the sphere, that the pressures on the curve surface of the upper and lower segments may be equal to one another ? The square root of 2, is 1.4142, and half the given diameter is 9 inches ; consequently, by the rule we have x =1.4142 X 9 12. 7278 inches, OF FLUID PRESSURE ON CIRCULAR PLANES AND ON SPHERES. 95 106. The preceding investigation applies to the particular case, in which the pressures on the curve surfaces of the segments are equal to one another ; but in order to render the solution general, we must investigate a formula to indicate the point of division, when the pressures are to one another in any ratio whatever ; for instance, that of m to n. By expunging the common factors from the equations (51) and (52), we obtain a : 2 { (D a) x + } (D *) 2 } : : m : n ; therefore, by equating the products of the extreme and mean terms, we get 2m{(Dx)x + J( D xY} = nx*, which, by expanding the bracketted expression, becomes nxt m^ x*}, or by transposition, we obtain (m -\- n) x* nz m D 2 , and finally, by dividing and extracting the square root, we have rm m+n (54). 107. The general equation just investigated, is sufficiently simple in its form for every practical purpose that is likely to occur ; it may therefore appear superfluous to reduce it to a rule, yet nevertheless, that nothing may be wanting for the general accommodation of our readers, we think proper to draw up the following enunciation. RULE. Divide the first term of the ratio by the sum of the termSj and multiply the square root of the quotient by the diameter of the sphere ; then, the product thus arising, will express the distance below the surface of the fluid, of that point through which the plane of division passes. It is unnecessary to propose an example for the purpose of illus- trating the above rule ; that which we have already given, where the values of m and n are equal to one another, being quite sufficient. CHAPTER V. OF THE PRESSURE OF NON-ELASTIC OR INCOMPRESSIBLE FLUIDS AGAINST THE INTERIOR SURFACES OF VESSELS HAVING THE ; FORMS OF TETRAHEDRONS, CYLINDERS, TRUNCATED CONES, &C. 1. WHEN THE VESSEL IS IN THE FORM OF A TETRAHEDRON. PROBLEM XVII. 108. Suppose a vessel in the form of a tetrahedron, or equi- lateral triangular pyramid, to be filled with an incompressible and non-elastic fluid : It is required to compare the pressure on the base with that upon the sides, and also with the weight of the fluid; the base of the vessel being parallel to the horizon. Let ABCD be the tetrahedron filled with fluid, of which ABC is the base parallel to the horizon, and ABD, c B D and ADC the sides or equal contain- ing planes. From D the vertex of the figure, let fall the perpendicular DP, upon the base or opposite side ABC; then will DP be the vertical depth of the centre of gravity of the base ABC, below the horizontal plane passing through D, the summit of the figure, or highest particle of the fluid. Bisect AD and AB, two of the adjacent edges of the figure, in the points m and n\ draw the straight lines BWI and DW, intersecting each other in the point r ; then is r the centre of gravity of the triangular plane ABD. Through the point r draw rs perpendicular to DP, the altitude of the vessel or pyramid; then is DS, the perpendicular depth of the centre of gravity of the triangular plane ADB, below the vertex D, or the uppermost particle of the fluid. OF FLUID PRESSURE UPON THE INTERIOR OF A TETRAHEDRON. 97 By the nature of the figure, the three containing planes A D E, ADC and BDC, are equal to one another, and they are also equally inclined to, or similarly situated with respect to the base ABC; consequently DS, the perpendicular depth of the centre of gravity, is common to them all. Now, by the property of the centre of gravity, we know, that or is equal to two thirds of D n ; therefore, by reason of the parallel lines np and rs, DS is also equal to two thirds of DP. Put a zn the area of the base and each of the other containing planes, I ~ the length of the side of each triangular plane, or the edges of the figure, d zz: DP, the perpendicular depth of the centre of gravity of the base ABC, 3 zr DS, the perpendicular depth of the centre of gravity of the side ADB ; P zz the pressure upon the b*ase, i ' p zz the pressure upon wzzthe weight of the fluid contained in the vessel, and s zz the specific gravity. Then, by the principles of Plane Trigonometry and the property of the right angled triangle, we have DP d~ \/4 sec.* 30, but by the arithmetic of sines, we know, that secJ30zzli-; consequently, by substitution, we have . i? d=-L V e. ' $ v; ::"::"; Now, according to the construction of the figure and the property of the centre of gravity, it follows, that DS is equal to two thirds of DP ; hence we get * 2J _ DszzrSz:: ^6. By the nature of the figure, it is manifest, that the area of each of the triangular sides is equal to the area of the base ; and by the prin- ciples of mensuration : The area of an equilateral triangle, is equal to one fourth the square of the side, multiplied by the square root. *f three. X-oTST-^ / or THE 98 OF FLUID PRESSURE UPON THE INTERIOR OF A TETRAHEDRON. Consequently, the area of the base, and each of the containing sides of the vessel, is expressed by therefore, the area of the three containing equilateral triangular planes, becomes hence, for the pressure upon the base, we have P = 4JV3"X iV6 X s- iP 5V % (55). and the pressure upon the three containing planes, is j> = JJV3 X JV6 X s=il*s^2; (56). consequently, by analogy, we shall have P :p ::Psi/iTi JJt/2; and this, by suppressing the common factors, becomes P : p : : 1 : 2. If the two equations marked (55) and (56) be added together, the sum will express the aggregate pressure upon the vessel ; therefore we have P + P=J>'=(i + 1) l*sV2 = $l*s^. (57). According to the principles of mensuration, the solid content of a tetrahedronal vessel, is equal to the area of its base, multiplied by one third of its perpendicular altitude ; therefore, we have IJViFx $1^/6 x i=ZTW2; now, the weight of the contained fluid, is manifestly equal to its magnitude multiplied by the specific gravity ; consequently, we obtain w = -frPsV2l (58). hence, by analogy, we get P : w : : %l s s j$ : ^s ^2, and this, by suppressing the common factors, gives P : w : : 3 : 1. COROL. It therefore appears, that the pressure upon the base, is to the pressure on the three sides, in the ratio of 1 to 2, and to the weight of the contained fluid in the ratio of 3 to 1 ; consequently, the weight of the fluid, the pressure on the base, and the pressure on the sides, are to one another as the numbers 1, 3 and 6. 99 2. WHEN THE VESSEL IS IN THE FORM OF A CYLINDER. PROBLEM XVIII. 109. If a cylindrical vessel be completely filled with an incompressible and non-elastic fluid, and so placed, that its bottom may be parallel to the horizon: It is required to compare the pressure against its bottom, with that against its upright surface, and also with the weight of the fluid which it contains. Let ABCD be a vertical section of a cylindrical vessel, filled with an incompressible and non-elastic fluid, whose surface AB is horizontal, and let it be required to compare the pressure exerted by the fluid on the bottom DC, with that upon the whole upright surface. Draw the diagonals AC and BD intersecting one another in the point p, and through the point p, draw the vertical line mn, meeting DC the bottom of the vessel in the point m, and A B the surface of the fluid in n; then is m the position of the centre of gravity of the bottom, and nm its perpendicular depth below the surface AB. Bisect AD in E, and through E draw EP parallel to A B or DC, and meeting the vertical line mn in p ; then is p the position of the centre of gravity of the upright surface, and n p its perpendicular depth below AB the surface of the fluid. Put D nz AB or DC, the diameter of the cylindrical vessel proposed, d ~ nm, the perpendicular depth of the centre of gravity of the bottom DC, below AB the surface of the fluid, 3 ~ nv, the perpendicular depth of the centre of gravity of the upright surface, Anr the area of the base or bottom of the cylinder, P the pressure upon it, nz the area of the curved or upright surface, p the pressure upon it, w zzz the weight, and s z= the specific gravity of the fluid. Then, by the principles of mensuration, the area of the base or bottom of the cylindric vessel, is A=r.7854D 2 , H 2 100 OF FLUID PRESSURE UPON TTTE INTERIOR OF A CYLINDER. and that of the upright surface, is a 3.1416DG?; consequently, the pressure on the bottom becomes P = . r 7S54v*ds, (59). and for the pressure upon the upright surface, we have p=:3.Ul6vdSs; (60). therefore, by analogy, we obtain P :p : : .7854 D*ds : 3.1416D^s; from which, by omitting the common factors, we get P :p :: D : 43; now, by the construction of the figure, we have 3 z= \d ; therefore 4 zz 2d, and by substitution, we obtain P : p : : D : 2d : : |D : d. Hence it appears, that the pressure upon the bottom of a cylindrical vessel, is to the pressure upon its upright surface, as the radius of the base is to the perpendicular altitude. Since the entire pressure sustained by a cylindrical vessel, is equal to the sum of the pressures on the bottom and the upright sides, it follows, that P -f p = p'= .7854 (D + 43) vds, or substituting \d for 3, we shall obtain p' = .7854 (D + 2d) vds. (61). It is demonstrated by the writers on mensuration, that the solid content of a cylinder, is equal to the area of its base, drawn into its perpendicular altitude ; therefore, we have where C denotes the solid content of the cylinder. Now, it is manifest, that the weight of an incompressible and non- elastic fluid, is equal to its magnitude drawn into its specific gravity; hence we have but this is precisely the expression which we have given in equation (59), for the pressure perpendicular to the bottom of the vessel ; con- sequently, the weight of the fluid, and the pressure on the bottom of the vessel, are equal to one another ; hence, the following inference. 110. When the sides of a vessel of any form whatever, are perpen- dicular, and its base parallel to the horizon : OF FLUID PRESSURE UPON THE ANNULI OF A CYLINDER. 101 The pressure perpendicular to the base of the vessel, is equal to the whole weight of the fluid which it contains. This is manifest, for the whole pressure of the fluid is sustained by the base and the sides together, and the sides being in the direction of gravity, sustain no part of the pressure which is exerted perpendi- cularly downwards ; consequently, the whole weight of the fluid is sustained by the base. 3. WHEN THE PRESSURE UPON THE ANNULI OF A CYLINDER IS TO BE DETERMINED. PROBLEM XIX. 111. If a cylindrical vessel whose bottom is parallel, and sides perpendicular to the horizon, be filled with an incom- pressible and non-elastic fluid: It is required to divide the concave surface, into any number n of horizontal annuli, in such a manner, that the pressure on each annulus shall be equal to the pressure on the bottom of the vessel. Let ABCD be a vertical section, passing along the axis of the cylinder, or vessel containing the fluid, whose surface is AB ; draw the diagonals AC and BD intersecting one another in the point p ; then is p the centre of gravity of the cylindrical surface. Through p the point of intersection, draw the ver- tical line mn parallel to AD or BC, and let a, b and c be the points, which with the extremities A and D of the side AD, terminate the several annuli : then, through the points a, b and c, and parallel to AB or DC, draw the straight lines af, be and cd, cutting BC the opposite side of the section in the points f t e and d. Draw the zigzag diagonals Af, fb, bd and dv, intersecting the vertical line mn in the points k, i, h and g ; then are the points thus determined, respectively the centres of gravity of the several annuli, into which the concave surface of the vessel is supposed to be divided ; and n k, ni, nh and ng, are the respective depths below the surface of the fluid, WP being the depth of the centre of gravity of the whole upright surface of the cylindrical vessel, and nm the vertical depth of the bottom. ' - i ^ o *y ' ' ' : ' aa &pO e *' 102 OF FLUID PRESSURE UPON THE ANNULT OF A CYLINDER. Put D = AB or DC, the diameter of the proposed cylindrical vessel, A =. the area of its bottom, d =nm, the whole perpendicular depth, or altitude of the cylinder, x ~Aa, the breadth of the first annulus, x' z=ab, the breadth of the second, x" c, the breadth of the third, x'" CD, the breadth of the fourth, and so on, to any number of annuli n, P the pressure on the concave surface of the cylinder, or the sum of the pressures on the several annuli into which it is divided, p the pressure on the bottom of the vessel, and each of the several annuli, TT =z 3.1416 the circumference of a circle whose diameter is unity, and s zr the specific gravity of the fluid. Then we have, nk \x ; ni in x -\- \x ; n h x -f- x' -|- \x", and ng x -\- x 1 -|- x" + k x> " an( * by the principles of mensuration, the area of the bottom of the vessel, is AZrjTTD*, and by Proposition (1), the pressure upon it, is p l7n>*ds= .7854 i>M$, (62). Again, by the principles of mensuration, the concave surfaces of the respective annuli are as follows, viz. For the first annulus, we have TT D x 3.1416 D x, the surface, second TTDO;' ~ 3.1416 D x', third TTDX" 3.1416 DX", fourth TT DB"' = 3.1416 Da? 1 ", &c. &c. &c. And by Proposition (1), the pressures perpendicular to these sur- faces, are respectively as below, viz. For the first annulus, the pressure isp=zl.5708Da: a s, second j0=3.1416D*' s(x+ J#'), third p 3.1416Dx"s(aj+x / +Ja: 1 '), fourth - - P =i3.U\6x'"s(x+x'+x"+%x f "), &c. &c. &c. Now each of these pressures, according to the conditions of the problem, is equal to the pressure upon the bottom, exhibited in the equation (62) ; consequently, by comparison, we have OF FLUID RRESSURE UPON THE ANNULI OF A CYLINDER. 103 1. 5708 vx*s :=. and casting out the common factors, we get 2a? ? = vd; therefore, by division and evolution, we have x=%^2nd'. (63). By proceeding in a similar manner for the breadth of the second annulus, we shall obtain and this, by expunging the common terms, becomes 4*'(* + J*') = Drf; therefore, by substituting for x, its value as expressed in equation (63), we shall get complete the square, and we have a' 2 -\-^2iTdx + %vd=: vd; and finally, extracting the square root and transposing, we obtain *' = 4(2^2) J~d. (64). Again, by performing a similar process for the breadth of the third annulus, we shall have 3.1416D3"s (x -f- x' + i") = .7854 Dd, from which, by casting out the common quantities, we get 4x"(x + x'+%x")=:i>d; therefore, by substituting for x and x', their values as expressed in the equations marked (63) and (64), and we shall obtain 4x" {J Jtod+ J(2 V2) V~^~d+ ix"} = Dd, and this, by a little reduction and proper arrangement, gives complete the square, and we obtain consequently, by extracting the square root and transposing, we get a /; -=JV6 2)^25- ( 65 > Pursuing a similar train of reasoning for the breadth of the fourth annulus, we shall obtain 3.1416D*"'s (x + x' + x" -f iO = .7854Drf, and by suppressing the common factors, we have 4*"' ( x + x' + x" 4- 1*'") = d ; 104 OF FLUID PRESSURE UPON THE ANNULI OF A CYLINDER. sustitute in this equation, the values of x, x and x" as represented in the equations marked (63), (64), and (65), and we get (2 -t/2) V D ^-KV6 2) V and this, by a little further reduction, becomes therefore, by completing the square, we obtain a;"'* 4- i/fiDd.x" 1 -f i Drf 2od; and finally, by extracting the square root and transposing, we have x 1 " = J (2 V~2 V 6) V^ (66). 112. And thus we may proceed to any extent at pleasure ; that is, to any number of annuli within the limit of possibility ; for it is mani- fest, from the nature of the problem, that impossible cases may be proposed, but the limit can easily be ascertained in the following manner. It is obvious, that the sum of the breadths of the several annuli, is equal to the whole depth of the vessel ; and that the sum of the pressures is equal to the pressure on the concave surface ; but in the problem immediately preceding, we have demonstrated that the pres- sure on the bottom of a cylindrical vessel, is to that upon its upright surface, as the radius of the base is to the perpendicular altitude. Now, according to the conditions of the question, the pressure on each annulus is equal to that upon the base ; consequently, in order that the problem may be possible, the depth of the vessel must be equal to the radius of the base, drawn into the number of annuli. If instead of D the diameter of the cylindrical vessel, we substitute 2R its equivalent in terms of the radius, the preceding equations (63), (64), (65), and (66), will become transformed into x = (/f v'O) y^, (67). *' (V2 /f) Va3, (68). ~ " (69). (70). From these equations the law of induction becomes manifest, and the general expression for the breadth of the n^ annulus, is (71). OF FLUID PRESSURE UPON THE ANNULI OF A CYLINDER. 105 113. And from the above general form of the equation, the follow- ing practical rule may be derived, for calculating the breadth of any proposed annulus, independently of the breadths of those which pre- cede it. RULE. From the square root of the number which expresses the place of the required annulus, subtract the square root of that number minus unity ; then, multiply the remainder by the geometric mean between the altitude of the vessel and the radius of its base, and the product will give the breadth of the required annulus. 114. EXAMPLE. A cylindrical vessel has the radius of its base, and its perpendicular depth, respectively equal to 4 and 24 feet; now, supposing the concave surface to be divided into 6 horizontal annuli, such, that the pressure upon each shall be equal to the pressure upon the base ; required the breadth of the fourth annulus ? By performing the operation as directed in the preceding rule, we shall obtain x'" (^ V3)X v/4 X 24 z=2.625 feet nearly. The annulus which we have just determined, corresponds to the fourth of the preceding class of equations, or that marked (69), and the distance of its centre of gravity below the surface of the fluid, or its position with respect to the bottom or top of the vessel, can easily be ascertained. The area of the cylinder's base, is A 3.1416B 9 ; the pressure which it sustains, is p = 3. 1416 R 2 d= 1206.3744, and this is equal to the pressure on the annulus. Now, according to the writers on mensuration, the area of the annulus, or the curved surface of a cylinder, whose radius is 4 feet and its perpendicular altitude 2.625 feet, is expressed as follows, viz. 6.2832 X 4 X 2.625 = 65.9736. If therefore, we divide the pressure on the base of the vessel, by the area of the annulus, the depth of its centre of gravity will become known ; thus, we have 106 4. WHEN THE VESSEL ASSUMES THE FORM OF A TRUNCATED CONE, THE BASE OF WHICH IS ALSO THE BOTTOM OF THE VESSEL, AND ITS AXIS PERPENDICULAR TO THE HORIZON. PROBLEM XX. 115. If a vessel in the form of the frustum of a cone, be filled with an incompressible and non-elastic fluid, and have its axis perpendicular to the horizon : It is required to compare the pressure on the bottom of the vessel with that upon its curved surface, and also with the weight of the fluid which it contains, both when the sides of the vessel converge, and when they diverge from the extremities of the bottom. Let ABCD represent a vertical section of a vessel in the form of the frustum of a cone, and filled with an incom- pressible and non-elastic fluid whose horizontal surface is AB ; produce AB both ways, to any convenient distance, and through D and c the extremities of the bottom diameter, draw Da and c b respectively perpendicular to D c, and meeting AB produced in the points a and b; then is abcv the vertical section, passing along the axis of the cylinder which circumscribes the conic frustum. Bisect AB and DC respectively in the points m and n, and draw the straight line mn\ then, because the figure ABCD is symmetrical with respect to the axis mn, it follows, that mn bisects the figure or trapezoid ABCD, and consequently passes through its centre of gravity. Draw the diagonal AC, dividing the figure ABCD into the two triangles ABC and ADC ; then it is manifest, that the common centre of gravity of the two triangles, and that of the trapezoid constituted by their sum, must occur in one and the same point ; therefore, bisect the diagonal A c in the point t, and draw A n and D t intersecting each other in the point r, and c m, B t intersecting in s ; then are r and s the centres of gravity of the triangles ADC and ABC; draw rs inter- secting mn in G, and G will be the centre of gravity of the trapezoid ABCD. Now, it is demonstrated by the writers on mechanics, that the centre of gravity of the surface of a conic frustum : OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. 107 Is situated in the axis, and at the same distance from its extremities, as is the centre of gravity of the trapezoid, which is a vertical section passing along the axis of the solid. Therefore, since by the construction, the point G has been shown to be the centre of gravity of the trapezoid, it is also the centre of gravity of the surface of the conic frustum, and TWO is its perpendicular depth below the surface of the fluid. Put AZZ: the area of the base or bottom of the vessel, whose diameter is DC, v=mn, the perpendicular depth of its centre of gravity, or the length of the axis of the vessel, a the curve surface of the conic frustum, d^nmo, the perpendicular depth of the centre of gravity, /3 zz: DC, the diameter of the base or bottom of the vessel, S zz: AB, the diameter of the top, Pzz: the pressure on the bottom, p HZ the pressure on the curve surface, wzz: the weight, and s the specific gravity of the fluid. Then, according to the principles of mensuration, the area of the lower base of the conic frustum, or the bottom of the vessel on which the fluid presses, becomes A = . 7854/3% and consequently, the pressure which it sustains, is Pzz:. 7854 /3 2 DS. (72). In the next place, the area of the curved surface of the conic frustum, or the sides of the vessel containing the fluid, is a= 1.5708 08 + 3)X V** + 1 (ft W I and therefore, the pressure which it sustains, is p = 1.5708 ()8 4- 5) ds VD* 4- 403 S) 2 . (73). Now, according to the writers on mechanics, the depth of the centre of gravity of the trapezoid ABCD, below the horizontal line AB, is obtained in the following manner : 3 (/3 4 3) : D : : 2/3 4 3 : d, and by equating the products of the extremes and means, we get therefore, dividing by 3 (/3 4- 3), we obtain _ = 108 OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. Let this value of d be substituted instead of it, in the equation marked (74), and we shall have for the pressure on the curved surface of the vessel p .5236 D s (2/3 4- 3) V^-4- J03 3)* ; (75). consequently, by comparing the equations (72) and (74), we get Pip:: .7854/3*DS : .5236 D s(2/3 + 3) J D 2 and this, by suppressing the common factors, becomes P : p : : 3/3* : 2 (2/3 + 3) V* + i (P - *)- (76). If J, the upper diameter of the frustum vanishes, the figure becomes a complete cone, and consequently, the pressure upon the base, is to that upon the curve surface, as three times the diameter of the cone, is to four times its slant height ; that is P:p:: 3/3: 4^+1^. (77). According to the principles of solid geometry, the capacity of the conic frustum, or the quantity of fluid which the vessel contains, is where c denotes the solid content of the vessel. But the weight of any quantity or mass of fluid, varies directly as the magnitude and specific gravity conjointly; consequently, the weight of fluid in the vessel, is expressed by w .2618 D s (/3 s + /3 J -f a 2 ). (78). Hence, if the equations marked (72) and (78), be compared with each other, we shall obtain p^i-.s/^os'-h/sa+a 2 ), (79). and when 3 vanishes, the vessel becomes a complete cone, and conse- quently, we get P : w : : 3 : 1, (80). It therefore appears, that the pressure against the bottom of a conical vessel, when filled with an incompressible and non-elastic fluid, (the bottom being downwards) : Is equivalent to three times the weight of the fluid which it contains. The solidity of the cylinder circumscribing the conic frustum, of which abci> is a vertical section, is c' =r.7854 f? D, where c' denotes the capacity of the cylinder circumscribing the vessel; OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. 109 and because the weight of any quantity or mass of fluid, is propor- tional to the magnitude and specific gravity jointly ; it follows, that if w' denote the weight of the circumscribing column of fluid, we obtain w/z=.7854/3 2 Ds. (81). COROL. Now this expression is precisely the same, as that which we obtained for the pressure on the bottom, indicated by the equation marked (72) ; hence it appears, that when the sides of the vessel converge from the extremities of the diameter of its base towards each other : The pressure on the base or bottom of the vessel, is equal to the weight of a column of the fluid, of the same magnitude as the cylinder circumscribing the conic frustum, or the vessel by which the fluid is contained. But the circumscribing cylinder is manifestly greater than the conic frustum ; consequently, the pressure upon the base or bottom of the vessel, is greater than the weight of the fluid which it contains ; and it is obvious, that the additional pressure arises from the re-action of the converging sides. 5. WHEN THE VESSEL REPRESENTS AN INVERTED TRUNCATED CONE, WITH ITS AXIS PERPENDICULAR TO THE HORIZON. 116. If the sides of the vessel diverge from the extremities of the base, as represented in the subjoined diagram ; then, it may be shown, that the weight of the fluid which the vessel contains, exceeds the pressure upon its" base. Let ABCD be a vertical section, passing along the axis of a vessel in the form of a conic frustum, and which is filled with an incompressible fluid whose horizontal surface is AB ; the greater base of the frustum being uppermost, or which is the same thing, the sides diverging from the extremities of the lower diameter. Bisect the diameters A B and c D respectively in the points m and n ; draw mn, and through the points D and c, the extremities of DC, draw the straight lines Da and cb respectively parallel to mn, and meeting AB in the points a and b; then is abcn a vertical section passing along the axis of the inscribed cylinder. Draw the diagonal AC, dividing the trapezoid ABCD, into the two triangles ABC and ADC; bisect the diagonal AC in the point t, and 110 OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. draw B t and D t, which will be intersected by the straight lines, c m and \n iu the points r and s; then are r and s respectively the centres of gravity of the triangles ABC and ADC. Join the points r and s, by the straight line rs, intersecting mn in the point G ; then it is obvious, that the common centre of gravity of the triangles ABC and ADC, (which coincides with that of the trapezoid ABCD), must occur in the line rs, which joins their respective centres. Now, because the trapezoid A B c D is symmetrically situated with respect to the axis mn, it follows, that its centre of gravity must occur in that line ; but we have shown above, that it also occurs in the line rs, it consequently must be situated in the point G, where these lines intersect one another ; hence, the centre of gravity of the surface of the conic frustum occurs at the point G, and m G is its perpendicular depth below AB the upper surface of the fluid. Put A =. the area of the lower base or bottom of the vessel, whose diameter is DC, P the pressure perpendicular to its surface, or the weight of a quantity of fluid equal to the inscribed cylinder, Dzzmw, the axis of the frustum, or the perpendicular depth of the centre of gravity of the bottom, a zz: the area of the curve surface of the vessel or conic frustum, p zz: the pressure perpendicular to the curve surface, d zz: m&, the perpendicular depth of its centre of gravity, 3 zz: DC, the diameter of the lower base or bottom of the vessel, ft zz: AB, the diameter of the top or upper base, w the weight, and s the specific gravity of the contained fluid. Then, by the principles of mensuration, the area of the lower base of the conic frustum, or the bottom of the vessel on which the fluid presses, is Azzr.78542 2 , and consequently, the pressure upon it, is Pzz:.7854j 2 Ds. (82). This equation, having 5 2 instead of /3 2 , is the same as that which we obtained for the pressure on the bottom in the preceding case, when the greater base of the vessel was downwards ; it therefore follows, since our notation is adapted to the same parts of the vessel, that not- withstanding the inversion, the pressure on the curved surface of the conic frustum, will still be expressed as in the equation marked (73) ; consequently, we have P :p: :.7854$ 2 ns:: 1.5708 (/3-f 2)ds VD' OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. Ill or by expunging the common quantities, we get P :p : : 3*D : 2(/3 + 3) dji>* + J(/3 3)'. But the writers on mechanics have demonstrated, that the depth of the centre of gravity of the trapezoid A BCD, below the horizontal line AB, is expressed as follows, viz. ' (83). let therefore, this value of d be substituted instead of it in the above analogy, and we shall obtain P : p : : 33* : 2 (ft + 23) V * + I (ft V- If , the diameter of the bottom or lower base should vanish, the vessel becomes a complete cone with its vertex downwards, in which case, the value of d as expressed in the equation marked (83), is Let this value of d be substituted instead of it, in the equation marked (73), and suppose 3 to vanish ; then, the pressure on the con- cave surface of a conical vessel with its vertex downwards, becomes p = .5236 ft D s V D 9 + J r /P. (84). The solid content of the inscribed cylinder, of which the vertical section passing along the axis is a6co, becomes c'=r.78543 2 D, and as we have already stated, its weight is proportioned to its mag- nitude drawn into the specific gravity ; hence we have / = . 7854 3*Ds; but this is the same expression which indicates the pressure on the bottom, as exhibited in the equation marked (82) ; hence it follows, that the pressure on the bottom or lower base of the conic frustum, when the sides diverge from the extremities of its diameter, Is equal to the weight of a column of the fluid, of the same magnitude as the cylinder inscribed in the conic frustum. But the solid content of the inscribed cylinder, and consequently its weight, is manifestly less than the content of the vessel ; hence we infer, that when the sides of the vessel diverge from the extremities of the diameter of its bottom, the pressure on the bottom is less than the weight of the fluid which it contains, the remaining weight being supported by the resistance of the diverging sides. 112 OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. 117. In order to compare the pressure on the bottom of the vessel, with the weight of the fluid which it contains, we must again have recourse to the principles of solid geometry ; from which we learn, that the solid content of a conic frustum, whose diameters are denoted by (3 and 5, and its perpendicular altitude by D, is and consequently, its weight becomes to = .2618 DS (0* + ft 3 + S 2 ) ; (85). therefore, by comparing this equation with that marked (82), we get . P : w : : 33* : (/3 2 + /3 3 + a 2 ). (86). Again, if we compare the equations marked (73 and (85) with one another, we shall have p : w :: 1.5708 (0 + 3) ds v 'D* -*%(& 3) : .2618 Ds(/3 2 + /33 + J') ; if therefore, we expunge the common quantities, from the third and fourth terms of the above analogy, and in the third term, substitute the value of d as it is expressed in the equation (83), then we shall obtain p : w : : 2(0 + 2 3) : (/3 2 + ft 3 + *). When 5 vanishes, or when the vessel becomes a complete cone with its vertex downwards, the preceding analogy gives p : w : : 2 ^ D* + |/3 2 : /3. In complying with the conditions of the 20th problem, the fore- going investigation has been conducted on the supposition, that the vessel in question is in the form of the frustum of a cone ; but the attentive reader will readily perceive, that the same mode of procedure will apply to the frustum of any other regular pyramid, and the result- ing formulae will partake of similar forms and combinations, differing only in so far as depends upon the constant numbers which express their respective areas and solidities ; it is therefore unnecessary to pursue the inquiry further, taking it for granted, that by a careful perusal of what has been done above, no difficulty will be met with in applying the same principles to any other case of form or condition that is likely to occur. COUOL. 1. By the preceding investigation, then, and the formulae arising from it, we learn, that by causing the sides of a vessel, which is filled with an incompressible and non-elastic fluid, to converge or diverge from the extremities of the base, supposed to be horizontal : The pressure on the base, may be greater or less than the weight of the fluid which the vessel contains, in any propor- tion whatever. OF FLUID PRESSURE USED AS A MECHANICAL POWER. 113 2. Upon these principles therefore, and others of a similar nature, which we have mentioned at the outset, is explained the paradoxical property of non-elastic fluids : That the pressure on the bottom of a vessel Jilted with fluid, does not depend upon its quantity, but solely upon the perpendicular altitude of its highest particles above the bottom of the vessel or the surface by which the pressure is sustained. 3. And from the property here propounded, is deduced the remark- able and important principle : That any quantity of fluid however small, may be made to balance, or hold in equilibrio, any other quantity, however great. Let the upright or vertical section of a vessel containing an incom- pressible and non-elastic fluid, be such as is repre- sented by A BCD in the annexed diagram, and let cdbe the corresponding section of a small pipe or tube inserted into its upper surface at the point d. Then, supposing the vessel and the tube to be filled with fluid as far as the point c ; it is manifest from the first case of the preceding problem, that the pressure upon DC the bottom of the vessel, is precisely the same as if it were entirely filled to the height acb\ for the pressure upon the bottom DC, is equal to the weight of a fluid column, the diameter of whose base is DC and its perpendicular altitude an or be; but this is evidently greater than the weight of the fluid in the vessel, and by increasing the height of fluid in the tube, the pressure on the bottom will be increased in the same proportion, while the actual increase of weight is very small, being only in proportion to the increase of pressure, as the area of a section of the tube is to the area of the bo torn. PROBLEM XXI. 118. Having given the diameter and perpendicular height of a cylindrical vessel, together with the diameter of a tube fixed vertically into the top of it : VOL. I. I 114 OF FLUID PRESSURE USED AS A MECHANICAL POWER. It is required to find the length of the tube, such, that when it and the vessel are filled with an incompressible fluid, the pressure on the bottom of the vessel may be equal to any number of times the fluid's weight. In resolving this problem, it will be sufficient to refer to the pre- ceding diagram, because a separate construction would exhibit no variety ; for this purpose then, Put D zz DC, the diameter of the base of the cylindrical vessel, A zz the area of its base, h zz A D, the altitude, or perpendicular depth of the vessel, C the capacity, or solid content, P the pressure on the bottom, d ed, the diameter of the tube inserted in the top of the vessel, a zz the area of its horizontal section, c zz the capacity, or solid content of the tube, w ~~ the weight of the fluid in the vessel, and w' the weight of that in the tube ; s zz the specific gravity of the fluid, n zz the number of times the pressure exceeds the weight, and x zz the required length of the tube. Then, according to the principles of mensuration, the area of the bottom of the vessel becomes AZZ.7854D 2 , and that of a horizontal section of a tube, is azz.7854d 2 ; and again, by the geometry of solids, the capacity of the vessel is Czz.7854D 2 A, and for the capacity of the tube, we have the respective weights being wzz.7854D 2 As, and w 1 zz .7854d 2 a-s; consequently, the whole weight of the fluid in the vessel and tube, is w + w' = .7854s (D 2 A, -f d*x). Now, we have shown above, that the pressure on the bottom of the vessel is equal to the weight of a fluid cylinder, whose diameter is DC and altitude D! P -^r- (89). And from the equation in its present form, we deduce the following practical rule. 120 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION RULE. Multiply the square of the diameter of the cylinder by the magnitude of the power applied, and divide the product by the square of the diameter of the forcing pump, and the quotient will express the intensity of the pressure on the piston of the cylinder. 125. EXAMPLE 2. If the diameter of the cylinder is 5 inches, and that of the forcing pump one inch ; what is the magnitude of the power applied, supposing the entire pressure on the piston of the cylinder to be 18750 Ibs. ? Here we have given D = 5 inches; d= 1 inch, and P= 18750 Ibs. ; therefore, by substitution, equation (88) becomes 5 a Xp 18750 X 1*; or p 750 Ibs. If both sides of the fundamental equation (88) be divided by D*, the general expression for the value of p, is _Pd 2 p ~ D 2 ' (90). And the practical rule which this equation supplies, may be expressed in words at length in the following manner. RULE. Multiply the given pressure on the piston of the cylinder, by the square of the diameter of the forcing pump, and divide the product by the square of the diameter of the cylinder for the power required. 126. EXAMPLE 3. The diameter of the forcing pump is one inch, and the power with which the plunger descends is equivalent to 750 Ibs. ; what must be the diameter of the cylinder, to admit a pressure of 18750 Ibs. on the piston ? Here we have given c?n= 1 inch; ^ = 750 Ibs., and P n= 18750 Ibs. ; consequently, by substitution, the equation marked (88) becomes 750 D 2 =l 8750 x I 2 ; hence, by division, we obtain consequently, by evolution, we have D ^ 25 5 inches. If both sides of the equation (88) be divided by p, and the square root of the quotient extracted, the general expression for the diameter of the piston, is OF THE HYDROSTATIC PRESS. 121 And the practical rule for the determination of D, may be expressed in words as follows. RULE. Multiply the pressure on the piston of the cylinder, by the square of the diameter of the forcing pump, and divide the product by the force with which the plunger descends ; then, the square root of the quotient will be the diameter of the cylinder sought. 127. EXAMPLE 4. The diameter of the cylinder is 5 inches, and the force with which the plunger descends, is equivalent to 750 Ibs. ; what must be the diameter of the forcing pump, in order to transmit a pressure of 18750 Ibs. to the piston of the cylinder ? Here we have given D r= 5 inches ; p 750 Ibs., and P zz 18750 Ibs. ; consequently, by substitution, equation (88) becomes 1 8750 d* = 750 X 5 2 , and by division, we shall have 750 X 25 therefore, by extracting the square root, we get d /f zzl inch. If both sides of the original equation marked (88), be divided by P, and the square root extracted, the entire pressure on the piston, the general expression for the value of d becomes = /l/ "P"' (92). And the practical rule which this equation supplies, may be expressed in words in the following manner. RULE. Multiply the force with which the plunger descends, by the square of the diameter of the cylinder, and divide the product by the entire pressure on the piston; then, extract the square root of the quotient for the diameter of the forcing pump. 128. The foregoing is the theory of the Hydrostatic Press, as restricted to the consideration of the diameters of the cylinder and forcing pump, and the respective pressures on the piston and plunger; but since the instrument is generally furnished with an indicator or 122 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION safety valve for measuring the intensity of pressure, the theory would be incomplete without considering it in connection with the diameters of the pump and cylinder. For which purpose Put 3 zz the diameter of the safety valve, expressed in inches or parts, and w the weight thereon, or the force that prevents its rising. Then, according to the principle announced in Proposition II., we obtain the following analogies, viz. D 2 : ^ : : P : w, d* : ^ : : p : w ; and from these analogies, by making the products of the extreme terms equal to the products of the means, we get D'wzz^P, (93). zudd*w = tfp. (94). Now, in order to pursue the expansion of these equations, we shall suppose the value of & to be one fourth of an inch, while the numerical values of the other letters remain the same as supposed for the several examples under equation (88) ; then, to determine the corresponding value of w, or the power which prevents the safety valve from rising, when all the parts of the instrument, or the several powers and pres- sures are in a state of equilibrium, we have the following examples to resolve according to the proposed conditions. 129. EXAMPLE 5. The diameter of the cylinder is 5 inches, that of the indicator or safety valve J of an inch, and the entire pressure upon the piston of the cylinder 18750 Ibs. ; what is the corresponding force preventing the ascent of the safety valve, on the supposition of a perfect equilibrium ? Here we have given D zz 5 inches ; 3 zz J of an inch, and P zz 1 8750 Ibs. ; consequently, by substitution, the equation (93) becomes 5 2 wzz.25 2 X 18750; from which, by division, we get .0625 X 18750 wzz zz46.875 Ibs. AQ But the general expression for the value of iv, as derived from the equation (93), becomes _J 2 P - V' (95). From which we derive the following rule. OF THE HYDROSTATIC PRESS. 123 RULE. Multiply the entire pressure on the piston of the cylinder, by the square of the diameter of the indicator or safety valve, and divide the product by the square of the diameter of the cylinder for the weight required. 130. EXAMPLE 6. The diameter of the safety valve is J of an inch, that of the cylinder 5 inches, and the weight on the safety valve 46.875 Ibs. ; what is the corresponding pressure on the piston of the cylinder ? Here we have given 3 = of an inch ; D zz: 5 inches, and w zz:46.875 Ibs ; therefore, by substitution, equation (93) becomes .25 2 P = 5 2 x 46.875, and by division, we obtain The general expression for the value of P, as derived from the equation marked (93), becomes * - (96). And the practical rule supplied by this equation, may be expressed in words as follows. RULE. Multiply the weight on the safety valve, by the square of the diameter of the cylinder, and divide the product by the square of the diameter of the safety valve, and the quotient will give the entire pressure on the piston of the cylinder. 131. EXAMPLE 7. The diameter of the cylinder is 5 inches, the entire pressure of the piston is 18750 Ibs., and the weight on the safety valve is 46.875 Ibs. ; what is its diameter ? Here we have given ozz:5 inches ; P zz: 18750 Ibs., and wzr46.875 Ibs. ; therefore, by substitution, equation (93) becomes 18750a 2 z=5 2 X 46.875, and from this, by division, we get and by extracting the square root, we obtain 3= -V/ .0625 zz: .25, or of an inch. The general expression for the value of 3, as derived from the equation (93), is as follows, viz. 124 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION *=y -p-- (97). And the practical rule which this equation affords, may be expressed in words in the following manner. RULE. Multiply the load on the safety valve by the square of the diameter of the cylinder ; divide the product by the entire pressure on the piston, and the square root of the quotient will give the diameter of the safety valve required. 132. EXAMPLE 8. The diameter of the safety valve is ^ of an inch, the load upon it 46.875 Ibs., and the entire pressure on the piston of the cylinder is 18750 Ibs. ; what is its diameter? Here we have given = J of an inch, w= 46.875 Ibs., and Pzr 18750 Ibs. ; consequently, by substitution, we have 46.875 D*=. 25* X 18750, from which, by division, we shall obtain .25 a X 18750 46.875 and finally, by extracting the square root, we get D = -v/ 25 5 inches. If both sides of the equation marked (93), be divided by w the weight on the safety valve, we get and by extracting the square root, the general expression for the value of D the diameter of the cylinder, becomes -y ^' (98). And from this equation we derive the following rule. RULE. Multiply the entire pressure on the piston of the cylinder by the square of the diameter of the safety valve, divide the product by the weight upon the safety valve, and extract the square root of the quotient for the diameter of the cylinder sought. 133. EXAMPLE 9. The diameter of the forcing pump is one inch, that of the safety valve is one fourth of an inch, and the power or force with which the plunger descends, is equivalent to 750 Ibs. ; what is the corresponding weight on the safety valve ? OF THE HYDROSTATIC PRESS. 125 Here we have given d 1 inch; Ji=^ of an inch, and p 750 Ibs. ; consequently, by substitution, the equation (94) becomes P X w = .25* X 750 ; that is, w = 46.875 Ibs,, the very same value as we derived from the fifth example. If both sides of the equation marked (94) be divided by d*, the general expression for the value of w becomes _3> ~d 2 * (99). And the practical rule supplied by this equation, may be expressed in words at length in the following manner. RULE. Multiply the force with which the plunger descends by the square of the diameter of the safety valve, and divide the product by the square of the diameter of the plunger ; then the quotient will express the load upon the safety valve. 134. EXAMPLE 10. The diameter of the safety valve is J of an inch, that of the forcing pump is one inch, and the load upon the safety valve is 46.875 Ibs. ; what is the power applied, or the force with which the plunger in the forcing pump descends ? Here we have given 3=J of an inch, d=:l inch, and w=r46.875 Ibs. ; consequently, by substitution, equation (94) becomes .25 2 j9=46.875 X I 2 , and from this, by division, we obtain ' ' The general expression for the value of p, as obtained from the equation marked (94), becomes _d*w : ~F* (100). from which we derive the following rule. RULE. Multiply the load on the safety valve by the square of the diameter of the forcing pump ; then, divide the product by the square of the diameter of the safety valve, and the quotient will give the force with which the piston descends. 135. EXAMPLE 11. The diameter of the plunger or the piston of the forcing pump is one inch, the force with which it descends is equivalent to 750 Ibs., and the load on the safety valve is 46.875 Ibs.; what is its diameter ? 126 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION Here we have given dnz 1 inch, p =. 750 Ibs., and w zz 46.875 Ibs. ; consequently, by substitution, we have 750 3"= 1 s X 46.875, and from this, by division, we obtain and finally, by evolution, we have a = V.0625 = .25 of an inch. , Let both sides of the equation marked (94) be divided by p, the power or force with which the piston of the forcing pump descends, and we shall have and by extracting the square root, we get (101). Hence, the following practical rule. RULE. Multiply the weight or load upon the safety valve, by the square of the diameter of the forcing pump, and divide the product by the force with which the plunger or piston of the forcing pump descends ; then, the square root of the quotient will be the diameter of the safety valve. 136. EXAMPLE 12. The diameter of the safety valve is one fourth of an inch, the weight upon it is 46.875 Ibs., and the power applied, or the force with which the plunger descends, is 750 Ibs ; what is the diameter of the forcing pump ? Here we have given J of an inch, w> 46.875 Ibs., and /?=z750 Ibs. ; consequently, by substitution, the equation marked (94) becomes 46.875d 2 =.25*x750; therefore, by division, we obtain .25^X750 : 46.875 Z and finally, by extracting the square root, we get , and by workmen, is usually denominated the ' Follower or ' Pressing Table.' B is the top of the frame into which the upright bars A A are fixed, and cc is the bottom thereof, both of which are made of cast, in preference to wrought iron, being both cheaper and more easily moulded into the intended form. The bottom of the frame cc, is furnished with four projections or lobes, with circular perforations, for the purpose of fastening it by iron bolts to the massive blocks of wood, whose transverse sections are indicated by the lighter shades at GG. The top B has two similar perforations, through which are passed the upper extremities of the vertical bars A A, and there made fast, by screwing down the cup-nuts represented at a and a. Fiy. 2 represents the plan of the top, or as it is more frequently termed, the head of the frame ; the lower side Fig. 2. or surface of which is made perfectly smooth, in order to correspond with, and apply to the upper surface of the pressing table E in^. 1 ; this correspondence of surfaces becomes ne- cessary on certain occasions, such as the copy- ing of prints, taking fac-similes of letters and the like ; in all such cases, it is manifest, that smooth and coincident surfaces are indispensable for the purpose of obtaining true impres- sions. The figure before us represents the upper side of the block, where it is evident, that the middle part B, (through whose rounded extremi- ties a and a, the circular perforations are made for receiving the upright bars or rods AA,^. 1), is considerably thicker than the parts on each side of it ; this augmentation of thickness, is necessary to resist the immense strain that comes upon it in that part ; for although the pressure may be equally distributed throughout the entire surface, yet it is obvious, that the mechanical resistance to fracture, must prin- cipally arise from that part, which is subjected to the re-action of the upright bars. 138 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION Fig. 3 represents the plan of the base or bottom of the frame ; it is generally made of uniform thickness, Fig. 3. and of sufficient strength to withstand the pressure, for be it understood, that all the parts of the machine are sub- jected to the same quantity of strain, although it is exerted in different ways.* The circular perforations cc correspond to a a in the top of the frame, and receive the upright bars in the same manner ; the perfora- tions dddd, receive the screw bolts which fix the frame to the beams of timber represented at GG, fig. \ ; the large perforation r receives the cylinder, the upper extremity of which is furnished with a flanch, for the purpose of fitting the circular swell around the perforation, and preventing it from moving backwards during the operation of the instrument. When the several parts which we have now described are fitted together, they will present us with that portion of the drawing in fig. 1 denominated " Elevation of the Press." A side view of the engine as thus com- pleted, is represented in fig. 4, where, as is usual in all such descriptions, the same letters of the alphabet refer to the same parts of the structure. F is the cylinder into which the fluid is injected ; D the piston, on whose summit is the pressing table E ; A one of the upright rods or bars of malleable iron ; B the head of the press, fixed to the upright bar A by means of the cup-nut a; c the bottom, in which the upright bar is similarly fixed ; and G a beam of timber supporting the frame with all its appendages. 147. But the Hydrostatic Press as here .described and constructed, must not be con- sidered as fit for immediate action ; for it is manifestly impossible to bore the interior of the cylinder so truly, and to turn the piston Fig. 4. * The upright bars, cylinders, and connecting tubes, resist by tension, the pistons by compression, and the pressing table, together with the top and bottom of the frame, resist transversely. OF THE HYDROSTATIC PRESS. 139 with so much precision, as to prevent the escape of water between their surfaces, without increasing the friction to such a degree, that it would require a very great force to counterbalance it. In order, therefore, to render the piston water-tight, and to prevent as much as possible the increase of friction, recourse must be had to other principles, which we now proceed to explain. The piston D is surrounded by a collar of pump leather oo, repre- sented in Jig. 5, which collar being doubled up, so Fig. 5. as in some measure to resemble a lesser cup placed within a greater, it is fitted into a cell made for its reception in the interior of the cylinder ; and when there, the two parts are prevented from coming toge- ^^m^ ther, by means of the copper ring pp, represented in Ft S- 6 - Jig. 6, being inserted between the folds, and retained in its place, by a lodgement made for that purpose on the interior of the cylinder. The leather collar is kept down by means of a brass or bell-metal ring mm, Jig. 7, which ring is received into a Fig. 7. recess formed round the interior of the cylinder, and the circular aperture is fitted to admit the piston D to pass through it, without materially increasing the effects of friction, which ought to be avoided as much as possible. The leather is thus confined in a cell, with the edge of the inner fold applied to the piston D, while the edge of the outer fold is in contact with the cylinder all around its interior circumference ; in this situation, the pressure of the water acting between the folds of the leather, forces the edges into close contact with both the cylinder and piston, and renders the whole water-tight ; for if the leather be properly constructed and rightly fitted into its place, it is almost impossible that any of the fluid can escape; for the greater the pressure, the closer will the leather be applied to both the piston and the cylinder. The metal ring mm is truly turned in a lathe, and the cavity in which it is placed is formed with the same geometrical accuracy ; but in order to fix it in its cell, it is cut into five pieces by a very fine saw, as represented by the lines in the diagram, which are drawn across the surface of the ring. The four segments which radiate to the centre are put in first, then the segment formed by the parallel kerfs, (the copper ring pp and the leather collar oo being previously introduced), and lastly, the piston which carries the pressing table. That part of the cylinder above the ring mm, where the inner 140 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION surface is not in contact with the piston, is filled with tow, or some other soft material of a similar nature ; the material thus inserted has a twofold use ; in the first place, when saturated with sweet oil, it diminishes the friction that necessarily arises, when the piston is forced through the ring mm; and in the second place, it prevents the admission of any extraneous substance, which might increase the friction or injure the surface of the piston, and otherwise lessen the effects of the machine. The packing here alluded to, is confined by a thin metallic annulus, neatly fitted and fixed on the top of the cylinder, the circular orifice being- of sufficient diameter, to admit of a free and easy motion to the piston . If a cylinder thus furnished with its several appendages be placed in the frame, and the whole firmly screwed together, and connected with the forcing pump, as represented in fig. 1, the press is completed and ready for immediate use ; but in order to render the construction still more explicit and intelligible, and to show the method of con- necting the press to the forcing pump, let jig. 8 represent a section of the cylinder with all its furniture, jr^. 3. and a small portion of the tube im- mediately adjoining, by which the connexion is effected. Then is FF the cylinder; D the piston; the unshaded parts oo the leather collar, in the folds of which is placed the copper ring pp, dis- tinctly seen but not marked in the figure; mm is the metal ring by which the leather collar is retained in its place ; nn the thin plate of copper or other metal fitted to the top of the cylinder, between which and the plate m m is seen the soft packing of tow, which we have described above, as performing the double capacity of oiling the piston and preventing its derange- ment. The combination at wx, represents the method of connecting the injecting tube to the cylinder : it may be readily understood by in- specting the figure; but in order to remove all causes of obscurity, it may be explained in the following manner. The end of the pipe or tube, which is generally made of copper^ has a projecting piece or socket flanch soldered or screwed upon it^ which fits into a perforation in the side or base of the cylinder, accord- OF THE HYDROSTATIC PRESS. 141 ing to the fancy of the projector, but in the figure before us the per- foration is in the side. The tube thus furnished, is forcibly pressed into its seat by a hollow screw w, called an union screw, which fits into another screw of equal thread made in the cavity of the cylinder ; the joint is made water- tight, by means of a collar of leather, interposed between the end of the tube and the bottom of the cavity. A similar mode of connection is employed in fastening the tube to the forcing pump, the description of which, although it constitutes an important portion of the apparatus, does not properly belong to this place ; the principles of its construction and mode of action, must therefore be supposed as known, until we come to treat of the con- struction and operation of pumps in general. Admitting therefore, that the action of the forcing pump is under- stood, it only now remains to explain the nature of its operation in connection with the Hydrostatic Press, the construction of which we have so copiously exemplified. 148. In order to understand the operation of the press, we must conceive the piston D Jig. 1, as being at its lowest possible position in the cylinder, and the body or substance to be pressed, placed upon the crown or pressing table E ; then it is manifest, that if water be forced along the tube b b b by means of the forcing pump, it will enter the chamber of the cylinder F immediately beneath the piston D, and cause it to rise a distance proportioned to the quantity of fluid that has been injected, and with a force, determinable by the ratio between the square of the diameter of the cylinder and that of the forcing pump. The piston thus ascending, carries its crown, and consequently, the load along with it, and by repeating the operation, more water is injected, and the piston continues to ascend, till the body comes into contact with the head of the frame B, when the pressure begins ; thus it is manifest, that by continuing the process, the pressure may be carried to any extent at pleasure ; but we have already stated, in developing the theory, that there are limits, beyond which, with a given bore and a given thickness of metal, it would be unsafe to continue the strain. When the press has performed its office, and it becomes necessary to relieve the action, the discharging valve, placed in the furniture of the forcing pump, must be opened, which will admit the water to escape out of the cylinder and return to the cistern, while the table and piston, by means of their own weight, return to their original position. 142 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION The method of calculating the power of the press, as well as every other particular respecting it, has been fully exemplified in the fore- going theory ; it is hence unnecessary to dwell longer on the subject : we shall therefore conclude our description of the press, and proceed with that of the Hydrostatic Bellows, which depends upon the same principle, viz. the quaqua versum pressure of non-elastic fluids. 2. THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION OF THE HYDROSTATIC BELLOWS. 149. In the preceding pages we have developed the theory and exemplified the application of the hydrostatic press ; and furthermore, in order to render the subject as complete as possible, we have given a minute and comprehensive description of its several parts, and for the purpose of guiding the practical mechanic in its erection, the instrument is exhibited in its complete and finished state, accompanied by the forcing pump and all its requisite appendages. The next subject, therefore, that claims our attention, is the Hydrostatic Bellows, an instrument of very frequent occurrence in philosophical experiments ; it is chiefly employed in illustrating the upward pressure of non-elastic fluids and the hydrostatic paradox, and consequently, it depends upon the same principle as the hydro- static press, admitting of a similar, but a more concise mode of dis- cussion and illustration. 150. The Hydrostatic Bellows consists of a tube or pipe FEI, of very small diameter, and of any convenient length at pleasure, connected by means of the elbow at ^ i, with a cylindrical vessel whose vertical section is CDGH, and whose sides are made of leather like a common bellows, represented by the waving lines AmD and BWK; the upper and the lower surfaces AB and DC, being formed of circular boards corresponding to the cylindrical form of the vessel. When the bellows is empty, it is manifest that the boards A B and DC are very nearly in contact, and would be completely so, but for the leather sides forming into folds and preventing a coincidence: in this state, when water or any other incompressible and non-elastic fluid, is poured into the tube, it flows into the bellows and separates the boards ; a heavy weight as w is then placed upon the upper OF THE HYDROSTATIC BELLOWS. 143 board, and by pouring more fluid into the tube, the moveable plane A B and its incumbent load w, will be raised and kept in equilibrio by the column of fluid in the tube ; and when the equilibrium obtains, we infer, that : The iveight of the supporting column of fluid in the tube, is to the weight upon the moveable plane, as the area of a section of the tube, is to the area of the plane. This is manifest, for the fluid at i, the lowest point of the vertical tube FEI, is pressed by a force varying as the altitude LI, and by the nature of fluidity, this pressure is communicated horizontally to all the particles in DC, and thence transmitted throughout the whole mass of fluid in the bellows ; consequently, the pressure upwards on the board AB, is equal to the weight of a column of the fluid, the diameter of whose base is DC, and altitude LI or GD ; but the actual weight of the fluid supported, is that of a column whose diameter is DC, and altitude EI or AD. Hence, the weight which maintains the equilibrium, will be that of a cylinder of fluid, whose base is A B and altitude A G ; consequently, the weight w, placed upon the moveable plane of the bellows, since it balances the column of fluid L E, is equivalent to the weight of a fluid cylinder, whose section along the axis is ABHG. Put D rz AB or DC, the diameter of the cylindrical vessel or bellows, d zz LM, the diameter of the vertical tube, w ~ the weight upon the moveable plane, and w'-=. the weight or pressure of the fluid in the column LE. Then, because by the principles of mensuration, the areas of circles are to one another as the squares of their diameters ; the foregoing inference gives w' : w : : d* : D 2 , and this, by equating the products of the extreme and mean terms, becomes tfw'=:d*w. (113). Let both sides of this equation be divided by the quantity D 2 , which is found in combination with the weight or pressure of the fluid in the tube, and we shall obtain , d*w W =^' (114). Here again, that singular property of non-elastic and incompressible fluids becomes manifest, viz. 144 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION That any quantity however small, may be made to balance any other quantity however great. 151. If the diameter of the tube, the diameter of the cylinder or bellows, and the weight upon the moveable board AB be given, the weight of the fluid in the tube, or its perpendicular altitude to main- tain the equilibrium, can easily be determined by means of the equa- tion (114), which affords the following practical rule. RULE. Multiply the square of the diameter of the tube by the load upon the moveable board, and divide the product by the square of the diameter of the bellows or cylinder ; then, the quotient will give the weight of the fluid by which the equilibrium is maintained. EXAMPLE. The diameter of the bellows or cylindrical vessel is 18 inches, that of the tube or pipe, through which the fluid is conveyed into the vessel, is one fourth of an inch, and the weight upon the moveable board is 5760 Ibs. ; what weight of water must be poured into the vertical tube, so that the whole may remain at rest ? In this example there are given, Dm 18 inches; c?=:| of an inch, and w = 5760 Ibs. ; therefore, by performing as directed in the rule, we shall have .25* X 5760 360 Here it appears, that a quantity of water weighing llbs., disposed in a tube of of an inch in diameter, is capable of balancing another quantity of 5760 Ibs., disposed in a cylinder of 18 inches diameter; it is therefore manifest, that the height of the one column must far exceed the height of the other, and the excess of altitude may be determined in the following manner. 152. It has been abundantly proved by experiment, that a cubic foot of distilled water, at the temperature of about 39 of Fahrenheit's Thermometer, weighs very nearly 1000 avoirdupois ounces, or 62 Jibs. ; consequently, the number of cubic inches in the column whose weight is l^lbs., is found by the following analogy, viz. 62 : 1728 :: l : 30f inches; hence, the solidity of a column which maintains the equilibrium is 30f- inches, and according to the conditions of the question, the diameter of its base or section, is one fourth of an inch, and con- sequently, the area of the base or section, is .25* X. 7854 = . 0490875 square inches. OF THE HYDROSTATIC BELLOWS. 145 Now, according to the principles of mensuration, the solidity of a cylinder is determined, by multiplying the area of its base into its perpendicular altitude ; consequently, if h denote the perpendicular height of the column, we have .0490875 h = 30.72; therefore, by division, we shall obtain OA (TO 153. The solution which we have here given, applies to the parti- cular example preceding, in which the data are assigned ; but in order to accommodate the theory to every case, it becomes necessary to draw up the solution in general terms ; for which purpose, we must recur to equation (114), where the weight of the equilibrating column has already been found ; then, according to the above analogy, we have 62i : 1728 : : f , ., where s denotes the solidity of the column. If in the above analogy, we make the product of the jnean terms equal to the product of the extremes, we shall have and from this, by division, we get tf (115). Therefore, if the solidity of the equilibrating column be divided by the area of its base, viz. the quantity .7854d 2 , the quotient will fur- nish the perpendicular altitude ; hence we have _ 35.2024 w D* (116). 154. From this it appears, that in order to determine the altitude of the equilibrating column, it is not necessary that its diameter should be known, for the equation is wholly independent of that element, the diameter of the bellows, and the weight upon tne moveable board only, entering into its composition. The following practical rule will therefore determine the altitude of the column by which the equilibrium is maintained. VOL. I. L 146 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION RULE. Divide 35.2024 times the load to be sustained upon the moveable board, by the square of the diameter of the bellows, and the quotient will be the altitude of the equi- librating column. We shall determine the perpendicular altitude by this rule, on the supposition that the diameter of the bellows and the weight upon the moving plane, are the same as in the foregoing example ; therefore we have . 35.2024X5760 _, Q10 . , ^ nz 625.8 19 inches. The equation (114) for the weight of the equilibrating column, was deduced from the equation (113), by simple division only, without the enunciation of any problem ; but in order to render the subject a little more systematic, we shall determine the other elements of the general equation, severally from the resolution of their respective and appropriate problems. PROBLEM XXII. 155. In a hydro-statical bellows of a cylindrical form, there are given, the diameters of the bellows and of the equilibrating tube, together with the weight of the fluid by which the equili- brium is maintained : It is required to determine the weight upon the moveable plane, at the instant when the equilibrium obtains. Let both sides of the general equation (113), be divided by d* the square of the diameter of the balancing tube, and we shall obtain _pV ~-~d^' (117). And this equation affords the following practical rule. RULE. Multiply the weight of the equilibrating fluid, by the square of the diameter of the bellows, and divide the product by the square of the diameter of the tube, for the weight upon the moveable plane. EXAMPLE. The diameter of a cylindrical bellows is 24 inches, the diameter of the balancing tube is one fourth of an inch, and the weight of the fluid in the tube is 2 J Ibs. ; what weight will this coun- terpoise on the moving board of the bellows ? OF THE HYDROSTATIC BELLOWS. 147 Here, by proceeding as directed in the rule, we obtain This is something more than 10 tons and a quarter, which is mani- festly a great load to be suspended by 2 Jibs.; but the altitude of the suspending column must be proportionably great, which circumstance, without the aid of some artificial force, would render the instrument very inconvenient for any practical purpose ; it was, no doubt, by viewing the matter in'this light, that Mr. Bratnah, senior, was led to apply the forcing pump, and thereby to produce that very powerful engine, which formed the subject of our last article. PROBLEM XXIII. 156. In a hydrostatical bellows of a circular form, there are given, the diameter of the bellows, the load suspended, and the weight of the suspending fluid : It is required to determine the diameter of the equilibrating tube, so that the instrument may be just in a state of equi- librium. Let both sides of the general equation (113), be divided by w the weight upon the bellows, and we shall obtain D 2 M/ w and from this, by extracting the square root, we get D 2 ^' w ' (118). _ And the practical rule which this equation supplies, may be expressed in words at length in the following manner. RULE. Multiply the square of the diameter of the bellows, by the weight of the fluid which maintains the equilibrium, and divide the product by the weight upon the bellows, then, the square root of the quotient will be the diameter of the equilibrating tube. * This equation for the diameter of the tube may be otherwise expressed ; thus * i 2 148 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION EXAMPLE. The diameter of the bellows or cylindrical vessel, is 24 inches, the weight of the suspending fluid is 2 Ibs., and the weight suspended on the bellows 8000 Ibs. ; what is the diameter of the tube? Performing according to the rule, we have and from this, by extracting the square root, we obtain 6?=V- 144 = - 38 of an i PROBLEM XXIV. 157. In a hydrostatic bellows of a cylindrical form, the diameter of the tube, the weight suspended, and the weight of the suspending fluid, are given : It is required to determine the diameter of the bellows, so that the whole may be in a state of equilibrium. Let both sides of the general equation (113), be divided by w' the weight of the suspending fluid, and we shall have __ w' from which, by extracting the square root, we get w 1 ' (119). And from this equation, we obtain the following practical rule. RULE. Multiply the square of the diameter of the suspend- ing tube, by the weight suspended, and divide the product by the weight of the fluid which maintains the equilibrium; then, the square root of the quotient will be the diameter of the cylinder sought. EXAMPLE. The diameter of the suspending tube in a cylindrical hydrostatic bellows, is half an inch, the weight of the suspending fluid is 2 Ibs., and the weight suspended on the bellows board is 12000 Ibs. ; what is the diameter of the bellows ? Here, by proceeding as directed in the foregoing rule, we get .5X. 5X12000 D*~ 1500, * This equation for the diameter of the bellows may be otherwise expressed ; thus : OF THE HYDROSTATIC BELLOWS. 149 and by extracting the square root, we have :rr 38.73 inches. 158. The foregoing problems and rules, unfold every particular respecting the calculation of the hydrostatic bellows, and from them we may infer, that in the case of an equilibrium, if more fluid be added : It will ascend equally in the suspending tube, and in the cylindrical vessel composing the bellows, whatever may be their relative magnitudes. The demonstration of this is very simple, for let A BCD be a vertical section, passing along the axis of the cylindrical vessel, and also along the axis of the suspending tube KI ; and suppose that F and c are the points to which the fluid rises in the vessel and the tube, when the bellows is in a state of equilibrium. Take ic equal to Da, and through the points a D C I and c let a horizontal plane be drawn, intersecting the vertical plane A BCD in the line ab ; then it is manifest, that the weight w in the position EF, is equivalent to the weight of the fluid column a&FE. Let more fluid be poured into the tube at K; the equilibrium will then be destroyed, and the weight w will ascend, until by discontinuing the supply, the equilibrium is restored, and the fluid in the vessel and the tube becomes again quiescent at the points n and K. Take IK equal to DA, and through the points A and K, let a hori- zontal plane be drawn, cutting the vertical plane A BCD in the line AB; then as before, the weight w in the position mn, is equivalent to the weight of the fluid cylinder, of which A&nm is a vertical section. Now, the weight w is not altered in consequence of the change of position from EF to mn\ therefore, because EF is equal to mn, it follows, that E is equal to mA; consequently, by taking away the common space ma, the remainders Em and a A are equal to one another; but by reason of the parallel lines ac and AK, the spaces a A and CK are equal to one another; therefore CK is equal to Em. From the principle here demonstrated, the resolution of the follow- ing problem may readily be derived. 150 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION PROBLEM XXV. 159. If a hydrostatic bellows of a cylindrical form, have a given quantity of fluid poured into the equilibrating or suspend- ing tube : It is required to determine through what space the weight on the moving board will ascend in consequence of the supply. Before we proceed to the resolution of this problem, it may be proper, as in the foregoing cases, to exhibit an appropriate notation ; for which purpose, Put D= AB or DC, the diameter of the cylindrical vessel or bellows, c?zz the diameter of the equilibrating or suspending tube, q HZ the quantity of fluid poured into the tube, and a; Em, the space through which the weight ascends by reason of the supply. Then, according to the principles of mensuration, the area of a transverse section of the cylindrical vessel or bellows, is and the area of the corresponding section of the tube, is where the symbols a and a' denote the respective areas. But by the property demonstrated above, the fluid rises equally in the bellows and in the tube ; therefore, the quantity of fluid which flows into the bellows in consequence of the supply, is and the quantity which remains in the tube, is where the symbols s and s' denote the solidities of the cylinders, whose diameters are D and d, and their common altitude x. Now, the sum of these quantities, is manifestly equal to the quantity of fluid poured into the tube ; hence we have and by division, we obtain - 9 (120). OF THE HYDROSTATIC WEIGHING MACHINE. 151 It therefore appears, that the space through which the weight ascends by reason of the supply : Is equal to the quantity of fluid which is poured into the tube, divided by the sum of the areas of a cross section of the tube and the cylindrical vessel or bellows. The practical rule, or method of applying the equation, may there- fore be expressed in words at length in the following manner. RULE. Divide the quantity of fluid which is poured into the tube, by .7854 times the sum of the squares of the dia- meters, and the quotient will give the quantity of ascent, or the space through which the weight is raised in consequence of the supply. EXAMPLE. The diameter of a cylindrical vessel is 20 inches, and that of the suspending tube is one inch ; now, suppose that an incom- pressible fluid is poured into the tube, until its weight sustain in equilibrio, a load of 8760 Ibs. upon the moveable bellows board ; then, how much higher will the load be raised, when 150 cubic inches of the fluid are superadded ? Here then, we have given D=r20 inches, d 1 inch, and '=.7854eZ 2 s (& + &' + 3), and this is in equilibrio with the pressure of the column ymnz, or the weight (w + w') ; consequently, we have .7854d 2 s (h + h' -f 3) : w -f w' : : .7854d 2 : .7854 D * ; or by suppressing the common factors, we have s (k-\-h' -M) : w + w' : : 1 : .7854o 2 ; therefore, by equating the products of the extremes and means, we get w -\- w/zz .7854D 2 s(/z- r -A'- r -S). (122). But we have seen above, equation (121), that wnr.7854/i5D 2 ; consequently, by substituting and separating the terms, we obtain W /i=.7854D 2 s(A / + a). (123). Now, it is manifest, that the descent of the cover in the vessel, and the rise of the fluid in the tube, must be to one another, inversely as the squares of the respective diameters ; therefore, we have Ztf^h'tf, * or by division, we get h'd* S = l^' and finally, by substitution, we obtain w' = .7854 h's (D 2 -f d 3 ). (124). 154 SCIENTIFIC DESCRIPTION OF THE HYDROSTATIC WEIGHING MACHINE. 161. If the fluid be water, whose specific gravity is represented by unity, the equation becomes somewhat simpler ; for in that case, we have w' = .7854 &' (D -f d 2 ). (125). From this equation the magnitude of the additional weight, or the measure by which it is expressed, can very easily be ascertained ; and the practical rule by which it is discovered, is as follows. RULE. Multiply the sum of the squares of the diameters, by .7854 times the rise of the fluid in the tube, or the eleva- tion above the first level, and the product will express the magnitude of the additional weight. EXAMPLE. The diameter of a cylindrical vessel is 16 inches, and that of the communicating tube one inch ; now, supposing the machine in the first instance, to be in a state of equilibrium, and that by the addition of a certain weight on the moveable cover, the water in the tube rises 6 inches above the original equilibrating level ; how much weight has been added ? By proceeding according to the rule, we have D 2 -|- d*= 16 2 + P i=256 + 1 = 257, and by multiplication, we obtain w' = .7854X6X257 zz 1211.0868 avoirdupois Ibs.* 162. If the additional weight, by which the water is made to rise in the tube be given, the distance above the first level to which it will rise, can easily be found ; for let both sides of the equation (125), be divided by the quantity .7854 (D* -\- d z ), and we shall obtain ,,_ ~.7854(D*4-d 2 )' And from this equation, we deduce the following rule. RULE. Divide the additional weight, by the sum of the areas of the moveable cover and the cross section of the communicating tube, and the quotient will give the height to which the fluid will rise above the first level. * It is manifest from the form of the equation which supplies the rule, that without paying particular attention to the nature of the load which produces the equilibrium in the first place, the value of vf is ambiguous, and may be read in ounces, Ibs., cwts., or tons ; and indeed, in any denomination of weight whatever ; but it must always be read in the same name as that by which the equilibrium is produced. EXPERIMENTS ON THE QUAQUAVERSUS PRESSURE OF FLUIDS. 155 EXAMPLE. The diameter of the moveable cover is 16 inches, and that of the communicating tube one inch ; then, supposing that the machine in the first instance is brought to a state of equilibrium, and that a load of 1211 Ibs. is applied on the cover, in addition to that which produces the equipoise ; to what height above the first level will the water ascend in the communicating tube? Proceeding according to the rule, we obtain .7854 (D 8 + d. And moreover, if the glass be still farther depressed, the fluid will ascend higher and higher, and the air will be compressed into a less and less space. Again, if the glass be inclined in any degree from the vertical position, as represented by EF and GH, taking care to have its mouth wholly immersed in the water, then it is evident, that the greater the degree of inclination, the greater is the quantity of fluid which enters, and the greater also is the condensation of the included air; but when the quantity of fluid which enters the glass is the same, both in the vertical and the inclined position, the density of the air is also the same, being compressed by the same force ; consequently, the water or fluid in which the glass is placed, exerts the same pressure in whatever direction it is propagated. One sees this experiment verified daily by empty casks having only one end, thrown into water. EXPERIMENTS. If the several tubes A, B, c, D A;B and E, bent at various angles, be .inserted in an empty vessel, or if they be held in the hand, and mercury be introduced at their lower extremities, in such a manner, as to come close to the ori- fices ; then let water be poured into the vessel, and it will be seen, that during the time of its filling, the mercury is pressed gradually from the 158 THE QUAQUAVERSUS PRESSURE OF FLUIDS lower towards the higher extremities of the tubes, which are supposed to rise to a height considerably above the surface of the water. Now, since the lower extremities of the tubes may be conceived to point in every possible direction, it follows, that the pressure of the superincumbent fluid is also propagated in every direction. But when it is required that the lower orifice should point directly downwards, in order to show the upward pressure of fluids, a straight tube must be employed, and the mercury which is introduced must be kept in by the finger, until the height of the water above the lower surface, is about fourteen times the height of the mercurial column ; for if the finger be removed before the water has attained that height, the mer- cury will fall out of the tube, since its weight is fourteen times greater than the weight of an equal bulk of water. If the finger be continued upon the orifice, until the height of the water be equal to fourteen times the height of the mercury, then, on removing the finger, and pouring in more water, the mercury will be seen to ascend in the tube, and will continue to rise higher and higher, according to the quantity of water poured in, thereby showing the upward pressure of the water. EXPERIMENT 4. The pressure of fluids at different points of their depths, may be very simply illustrated in the following manner : let K be a bag of leather, or some other tough and flexible material, filled with mercury, and attached to the extremity of a glass tube zi, in such a manner, that the mercury may just enter the tube when the bag is held in air. Then, if the bag be immersed in water, it is manifest that the pressure of the fluid will cause it to collapse, and the mercury will ascend in the tube to a certain height, corresponding to the pressure exerted by the water, at the depth where the bag is placed. If the bag continue to be lowered in the water, it will become more and more collapsed in consequence of the increased pressure, and the mercury will ascend higher and higher in the tube, and the heights to which it rises, will indicate the magnitude of pressure at different depths. EXPERIMENT 5. There is a very simple and amusing experiment, by which the propagation of pressure through fluids is illustrated, called the " Cartesian Devil" from M. Descartes, the celebrated French philosopher, by whom it was discovered ; it is as follows. ILLUSTRATED BY EXPERIMENTS. 159 Let the little figure in the inverted jar AB represent the " Cartesian Devil," surmounted by a bag-like crown of great size in proportion to his body, filled with some very light substance, such as air, and we shall therefore suppose that air is the body which it contains. The imp himself must be constructed of glass or enamel, so as to possess the same specific gravity as water, and therefore to remain suspended in the fluid. A At the bottom of the vessel or jar, is placed a diaphragm or bladder, that can be pressed upwards by applying the finger to the extremity of a lever eo, moving round o as its fulcrum or centre of motion. The pres- sure applied at a is communicated through the water to the bag of air at m, which is thus compressed, and consequently, the specific gravity of the figure is increased, by which it sinks to the bottom of the jar. By removing the pressure on the dia- phragm at a, the figure will again ascend, so that it may be made to oscillate, or rise upwards and sink downwards alternately, and to dance about in the jar, without any visible cause for its movements. Other figures, such as fishes made of glass, are sometimes employed in this experiment, but the principle is nevertheless the same, and when a common jar is used, the pressure is applied to the upper surface of it, as at A. EXPERIMENT 6. The pressure of fluids at very great depths, is beautifully illustrated by an experiment which has often been made at sea, where the water is sufficiently deep to admit of the principle being accurately put to the trial. The experiment is this : an empty bottle well corked is made to descend to a great depth, on which the pressure of the fluid becomes so great as to drive in the cork, and the bottle when brought up is always filled with water. Several methods have been employed to prevent the cork from being driven inwards, but although this has been effected, yet the bottle on being brought to the surface, is con- stantly filled with the fluid in which it has been sunk. The following experiments of this sort, are detailed by Mr. Campbell, the author of " Travels in the South of Africa," published at London in the year 1815; the experiments were tried on his voyage home- wards from the Cape of Good Hope. He drove very tightly into an 160 THE QUAQUAVERSUS PRESSURE OF FLUIDS. empty bottle, a cork of such a large size, that one half of it remained above the neck : a cord was then tied round the cork and fastened to the neck of the bottle, and a coating of pitch was put over the whole. When the bottle was let down to the depth of about fifty fathoms, he perceived by the additional weight, that it had instantly filled ; and on drawing it up, the cork was found in the inside of the bottle, which of course was filled with water. Another bottle was prepared in a similar manner ; but in order to secure the cork, and to prevent it from being pressed within the bottle, a sail needle was passed through it, so as just to rest on the margin of the glass, and the whole was carefully covered with a coating of pitch. When the bottle had descended to the depth of about fifty fathoms, as in the former case, it was again perceived to have been filled with water; and on bringing it to the surface, the cork and needle were found in the same position, and no part of the pitch appeared to be broken, although the bottle was completely filled with water. Here the water must have insinuated itself through the pores of the pitch and the cork, and not as the experimentalist supposes, through the pores of the glass. The equality of fluid pressure in every direction, is very easily demonstrated in the following manner. EXPERIMENT 7. If a piece of very soft wax, as GUI, and the egg E, be placed in a bladder, or some other flexible vessel filled with water, and if the bladder be put into a brass box, and a moveable cover laid upon the bladder so as to be wholly supported by it. Then, if one hundred, or one hun- dred and fifty Ibs. be laid upon this cover, so as to press upon the bladder and its contained fluid ; this enor- mous force, although propagated throughout the fluid, and acting upon the soft wax and the egg, will produce no effect, the wax will not change its form, and the egg will not be broken. And in like manner, if a living fish should be put into the cylinder of a hydrostatic press, when under a very high degree of pressure, it will not suffer the least incon- venience ; from which it is obvious that every particle of the fluid is equally pressed, and presses equally in all directions. Numerous other examples might be adduced for proving the same thing, but since the principle is manifest, it is needless to dwell longer on the subject. CHAPTER VII. OF PRESSURE AS IT UNFOLDS ITSELF IN THE ACTION OF FLUIDS OF VARIABLE DENSITY, OR SUCH AS HAVE THEIR DENSITIES REGULATED BY CERTAIN CONDITIONS DEPENDENT UPON PAR- TICULAR LAWS, WHETHER EXCITED BY MOTION, BY MIXTURE, OR BY CHANGE OF TEMPERATURE. IN the former part of this treatise, we have displayed the nature of pressure as it occurs in the action of non-elastic fluids of uniform density, and in addition, we have investigated the theory and exem- plified the application of the Hydrostatic Press, the Hydrostatic Bellows, and the Hydrostatic Balance or weighing machine ; instru- ments whose operations depend upon the quaqua-versus principle of non-elastic and incompressible fluids: We come therefore in the next place, to consider pressure as it unfolds itself in the action of fluids of variable density, or such as have their densities regulated by certain conditions, dependent upon particular laws, whether excited by motion, by mixture, or by change of temperature. In mechanical science density is used as a term of comparison, expressing the proportion of the number of equal moleculee in the same bulk of another body; density, therefore, is directly as the quantity of matter ; and inversely as the magnitude of the body* We cannot by means of our senses discover the figure and magnitude of the ele* mentary particles of matter. Mechanical inventions have wonderfully magnified objects invisible to the unassisted eye; but no microscopical assistance has yet en- abled us to assume that we have seen an elementary particle of matter. A number of elementary particles uniting bv the power of cohesion form greater particles, and these again uniting, by the same power, form still greater; and we may consider the aggregate of many such formations to become at length an atom of a sensible bulk. All bodies seem to be composed of these derivative corpuscles, which, formed of more or fewer of these repeated unions, compose bodies more or less dense. These derivative corpuscles are sometimes similar, as the coloured rays of a beam of light, VOL. I. M 162 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY separated by the prism ; mercury, when squeezed through the pores of leather, or raised in fume and received upon clean glass, which exhibits globules similar and undistinguishable. In short, every mass of matter is divisible into particles, which we designate by the Greek term atom, or that which is so exceedingly minute that it cannot be further cut or divided, and which therefore, as far as sense is concerned, is the ultimate resisting particle. It must be obvious, that the density or quantity of atoms which exist in a given space is very different in different substances. Hence, if it be asked why bodies are called dense ? the answer is, Because they contain more atoms than others of the same size. There are more atoms in a cubic inch of lead than in a cubic inch of cork : the former is forty times heavier than the latter. A cubic foot of rain water weighs 62 Ibs. ; but an equal volume of mercury, which is fourteen times heavier than water, weighs (62^x14) =875 Ibs. Density must depend on three circumstances, to which we should carefully attend in all our disquisitions : first, the size or weight of the individual atom ; secondly, on porosity, or the arrangement of the atoms by cohesion, or mechanical and physical arrangement ; thirdly, the proximity of the atoms determined by the substance of which they are constituent particles, possessing tenacity and incompressibility. Thus, heat dilates some bodies and contracts others. A pound of tin and a pound of copper melted together form bronze ; but this new mass occupies less space by one fifteenth than the two masses did when separate j proving that the atoms of the one are partially received into what were empty spaces of the other. In other words, the affinity of cohesion is one fifteenth greater in the bronze than in the tin and copper separately. Two pounds of brine are made out of a pound of salt and a pound of water ; but the mass is of less bulk than the aggregate of the ingredients apart. Water, we have seen, resists compression very powerfully, but at the depth of 1000 fathoms yielding a very small part of its bulk at the surface, shows the particles not to be in contact, and that the fluid may acquire density in propor- tion to its depth. Wood swims in water, because the water has more atoms in the same bulk than the wood, and therefore more weight or central force than the wood ; consequently, the water falls first and leaves the wood behind; in other words, the wood floats upon the water the wood is borne on the surface of the water with a force exactly proportional to the difference between its weight and that of an equal bulk of water. The pressures which the fluid exerts in supporting the wood are together equivalent to a force directed upwards through the centre of gravity of the fluid displaced, and equal to the weight of a quantity of the fluid so displaced by the immersed part of the body. But it is not necessary here to dwell further on this topic, the density of water. We therefore pass on to another character it possesses, viz. gravity or weight : and it is, in fact, by comparing the weight of a body with the force which holds it up in the fluid, that the comparative weights or specific gravities are found, as of metals compared with water, and of admixtures of metals for the purpose of ascertaining at once the proportion of each in the compound mass. Water is the common standard with which all other substances are compared, whose weight we would fix and record in tables of specific gravities. When we say, therefore, that gold is of the specific gravity of 19, and copper of 9, and cork of one seventh, we mean that these substances are just so much heavier or lighter than their bulk of pure water in its densest form, viz. at the temperature of 40 degrees of Fahrenheit's thermometer. It appears, therefore, that the terms 0f THe ^\ ( UNIVERSITY J VSi^WX UPON THE SIDES AND BOTTOM OF A CYLINDRICAL VESSEL. 163 density and specific gravity express the same thing" under different aspects ; the former being' more accurately restrained to the greater or less vicinity of particles, the latter to a greater or less weight in a given volume ; hence, as weight depends upon the closeness of the particles, the density varies as the specific gravity, and the terms may in most cases be indiscriminately used. The specific gravities of fluids are usually considered without any regard to the empty spaces between the particles, though if the particles of fluids are spherical, the vacuities make at least one fourth of the whole bulk. But it is sufficient that we know precisely in what sense the specific gravities of fluids are understood. PROBLEM XXV. 163. A cylindrical vessel whose sides are perpendicular to the horizon, has a certain quantity of fluid in it; which fluid, by reason of a sudden change of temperature, has its magnitude or bulk increased by a certain part of itself : It is therefore required to determine what will be the alteration of pressure on the sides and bottom of the vessel. Let ABCD, and abed respectively, represent vertical sections of the cylindrical vessel, of which the sides are perpendicular and the base parallel to the horizon ; then in the first instance, let E F be the height to which the vessel is filled, and ef the height to which the fluid rises, by reason of the change that takes place in the temperature. Draw the diagonals EC, FD and ec, fd intersecting respectively in the points G and g, and through the points G, g, draw the Vertical lines MN and mn ; then are MG and mg, the respective depths of the centres of gravity of the cylindric surfaces, in contact with the fluid before and after the expansion, and MN, mn, are the depths of the centres of gravity of the bases or bottoms D c and dc. Through the points G and g, draw the straight lines GT and gs, parallel to the horizon and to one another; then is GS or rg the height which the centre of gravity of the cylindric surface is elevated, by reason of the expansion of the fluid. Put d = DC or dc, the diameter of the cylindric vessel, h = MN, the height to which the vessel is originally filled, h'mn, the height at which the fluid stands in the vessel after expansion, M 2 164 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY Put P zz: the pressure on the bottom DC, by the fluid in its original state, p zz: the corresponding pressure on the cylindric surface, P'zz: the pressure on the bottom dc, by the fluid after expansion, p' zz: the corresponding pressure on the cylindric surface, s zz: the specific gravity of the fluid before expansion, s' z= the specific gravity of the fluid after expansion, a zz: the area of the base or bottom of the vessel in both cases ; and $' the cylindric surfaces, and zz: the part of its bulk by which the fluid is increased. Then, since d denotes the diameter of the bottom, the area accord- ing to the principles of mensuration, becomes and the pressure exerted by the fluid in its original state, is (126). Again, according to the principles of mensuration, the cylindric surface in contact with the fluid before expansion, is and consequently, the pressure upon it, is /;zz:3.1416d/iXjAXszz:1.5708dA 2 s. (127). Now, it is manifest, that since the diameter of the vessel is the same both before and after the expansion of the fluid, the capacity and the altitude must vary directly as each other; consequently, because the capacity or bulk is increased by ^th part of itself, it follows, that the altitude is increased in the same proportion ; there- fore we have but when the weight of the fluid remains the same, the density, and consequently the specific gravity, varies inversely as the magnitude. The specific gravity of the fluid, after it has expanded by reason of an increase of temperature, is therefore, *J> ') .ft ........-' n n-j-l ' hence, the pressure on the bottom of the vessel, after the fluid has increased by expansion, becomes P' = .7864dAV; that is, 'Vv ON SEMICIRCULAR PLANES IMMERSED VERTICALLY. 165 P' .7854ef X h - X --7 = .7854d h s. (128). \ n ' n -f- 1 The cylindric surface in contact with the fluid after expansion, may be expressed as follows, viz. but it has been shown above, that 3.1416eM; therefore, by substitution, we obtain n and consequently, the pressure becomes If therefore, the equations (126) and (128) be compared with one another, it will be found that the pressure is the same, and equal to the weight of the fluid in both cases; but if the equations (127) and (129) be compared, the pressure in the one case, is to that in the other, in the ratio of n : n + 1 ; that is, p : p' : : n : n -\- I . PROBLEM XXVI. 164. A semi-circular plane is vertically immersed in a fluid whose density increases as the depth, and in such a manner, that the horizontal diameter coincides with the upper surface of the fluid : It is required to determine, on which chord parallel to the horizon, the pressure is a maximum, or greater than the pressure on any other chord. Let A BCD represent a vertical section of a mass of fluid, of which AB is the surface, and whose density varies directly as its depth ; and let jmr aGFHB be the semi-circular plane im- mersed in it, in such a manner, that the horizontal diameter a b, coincides with AB the upper surface. Let m be the point in the vertical 166 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY radius EF, through which the chord of maximum pressure is supposed to pass; draw the chord GH, and the radius EG; then, because the chord GII is parallel to the diameter ab y it follows, that GH is bisected in m by the vertical radius EF; consequently, m is the place of the centre of gravity of the chord GH, and Em is its perpendicular depth below AB, the upper surface of the fluid. Put r IZTEG, the radius of the semi-circular plane, = GF, half the arc subtended by the required chord GH, and x Em, the distance of the chord below the surface of the fluid. Then, because by the conditions of the problem, the density of the fluid varies directly as its depth ; it follows, that the pressure on the chord GH varies directly as Gra drawn into Em 2 ; that is, where p denotes the pressure upon the chord ; but this, by the condi- tions of the problem, is to be a maximum ; therefore, we have x* \/ r 2 # 2 n: a maximum, from which, by equating the fluxion with zero, we get or by transposing and expunging the common factors, we obtain 3z 2 = 2r 2 ; therefore, by division, we have z'zn-fr 2 , and finally, by evolution, it becomes * = rV?. (130). The same result, however, may be otherwise determined ; for by the arithmetic of sines, we have, to radius unity Gmzzsin.0, and Emzncos.0 ; but in order to accommodate these quantities to the radius r, it is Gmmr sin.0, and Em 2 zziiK 2 z=:r 2 cos. 2 ^; consequently, by multiplication, we obtain and this, by the conditions of the problem, is to be a maximum ; hence we get r*sin.0 cos. 2 ^ = a maximum, which being thrown into fluxions, becomes r 3 (0- cos. 3 < 20- sin. 2 ^ cos.^) ; ON SEMICIRCULAR PLANES. THE CHORD OF MAXIMUM PRESSURE. 167 therefore, by transposing and casting out the common terms, it is * cos. 2 0nz2sin. 2 0. But according to the principles of Plane Trigonometry, we have 2sin. 2 = 2 2cos. 2 <; consequently, by substitution, the above expression becomes cos. 2 zz 2 2 cos. 2 ; therefore, by transposition and division, we obtain cos. 2 0=rf, and by extracting the square root, we get cos.0rz:-v/-| ; finally, let both sides of this expression be multiplied by r, for the purpose of adapting it to the proper radius, and we shall have # = rcos.0:zrr|/!-, the same as above. 165. The practical rule for reducing this equation, may be expressed in words at length in the following manner. RULE. Multiply one third of the radius of the given semi- circular plane by the square root of 6, or by the constant number 2.44947, and the product will give the distance of the point on the vertical axis below the surface of the fluid, through which the chord of maximum pressure passes. 166. EXAMPLE. The radius of a semi-circular plane, immersed in a fluid agreeably to the conditions of the problem, is 27 inches ; at what distance below the surface of the fluid must a horizontal chord be drawn, so that the pressure which it sustains may be greater than the pressure sustained by any other chord drawn parallel to it? By operating according to the rule, we shall obtain x 9 ^6*= 9 X2.44947 = 22.04523 inches. 167. The same example admits of a very simple and elegant geo- metrical construction, which may be effected in the following manner. Let ACS be the semi-circular plane, of which the diameter AB is parallel, and the radius D c perpendicular to the horizon ; draw the chord BC, and from c as a centre, with the radius CD, describe the circular arc DKH cutting BC produced in the point H. Through the point c draw the tangent c K, and let fall the perpendicular H i meeting c K in i; join DI to intersect the arc A EC in E, and through the point E 168 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY. draw the chord EGF parallel to AB the diameter of the semicircle; then is EGF the chord, on which the pressure is a maximum. That the line DG corresponds with x in the equation marked (168), may be thus demonstrated. By reason of the parallel lines AB and KC, the angles ABII and KCH are equal to one another ; but the angle ABH is manifestly equal to half a right angle or 45 degrees, therefore, the angles KCH and CHI, are each of them equal to half a right angle, and the lines ci and HI are equal, being respectively the sine and cosine of 45 degrees to the radius CH or CD. Now, according to the principles of Plane Trigonometry, the sine and cosine of 45 degrees to the radius unity, are respectively expressed by \ V% > hence we have ci^JvC and by the property of the rightangled triangle, it is Dizr- v /ci*-4-DC 2 =:r- v /|-, and by similar triangles, we have r V i : r ' : ' r ' D G - = - r V $ The length of the chord line EF is very easily found, for by reason of the right angled triangle EDG, of which the two sides DE and DG are known, it is E G 2 ~ D E ? D G 2 ; but by the elements of geometry, the square of a line is equal to four times the square of its half, therefore, we have EF 2 H=4(DE 2 DG 2 ); hence, by extracting the square root, we get E F Zr 2 ^ D E 2 D G 2 J now D L 2 1=1 r*, and DG*I=: f r 9 ; therefore it is EF=z|rV~3. (131). Wherefore, if we take the radius of the semi-circle equal to 27 inches, as in the preceding example, the whole length of the chord will be 18X1. 7321=31. 176inches. PROBLEM XXVII. 168. If a given conical vessel be filled with fluid, and sup- ported with its axis inclined to the horizon at a given angle : It is required to determine, on vjhat section parallel to the base the pressure is a maximum. MAXIMUM PRESSURE ON A SECTION OF A CONICAL VESSEL. 169 Let ABC be a section passing along the axis of the conical vessel, of which c is the vertex, and AB the diameter of its base. Conceive AI to be horizontal, and produce the axis CD to meet the horizontal line AI in the point i ; then is A ic the angle of inclina- tion between the axis and the horizontal line AI. Let G be the point in the axis through which the plane of the required section is supposed to pass, arid through G draw the straight line EF parallel to AB, and GH perpendicular to AI; then is EF the dia- meter of the section, and GH the perpendicular depth of its centre of gravity below A, the highest particle of the fluid. Put Rizr AD, the radius of the base of the conical vessel, H c i), the axis or height, r ZZTEG, the radius of the section on which the pressure is a maximum, a ~ the area of the section, d ~ GH, the perpendicular depth of its centre of gravity, p the pressure perpendicular to its surface, zr: A ic, the angle of inclination between the axis of the cone and the horizon, x zr CG, the distance between the section and the vertex of the cone, and s z= the specific gravity of the fluid. Then, because of the rightangled triangle ADI, and from the prin- ciples of Plane Trigonometry, we have R : DI : : tan.0 : 1, and from this, we obtain consequently, by adding the axis, we get cizz R cot. -f-H, and again by subtraction, it is G i ~ R cot.p 4" H x. But the triangle GHI, is by construction right angled at H ; there- fore, by Plane Trigonometry, we have GHmc?zz:{Rcot.^ -j~ H a?}sin.0. Again, the triangles CD A and CGE are similar to one another; therefore, by the property of similar triangles, we have 170 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY H 2 : R 2 : : x* : r 2 ; or from this, by equating and dividing, we get r *-^. ~ H 2 ' consequently, by the principles of mensuration, the area of the section, on which the pressure is proposed to be a maximum, becomes 3.1416R 2 * 2 -rf -- ; therefore, the pressure upon its surface is 3.1416R a * 2 s Now, because the whole of the quantities which enter this equa- tion are constant, excepting x, and the bracketted expression {Rcot.^-f H x} which is affected by it; it follows, that the value of p varies as rc 2 {RCOt.0-|- H x}, and consequently, is a maximum, when the quantity which limits its variation is a maximum ; hence we have a? 2 {R cot.^ -|- H x} zz a maximum. Let the above expression for the maximum be thrown into fluxions, and we shall obtain 2 (R cot.0 -f- H) xx 3a? a i =. ; therefore, by transposing and expunging the common quantities, we get 3a?i=2(RCot.0+ H), and finally, by division, we obtain a? = |-(RCOt.^ + H). (132). 169. The practical rule for reducing this equation, may be expressed in words at length in the following manner. RULE. Multiply the natural cotangent of the angle which the axis of the cone makes with the horizon, by the radius of the vessel's base, and to the product add the altitude or axis of the cone ; then, two thirds of the sum will give the distance of the section, on which the pressure is a maximum, from the vertex of the cone. 170. EXAMPLE. A conical vessel whose altitude is 20 inches, and the radius of its base 8 inches, is filled with fluid and so inclined, that its axis makes with the horizontal line passing through the extremity of the diameter of its base, an angle of 48 degrees ; on what section, parallel to the base is the pressure a maximum ? ON A GLOBULAR BODY OF CONDENSIBLE AND ELASTIC MATTER. 171 Here we have given, RZH 8 inches, H m 20 inches, and 0m 48 degrees, of which the natural cotangent is 0.9004 very nearly; con- sequently, by the rule, we have x = $ {8X0.9004 + 20} =18.1355 inches. If therefore, 18.1355 inches be set off from the vertex, a straight line drawn through that point parallel to the base, will be the diameter of the section on which the pressure is a maximum. 171. In any fluid the particles towards its base support those that are immediately above ; these again bear the load above them, and so on to the surface, where the whole mass supports the super- incumbent atmosphere. There is therefore a pressure among the successive strata of an homogeneous fluid increasing in exact propor- tion to the perpendicular depth. Hence a bubble of air or of steam, set at liberty far below the surface of water, is small at first, and gradually enlarges as it rises. This phenomenon shows that the compressive power of the fluid slackens by ascent. Experiment and calculation most readily demonstrate the compressibility of water : and the next problem exhibits the striking effects from the increase of pressure at great depths of the sea. PROBLEM XXVIII. 172. A globular body of condensible and elastic matter, is suffered to ascend vertically from the bottom to the surface of the sea : It is required to determine its diameter at the surface, the depth of the sea, and the diameter at the bottom being given. Let AB be the surface of the sea, ab its bottom, and ecf, ECF, two positions of the globular body in its ascent from the bottom to the surface, and A sea, B vfb the curves described by the extremities of the diameter. Through G and g, the centre of the globe in the two positions, draw the vertical line cc, which is manifestly the abscisse to the curve, the radii ge and GE, as well as ca and CA, being ordinates. Produce the abscisse cc to D, and make CD equal to the height of a column of sea water, which would be in equilibrio with the pressure 172 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY of the atmosphere ; through the point D draw the straight line 11 1 parallel to the surface of the sea, and HI will manifestly be the asymptote to the curve described by the extremities of the diameter. Put r=ac, the radius of the globe at the bottom of the water, d cc, the depth of the water at the place of immersion, 7i CD, the height of a column of water equal to the weight of the atmosphere, x rr CG, any abscissa, y nz G E, the corresponding ordinate, or radius of the globe. Then, because the magnitude of the globular body, is inversely as the density, (the weight and the quantity of matter remaining the same,) and the density is directly as the pressure ; it follows, that the magnitude of the body at different points of its ascent, is inversely as the pressure at those points, and the pressure is directly as the depth ; therefore, we have DC : DG : : GE S : ca 8 ; but according to the foregoing notation, we have d+h : h + x:: y* : r 3 ; from which, by equating the products of the extremes and means, we get hence, by division, we obtain = (*+*) and by extracting the cube root, it is ,/(d + h) r y (T+T)' (133). The equation in its present form, exhibits the nature of the curve described by the diameter of the body during its ascent; or it ex- presses generally, the value of the ordinate or radius corresponding to any depth ; but in order to determine the radius at the surface, which is the primary demand of the problem, we must suppose the quantity x to vanish, in which case, the above equation becomes d+h h (134). 173. The practical rule supplied by, or derived from this equation, may be expressed in words at length in the following manner. ON A GLOBULAR BODY OF CONDENSIBLE AND ELASTIC MATTER. 173 RULE. To the given depth of the sea, add the height of a column of sea water, which is equal to the weight or pressure of the atmosphere ; divide the sum by the height of the atmo- spheric column, and multiply the radius of the body at the bottom of the sea by the cube root of the quotient, and the product will give the radius at the surface. 174. EXAMPLE. The radius of a globe of elastic arid condensible matter, when placed at the depth of 75 fathoms in sea water, is equal to 4 inches ; what will be the radius on ascending to the surface, the atmospheric column being equal to 33 feet ? Here we have given d~ 75 fathoms, or 75 x 6 m 450 feet ; h zz 33 feet; fizz 4 inches; consequently, by the rule, we have =v - izz 9. 785 inches nearly. 33 175. From this it appears, that if a globe of condensible matter, whose radius is 9.785 inches, be immersed in the sea to the depth of 450 feet, its radius will be decreased to 4 inches ; this circumstance may suggest some easy and accurate methods of determining the depth of the ocean, when it is so great as to preclude the application of other methods. 176. In order, however, to adapt our equation to the determination of the depth, we must consider the radii at the surface and at the bottom, together with the height of the atmospheric column, to be accurately known at the time of trial ; then, by a very obvious trans- formation, the depth of descent may be ascertained ; for let R be substituted instead of y in the foregoing equation, to denote the radius at the surface, and we shall have R r V ~T~~' in which equation, d is the unknown quantity. Let both sides of the equation be divided by r, the radius of the globe at the bottom of the sea, and we shall obtain (d+h), ~T~~' and cubing both sides, it becomes multiply by h, and we obtain 174 OF THE PRESSURE OF UNMIXABLE FLUIDS OF DIFFERENT DENSITIES, **=- + *. and finally, by transposition, we have r 3 (135). 177. The practical rule for reducing the above equation, may be expressed in words at length in the following manner. RULE. Multiply the difference of the cubes of the radii) by the height of the atmospheric column, and divide the product by the cube of the lesser radius for the depth required. EXAMPLE. The radius of a globe of condensible matter is 10 inches before immersion, and it is suffered to descend so far as to have its radius diminished to 3 inches ; required the depth of descent, the atmospheric column at the time of the experiment being equivalent to 33 feet. Here we have given R 10 inches, r 3 inches, and h 33 feet ; therefore, by proceeding according to the rule, we have R s_ r8 __ 10QO _27 973 ; consequently, multiplying by 33 feet, we obtain 973x33 32109, therefore, by division, it is PROBLEM XXIX. 178. Let a vessel of any form whatever, whose base is hori- zontal, be filled with fluids of different densities which do not mix : It is required to determine the pressure on the bottom of the vessel, supposing the fluids to succeed each other in the order of their densities. Let ABGH represent a vertical section of the vessel, containing fluids of different densities or specific gravities, as indicated by the shading of the several strata AC, DF and EG; and for the sake of simplicity of investigation, let the bottom HG be parallel, and the sides AH, BG perpendicular ON THE BOTTOM OF ANY VESSEL. 175 to the horizon. Then are AB, DC and EF, the respective surfaces of the several fluids, as mercury, water, and olive oil, also parallel to the horizon ; for, as we have elsewhere stated : The common surface of two fluids which do not mix, is parallel to the horizon. Now, it is manifest, (since the sides A H and B G are perpendicular to the base HG), that the pressure upon the base HG, is equal to the pressures or weights of the several fluids contained in the vessel ; therefore Put d zz EH, the perpendicular depth of the lowest stratum EG, d' =. DE, the perpendicular depth of the middle stratum DF, d"=: AD, the perpendicular depth of the upper stratum AC, p zz the pressure of the stratum E G upon the line H G, jt/zz the pressure of the stratum DF upon the line EF, p" the pressure of the stratum A c upon the line D c ; and let s, s' and s" denote the specific gravities of the respective fluids. Then, since the pressure upon any surface, is equal to the area of that surface, drawn into the perpendicular depth of its centre of gravity; it follows, that the pressure upon HG, occasioned by the fluid in EG, is ^ZZTHGX^S, and in like manner, the pressure upon EF, is j/ = EFXdV, and lastly, the pressure upon D c, is But the total pressure upon H G, is manifestly equal to the sum of these pressures ; therefore, if P denote the entire pressure on the line H G, we have P but the lines HG, EF and DC, are equal among themselves, therefore we get P = HG (ds + d's' -f- d"8 u ). (136). 179. In the preceding investigation, we have considered three fluids of different densities to be contained in the vessel; but the same mode of procedure will extend to any number whatever, and what we have done respecting three fluids is sufficient to discover the law of induction for any other number. It is this : 176 OF THE PRESSURE OF UNMIXABLE FLUIDS OF DIFFERENT DENSITIES. The perpendicular pressure upon the horizontal base of a vessel containing any number of fluids of different densities, which do not mix in the vessel : Is equal to the area of the base, multiplied by the sum of the products of the specific gravities drawn into the altitudes of the several fluids . But the pressure upon the base, will manifestly be the same, if we suppose the vessel to be filled with a fluid of uniform density, arising from the composition of the densities of the several fluids according to their magnitudes ; or if the magnitudes are equal, the uniform density will be a medium between the several given densities. 180. EXAMPLE. A cylindrical vessel, whose diameter is 6 and alti- tude 24 inches, is filled with mercury, water and olive oil, in the following proportions, viz. mercury 7, water 8, and olive oil 9 inches; what is the pressure on the bottom of the vessel, the specific gravities being 13598, 1000 and 915 respectively? Here, by the principles of mensuration, the area of the bottom of the vessel containing the fluids, is 36 X .7854 28.2744 square inches ; consequently, the pressure produced by the mercury, is ;? = 28.2744x7X13598 = 2691327.0384, and in like manner, the pressure of the water, is p 1 = 28.2744 X8 X 1000 = 226195.2, and lastly, the pressure produced by the oil, is p"~ 28.2744x9x915 = 232839.684 ; and the sum of these is manifestly the whole pressure ; hence we get P =. 2691327.0384 + 226195.2 + 232839.684 z=z 3150361 .9224. If the pressure as here expressed be divided by 1728, the number of solid inches in a cubic foot, we shall have 181. Again, suppose the dimensions of the vessel to remain as above, and let it be filled with the same fluids in equal quantities ; that is, 8 inches of mercury, 8 of water, and 8 of olive oil ; what then is the pressure upon the bottom ? Here, by proceeding as above, we have for mercury, p = 28.2744 X 8 X 1 3598 = 3075802,3296 ; ON THE CONCAVE SURFACE OF A VESSEL. 177 for water, it is p' zz 28.2744X 8 X 1000 226195.2, and for olive oil, it is p" zz 28.2744 X 8 X915 zz 206968. 608 ; hence by summation, the entire pressure on the bottom, is P zz 3075802.3296 -f 226195.2 4. 206968.608 zz 3508966.1376, and lastly, dividing by 1728, we obtain 172o The pressure which we have found in this last instance, is the very same as that which would arise, if the vessel were filled with fluid of a medium density ; for we have i (13598 4- 1000 + 915) zz 5171 medium density ; hence, the entire pressure on the bottom, is P 28.2744 X 24X5171 3508966.1376; which by division, gives 3508966.1376 P - r^rr - zz 2030.6517 ounces, the same as before. 182. Let the conditions of the problem remain as above, and let it be required to determine the pressure on the concave surface of the vessel, and to compare it with that upon the bottom. Let ABGH, as in the preceding case, represent an upright section of the vessel, of which the base HG is parallel, and the sides AH, BG perpendicular to the horizon; and suppose the fluids of different densities to be contained in the strata AC, DF and EG. Bisect the surface and base A B and HG, in the points m and n, and join mn ; then do the centres of gravity of the several cylindric surfaces occur in that line. Draw the diagonals AC, DF and EG, cutting the vertical line mn in the points c, b and a, which mark the places of the respective centres of gravity. Put D zz AB or HG, the diameter of the vessel containing the fluids, \d zz ea, the depth of the centre of gravity of the lower cylindric surface, \d' zz db, the depth of the middle cylindric surface, x)} ; or multiplying by s, we have s vers.(2^> ar) s vers.(^> x} zz: s' vers.a? s' vers.(0 x). Now, by substituting for the several versed sines, their values in terms of the cosines and radius, we shall obtain s{cos.(0 x} cos. (2^ a:)} s'{cos.(0 x) cos. a- } from which, according to the arithmetic of sines, we get s {cos.0 cos.a: -f- sin.0 sin.a; cos.2^ cos. x sin.20 sin. a;} zz: s' {cos.< cos. a; -j- sin.0 sin. a? cos. a;}. Let all the terms of this equation be divided by cos. a;, and it becomes transformed into s cos.0 -{- s sin.0 tan. a; s cos.20 s sin.2^> tan. a: zz: s r cos.0 -f~ s/ X sin. ^ tan. x 5'; therefore, by bringing to one side, all the terms that involve tan. a:, we shall have * sin.^(s' s)tan.a; 5 sin. 20 tan. x n: scos.2^ -|- (*' s)cos.^> s' ; hence, by division, we shall obtain s cos.2<6 4- (s' s) cos.0 5' tan x "~~ - s sin. 20 + ( s> s ) sin.0 (142). If the equation which we have just obtained, be compared with that numbered (141), it will readily appear, that the one might have been deduced immediately from the other, by simply substituting s' for m, and s for unity, in the several terms of the numerator and denominator; but in order to render the formation of the formula more intelligible, we have thought proper to trace the steps throughout. 200. The practical rule for reducing the above equation, will require a different mode of expression from that which we have given in the rule to equation (141), but it will not be more operose ; the rule is as follows. 188 OF THE PRESSURE OF UNMIXABLE FLUIDS OF DIFFERENT DENSITIES RULE. Multiply the difference of the yiven specific gravities, by the natural cosine of the circular space in contact with one of the fluids ; to the product, add the natural cosine of the whole circular space drawn into the less specific gravity , and from the sum subtract the greater specific gravity for a dividend. Again. Multiply the difference between the specific gravi- ties, by the natural sine of the circular space in contact^with one of the fluids, and to the product, add the natural sine of the whole circular space drawn into the less specific gravity, and the sum ivill be the divisor. Lastly. Divide the dividend by the divisor, and the quotient will give the natural tangent of a circular arc, which being found in the tables, enables us to assign the actual position of the fluids when in a state of equilibrium. 201. EXAMPLE. On the inner surface of a circular tube containing mercury and rectified alcohol, it is observed, that when the tube is held in a vertical plane, and the fluids in a state of equilibrium, a space of 75 degrees of the circumference, is occupied by, or in contact with each fluid ; it is required to determine the position of the fluids at the instant of observation, their specific gravities being 14000 and 829 respectively ? In this example there are given /= 14000; s 829; ^ 7 7 its natural sine and cosine equal to .96593 and .25882; 2^>rr: 150, its natural sine being .50000, and its cosine .86603 ; consequently, by proceeding according to the directions contained in the foregoing rule, we shall obtain For the dividend scos.2^ -f (s' s)cos.< s'= 829 X .86603 -|- (14000 829) x .25882 14000 = 11 309 .02065 ; For the divisor s sin .2< (V s) sm.^> 829 X .50000 (14000 829) X .96593 = 1 3136.76403 ; consequently, by division, we obtain ta "-* = = - 86086 = at - tan - 40- 43' 25'. 202. The positions of the fluids in this example, are manifestly very different from what they are in the preceding, the point F in the vertical diameter falling on the other side of the centre; but in this case, we shall leave the construction for the reader's amusement, and proceed to inquire what changes the general formula will undergo, in ORIGINATING THE CONSTRUCTION OF A HYDROSTATIC QUADRANT. 189 consequence of certain assumed spaces of the inner surface, being in contact with each of the contained fluids. If 0:zzhalf a right angle, that is, if each fluid cover a space of 45 degrees; then 20 zz 90, and consequently, sin.20 zr 1 and cos.20:zrO, while sin. 0:=! ^/ 2, and cos.0:=r | ^2 ; therefore, by substitution, equation (142) becomes l(s' - J(s ' Now, let the fluids be mercury and rectified alcohol, as in the pre- ceding example, then we shall have 829 V2- i (14000829)^2 -f 829 which answers to the natural tangent of 28 55' nearly. Again, if 0zza right angle, that is, if each fluid cover a space of 90 degrees on the inner surface of the tube; then, 20zzl80, of which the sine and cosine are respectively and 1, while the sine and cosine of 0, are respectively 1 and ; consequently, by substitu- tion, equation (142) becomes s' -4- s tan.zzz-7-- . /IAA\ s 1 s (!44). 203. This is a very neat and obvious expression, and the practical rule derived from it, may be enunciated in the following manner. RULE. Divide the sum of the specific gravities of the two fluids by their difference, and the quotient will give the natural tangent of an arc, which being estimated from the lowest point of the tube, will indicate the highest point of the heavier fluid. If therefore, the contained fluids be mercury and rectified alcohol, as in the preceding cases, we shall have 14000 -|- 829 ta "-*=14000-829 =U2587 - which answers to the natural tangent of 48 23' 19''. We might assume other particular values of the spaces in contact with the fluids, and thereby deduce corresponding forms of the equa- tion ; but what we have already done on this subject is quite sufficient. CHAPTER VIII. OF THE PRESSURE OF NON-ELASTIC FLUIDS UPON DYKES, EM- BANKMENTS, OR OTHER OBSTACLES WHICH CONFINE THEM, WHETHER THE OPPOSING MASS BE SLOPING, PERPENDICULAR OR CURVED, AND THE STRUCTURE ITSELF BE MASONRY OR OF LOOSE MATERIALS, HAVING THE SIDES ONLY FACED WITH STONE. 1. OF FLUID PRESSURE AGAINST MASONIC STRUCTURES. 204. BEFORE we proceed to develope the theory of Floatation, and to explain the method of weighing solid bodies by immersing them in, or otherwise comparing them with liquids ; it is presumed that it will not be considered out of place, to take a brief survey of the circum- stances attending the pressure of non-elastic fluids, when exerted against dykes or other obstacles, that may be opposed to the efforts which they make to spread themselves. This is an interesting and important subject in the doctrine of Hydraulic Architecture, and since the principles upon which it is founded, depend in a great measure on Hydrostatic pressure, it cannot properly be omitted in unfolding the elementary departments of the Mechanics of Fluids, which come so directly before our view in what is called level cutting in the practice of canal making. Every one knows that in cutting a canal, no further excavation is required than that which will hold the water at a given depth and breadth ; when a bank is made on both sides with the earth excavated, the 'level sur- face of the canal may be elevated above the natural surface of the adjacent land, and in this case great part of the cost of excavation will be saved. But when the canal is to be carried along wholly within embankments, too much attention cannot be paid to the prin- ciples of fluid pressure, if we would avoid unnecessary expense, and at the same time complete the work with systematic regard to its permanent durability ; this therefore is the object of the present OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 191 section, intended as a preliminary article to our Inland Navigation, which will consequently form a part of Hydraulic Architecture. 205. When an incompressible and non-elastic fluid presses against a dyke, mound of earth, or any other obstacle that it endeavours to displace, there are two ways in which the obstacle thus opposed may yield to the effort of the fluid. 1 . It may yield by turning upon the remote extremity of its base. 2. It may yield by sliding along the horizontal plane on which it stands. In either case, the effort to overcome the obstacle, arises from the force which the fluid exerts in a horizontal direction ; and the stability of the obstacle, or the resistance which it opposes to being overcome or displaced, arises from its own weight, combined with the vertical pressure of the fluid upon its sloping surface. 206. When the vertical pressure of the fluid is considered, the investigation, as well as the resulting formulae, a're necessarily tedious and prolix ; but when the effect of the vertical pressure is omitted, the subject becomes more easy, and the computed dimensions are better adapted for an effectual resistance ; but in order to render the inves- tigation general, it becomes necessary to include its effects. Now, it is manifest from the nature of the inquiry, that when an equilibrium obtains between the opposing forces, the momentum of the horizontal pressure must be equal to the momentum of the vertical pressure, together with the weight of the body on which the pressure is exerted ; and for the purpose of showing when this condition takes place, let A BCD represent a vertical section of the dyke, whose resistance is opposed to the pressure of the stagnant fluid, of which the surface is ME and the perpendicular depth EF. Let AB and DC be parallel to the hori- zon, and consequently parallel to one another; and from the points E, A and B, demit the straight lines EF, AK, and BL, respectively perpendicular to DC the base of the section. Take EG any small portion of the sloping side AD, and through the point G, draw the lines GH and 01, respectively parallel and perpendicular to the horizon, constituting the similar triangles EHG, EFD and GID. The figure being thus prepared, it only remains to establish the proper symbols of reference, before proceeding with the investigation. 192 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKM EN TS. Put b zz: DC, the breadth of the section's base, or the thickness of the dyke at the foundation, D zz AK or BL, the perpendicular altitude or height of the section, d =z EF, the perpendicular depth of the fluid whose surface is at EM, I zz: DF, the distance between the near extremity of the base at D, and the perpendicular E F, c zz: DK, the measure of the slope AD, or the distance between the near extremity of the base at D, and the perpendi- cular from the extremity of the opposite side at A, c zz: CL, the distance between the remote extremity of the base at c, and the perpendicular from the extremity of the opposite side at B, or the measure of the slope BC, a zz: ABCD, the area of a vertical section of the obstacle to be displaced, p the horizontal pressure of the fluid on the increment of EG, / zz: the force with which the horizontal pressure operates to overcome the resistance of the dyke, m zz: the momentum of that force, p' = the vertical pressure of the fluid on the increment of EG, /' zr the force with which the vertical pressure resists the dis- placement of the obstacle, tfi'nr the momentum of that force, w zz: the symbol which denotes the weight of the dyke or obstacle of resistance, F =z the force with which it opposes the horizontal pressure of the fluid, M zz the momentum of that force, s zz- the specific gravity of the fluid, s' zz: the specific gravity of the dyke, or opposing body, z zz: EG, any small portion of the sloping side AD on which the fluid presses, z zz: the increment or fluxion of that portion, y zz: EH, the perpendicular depth of the point G, y z= the increment or fluxion of y, x zz: G H, the ordinate or horizontal distance, and x z= the increment or fluxion of the horizontal ordinate or dis- tance G H. Then, since the pressure upon any line or surface, is equal to, or expressed by the magnitude of that line or surface, multiplied by the OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 193 perpendicular depth of its centre of gravity, and again by the specific gravity of the fluid ; it follows, that the horizontal pressure on the increment of EG, is but by the principles of mechanics, the aggregate or accumulated force, with which the horizontal pressure operates to overturn or remove the dyke, is and by taking the fluent of this, it is /= %sy*. But the perpendicular distance from E, at which this force must be applied, is manifestly equal to %y ; for the centre of gravity of the triangle EHG, occurs in the horizontal line passing through that point; therefore, the length of the lever on which the force operates to over- turn the dyke is consequently, for the momentum of the force, we have and when y becomes equal to d, the whole height of the fluid, it is m sd\ (145). Again, the vertical pressure exerted by the fluid on the increment of EG, is obviously equal to the weight of the incumbent column ; that is and this pressure expresses the force, with which the fluid operates vertically to retain the obstacle in its position, or to prevent it from rising to turn about the point c ; consequently, p'=f'=syx. Now, the length of the lever on which this force acts, is evidently equal to ic, the distance between the fulcrum c, and the point i, where the perpendicular passing through G cuts the base DC ; but ic accord- ing to the figure, is equal to DC DF -f i F ; that is ic = b + *; consequently, the momentum of the force /', is m' = syx(b 3 -far), or taken collectively, the momentum on EG, is VOL. I. 194 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. but by reason of the similar triangles EHG and EFD, we have the following proportion, viz. d : 3 : : y : x, from which we obtain : and because the fluxions of equal quantities are equal, it is Let these values of x and x, be substituted instead of them in the preceding value of m', and we shall obtain consequently, by taking the fluent, it becomes there being no correction, since the whole expression becomes equal to nothing when y is equal to nothing. When y becomes equal to d the whole perpendicular height of the fluid, then the foregoing value of m' becomes wf = 8d(J6 >8). (146). The foregoing equations (145) and (146), exhibit the horizontal and vertical momenta of the pressure exerted by the fluid on the sloping side of the obstacle ; and it is manifest from the nature of their action, that they operate in opposition to one another ; the horizontal pressure, endeavouring to turn the body round the point c as a fulcrum or centre of motion, and the vertical pressure tending to turn it the contrary way round the same point, or otherwise to render it more stable and firm on its foundation. 208. But the stability of the dyke is farther augmented by means of its own weight, which being conceived to be collected into its centre of gravity, opposes the horizontal pressure of the fluid with a force, which is equivalent to its own weight drawn into a lever, whose length is equal to the perpendicular distance between the centre of motion, and a vertical line passing through the centre of gravity of the section ABCD. Now, it is manifest from the principles of mensuration, that if the transverse section of the dyke be uniform throughout, the weight is proportional to the area of the section, multiplied into the specific OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 195 gravity of the material of which it is composed, and again into ijs length ; but the length of the dyke is the same as the length of the fluid which it supports ; consequently, the weight is very properly represented by the area of the section and the specific gravity of the material ; thus we have w as'. (147). But according to the writers on mensuration, the area of the trape- zoid A BCD, is equal to the sum of the parallel sides AB and DC, drawn into half the perpendicular distance AK or BL ; hence we have a=(AB -f- DC)XjAK, but by the foregoing notation, it is A B b (c -{- e) ; consequently, by addition, we have A B -f DC = 26 (c + e) ; therefore, the area of the section is a = ID (26 c e); let this value of a be substituted instead of it in the equation marked (147), and it becomes w =. |DS' (26 c e) ; but the weight of the dyke is equivalent to the force whose momentum, combined with that of the vertical pressure of the fluid, counterpoises the momentum of the horizontal pressure, which force we have repre- sented by F ; hence we have F IDS' (26 c e\ and the momentum of this force, is IF M zz JD Is' (26 c e), where / denotes the lever whose length is equal to the distance between the fulcrum, or centre of motion at c, and the vertical line passing through the centre of gravity of the section A BCD ; conse- quently, in the case of an equilibrium, we have mi^.m' -j- M, and this, by restoring the analytical values, becomes Lsd* = s3d(tib 3) -{- ID/*' (26 c e). (148). 209. This is the general equation which includes all the cases of rectilinear sloping embankments, but it has not yet obtained its ultimate form ; for the value of I has still to be expressed in terms of the sectional dimensions, and in order to this, a separate investi- gation becomes necessary. o2 196 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. Thus, let A B c D be a vertical section of the dyke as before, and bisect the parallel /Hn\ sides AB and DC in the points m and w, and join mn\ then, the straight line mn will pass through the centre of gravity of the figure ABCD. Take g a point such, that mg is to ng, as 2n c -j- A B is to BC -}- 2AD, and g will be the centre of gravity sought ; through the points m and g, draw the straight lines mr and gs, respectively perpendicular to DC the base of the section, then is sc the length of the lever by which the weight of the dyke or embankment opposes the horizontal pressure of the fluid. From the points A and B, draw the straight lines AK and BL, re- spectively perpendicular to D c ; then it is manifest from the principles of geometry, that rw= (CL DK), and this, by restoring the symbols for CL and DK, becomes rn~^(e c). But mrzrD; consequently, by the property of the right angled triangle, we have m n* m m i* -f- n r 3 ; or by restoring the analytical values, it is raw 2 =iD 9 4- \(e cf\ therefore, by extracting the square root, we have m n J -Y/ 4o 2 -|- (e -= c)". By the property of the centre of gravity, and according to the foregoing construction, the point g is determined in the following manner. 2DC-{- AB = 36 c e 36 2c 2e 66 3c 3e : i^/4D 2 4-(e c) 8 : : 36 2c 2e : gn, from which, by reducing the analogy, we get (36 2c 2e) v /4o 2 -f- (e c) s 9 n -~ 6 (26-c- e ) and by the property of similar triangles, it is 2c OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 197 wherefore, by reducing the analogy, we obtain _(e c)(3b 2c 2e) 6(26 c e) ' But by referring to the diagram, it will readily appear that scur sn -\- nc ; therefore, by addition, we obtain .36(26 c e) + (e c)(36 2c 2e) 6(26 c e) Let this value of I be substituted instead of it in the equation marked (148), and we shall obtain and this being reduced to its simplest general form, becomes sd 8 ~das(36 S} + 3bvJ(b c)-hDs'(c 8 e 2 ). (149). 210. The general equation in the form which it has now assumed, is very prolix and complicated ; but its complication and prolixity, as we have before observed, are much increased by the introduction of the vertical pressure; if that element be omitted, the equation becomes sd 3 = 3bDs'(b c) + DS'(C* e 8 ). (150). An expression sufficiently simple for every practical purpose ; but it must be observed, that if e 2 be greater than c 2 , the term in which it occurs will be subtractive. We shall not attempt to express these equations in words, or to give practical rules for their reduction ; the combinations are too complex, to admit of this being done in a neat and intelligible manner; it is necessary, however, to illustrate the subject by proper numerical examples, for which purpose, the following are proposed in this place. 211. EXAMPLE 1. The water in a reservoir is 24 feet deep, and the wall which supports it is 30 feet in perpendicular height, the slope of the side next the water being one foot, and that of the opposite side one foot and a half; it is required to determine the transverse section of the wall or dyke, supposing it to be built of materials whose mean specific gravity is 2 J, that of water being unity ? By contemplating the conditions of the question as here proposed, it will readily be observed, that the breadth of the section at the base, is the first thing to be determined from the equation ; for since the quantity of the slopes, as well as the perpendicular height are given, the breadth of the dyke at top can easily be found, when the breadth at the foundation is known. 198 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. In the first place then, let us take into consideration the effect produced by means of the vertical pressure of the fluid ; this will refer us to equation (149), but previously to the substitution of the several numerical quantities, it becomes necessary to assign the nu- merical value of S, which is not expressed in the question, but is determinable from the perpendicular altitudes of the wall and the fluid, together with the slope of that side on which the fluid presses : thus, '-' 30 : 24 : : 1 : 8 = of a foot. Let therefore, the several given numbers replace their representa- tives in equation (149), and we shall have 24 8 =24XX2K3& i)4-3X30x2J(6 1)6 30X2|(1F I 2 ), in which expression, b is the unknown quantity. If the several terms be expanded, collected, and arranged, accord- ing to the dimensions of the unknown quantity, we shall have 2256 s 816=13956.15; complete the square, and we get 2025006* 729006 X 8 1 2 = 12567096, and extracting the square root, it is 4506 81= |/1 2567096 = 3545 nearly; therefore, by transposition and division, we get b 8.05 feet. Consequently, if from the breadth of the foundation as above determined, we subtract the sum of the slopes, the remainder will be the breadth of the dyke at the top ; hence, the section can be deli- neated. 212. The above is the method of performing the operation, when the effect produced by the vertical pressure of the fluid is taken into consideration ; but when that effect is omitted, the process is consi- derably shortened ; for in the first place, there is no occasion to cal- culate the value of S, that term not occurring in equation (150), and in the next place, there are fewer quantities to be substituted for ; this greatly abbreviates the labour of reduction ; but the equation is still of the same degree, and consequently, it must be resolved in the same manner. Let the several given quantities remain as in the preceding case, and let them be respectively substituted in the equation (150), and we shall obtain 2256* 2256= 13917.75; OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 199 if all the terms of this equation be divided by 225, the co-efficient of b*, we shall get b* 6 = 61.856, and this, by completing the square, becomes therefore, by extracting the square root and transposing, we have b rz 8 . 1 4 feet nearly. COROL. It therefore appears, that under the same circumstances, the computed breadth of the foundation differs very little, when the vertical pressure of the fluid is considered, from what it is when the pressure is omitted ; and what is very remarkable, the difference, whatever it may amount to, leans to the side of safety and conve- nience in the case of the omission ; it will therefore be sufficient in all cases of practice, to employ equation (150), but under certain circumstances of the data, it will admit of particular modifications. 213. When the slopes c and e are equal ; that is, when the vertical transverse section of the dyke or embankment, is in the form of the frustum of an isosceles triangle, as represented by ABC D in the annexed diagram ; then, the general equation (149), be- comes transformed into sd*=idSs(3b S) + 36D/(6 c). (151). If the perpendicular height of the dyke, and the depth of the fluid, are equal to one another ; that is, if the water is on a level with the top of the wall ; then, rfzz D and 3 = c, and the above equation becomes sd* = cs(3b c) + 3bs'(b c). (152). Again, if we neglect the effect of vertical pressure, and express the specific gravity of water by unity, we get 3s'(tf cb) = d\ (153). And finally, if both sides of the equation be divided by the quantity 3/, we shall obtain b *- cb = 37- (154). The method of applying this equation is manifest, for we have only to substitute the given numerical values of c, d and /, and the value of b will become known by reducing the equation. 214. EXAMPLE 2. The dyke or embankment which supports the water in a reservoir, is 20 feet in perpendicular height, and it slopes equally on both sides to the distance of 2 feet ; what is the breadth of 200 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. the base, supposing the water to be on a level with the top of the wall, the specific gravity of the materials of which it is built being If, that of water being unity ? Let these numbers be substituted for the respective symbols in the above equation, and we get b* 26 61.619, complete the square, and it becomes &* 26+ 162.619, from which, by evolution and transposition, we get b 8.9 13 feet nearly. Here then, the transverse section of the dyke is 8.913 feet across at the bottom, and consequently it is 8.913 4 4.9 13 feet broad at the top ; hence the delineation is very easily effected. 215. If the slope c should vanish; that is, if the side of the dyke on which the fluid presses A.JB be vertical, as represented by A BCD in the an- nexed diagram ; then S vanishes also, and the equation marked (149) becomes (155). where it is manifest there is no vertical pressure on the dyke, the whole effect of the fluid being exerted in the hori- zontal direction, tending to turn the wall about the remote extremity of its base. When the perpendicular altitude of the wall or dyke, and the depth of the water are equal ; then dizzo, and admitting that the value of 5, or the specific gravity of water is represented by unity, we obtain and this, by transposition and division, becomes *-=^' and lastly, by extracting the square root, we get ' y (156). Let the slope of the dyke be two feet, its perpendicular altitude, or the depth of the fluid 20 feet, and the specific gravity of the material If, as in the preceding example; then, by substitution, we obtain 6= i / 400 +" = 8.804 feet. V 3X1.75 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 201 COROL. The breadth at the base, as determined by this and the preceding equation, exhibits but a small difference, being in excess in the former case, by a quantity equal to 0.109 of a foot; but the breadth at the top in the latter case, exceeds that in the former, by a quantity equal to 1.891 feet; and the difference in the area of the section, is 17.82 feet: it is consequently more expensive to erect a dyke or embankment, with the side next the fluid perpendicular, than it is to erect one of equal stability with both sides inclined or sloping outwards. 216. If the slope e should vanish; that is, if the side of the dyke, opposite to that on which the fluid presses, be perpendicular to the horizon, as represented by A BCD in the annexed diagram, then, the equa- tion (149) becomes sd*=d$s(3b )-}-3Ds'(6 ? c)4-DcY. (157). But when the effect of the vertical pressure of the fluid is omitted, we obtain sd s =i SDS' (b* c&) -f- DC 9 s', (158). and by supposing the altitude of the dyke, and the depth of the fluid to be equal (the specific gravity of the fluid being expressed by unity) ; then we have d D, and the foregoing equation becomes d* = 3s'(b* c&)4-cV; consequently, by transposition and division, we get A* I * CV ~~37~ (159). from which equation, the value of b is easily determined. Let the slope of that side of the dyke on which the fluid presses, be equal to 2 feet, and the perpendicular altitude of the dyke, or the depth of the fluid 20 feet, the specific gravity of the material being If as before ; then by substitution, the foregoing equation becomes ,-==$!!=,, by completing the square, we obtain p 2b + 1 = 74.8571 4- 175.8571 ; consequently, by extracting the square root and transposing, we get b = 9.709 feet. In this case, the dyke has less stability than it has when the perpen- dicular side is towards the water, as is manifest from its requiring a greater section, and consequently, a greater quantity of materials to resist the effort of the pressure which tends to overturn it. 202 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. The sectional area in the one case, is 7. 804 X 20 =: 156.08 square feet, and in the other, it is 8.709x20=: 174.18 square feet, being a difference of 18.1 square feet in favour of magnitude in the latter form, where the sloping side is adjacent to the fluid ; and this being multiplied by the length of the dyke, will give the extra quantity of materials necessary for obtaining the same degree of stability. 217. If both the slopes c and e become evanescent; that is, if the section of the dyke be rectangular, having both its sides perpendicular to the horizon, as represented by ABC D in the annexed diagram ; then, the general equation (149), becomes transformed into sd a = 3vs'b\ (160). Then, by supposing the depth of the fluid, and the perpendicular altitude of the dyke to become equal, (the specific gravity of water being expressed by unity,) we have 3s'& 2 zrd 8 ; and this by division becomes -I- consequently, if the square root of both sides of this equation be extracted, we shall have '- (161). 218. This is indeed a very simple form of the, equation, applicable to the very important case of rectangular walls ; it is however accu- rate, and corresponds in form with that investigated by other writers for the same purpose, and by different methods ; the mode of its reduction is simply as follows. RULE. Divide the specific gravity of the fluid to be sup- ported, by three times the specific gravity of the dyke or embankment, and multiply the square root of the quotient by the perpendicular altitude of the dyke, for the required thickness. Let the perpendicular depth of the water, or the altitude of the dyke be equal to 20 feet, and the specific gravity of the materials of which it is built If, as in the foregoing cases; then, by proceeding according to the rule, we have OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 203 219. There is still another case of very frequent occurrence that remains to be considered, viz. that in which the section is in the form of a right angled triangle, having its vertex on the same level with the surface of the fluid. This case will also admit of two varieties, according as the perpen- dicular side of the dyke is, or is not in contact with the fluid ; when it is in contact with it c vanishes, and since the section is in the form of a triangle, the breadth of the base b is equal to the remote slope e, and the vertical pressure of the fluid on the dyke is evanescent; con- sequently, the equation marked (149) becomes sd* = 3Ds'P DsV; (162). but by the nature of the problem e 2 is equal to 6 2 , and by the hypo- thesis of equal altitudes c?=z= D ; therefore, in the case of water, whose specific gravity is expressed by unity, we obtain Zs'tfi^d*; and from this, by division, we get and finally, by extracting the square root, it is b d\/ _L V 2s'' (163). 220. This is also a very simple expression for the base of the section, and the rule for its reduction is simply as follows. RULE. Divide the specific gravity of the incumbent fluid, by twice the specific gravity of the dyke or embankment, and multiply the perpendicular depth of the fluid by the square root of the quotient, for the required thickness of the dyke. Let the perpendicular altitude and the specific gravity of the wall, be 20 feet and 1 j respectively, as in the foregoing cases, and we shall have 6 = 2 nJ(2b c e). (168). This is the equation of equilibrium, or that in which the resistance of the dyke is counterpoised by the horizontal pressure of the fluid, the effect of the vertical pressure not being considered ; but in order to express the breadth of the base in terms of the other quantities, let both sides of the equation be divided by ons', and it becomes D7ZS consequently, by transposition and division, we obtain d*s 6 = 2^7+ i(c + e): (169). and finally, if the perpendicular depth of the fluid and the height of the dyke are equal, we shall have - - . (170). 225. In order therefore, to illustrate the reduction of the above equation;1by means of a numerical example, we must assume a value to the letter n, having some relation to the nature of the materials of which the resisting obstacle is constructed ; now, it has been found by numerous experiments, that when rough and uneven bodies rub upon one another, or when a heavy body composed of hard and rough materials, is urged along a horizontal plane, the effect of the friction is equivalent to about one third of the weight of the body moved ; or in other words, it requires about one third part of the force applied to overcome the effects of the friction ; and moreover, in the case of a wall built of masonry, there is, in addition to the friction, the adhesion of the materials to the plane on which the wall is built. If therefore, we consider the effect of adhesion to be equivalent to the effect of friction, it is manifest, that their conjoint effects will destroy about two thirds of the force applied ; consequently, in the case of masonry, we may suppose that the value of w, is very nearly equal to 1J, but for other materials it will vary according to the specific gravity or weight. OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 207 Having thus assigned a particular value to the letter n, we shall next proceed to illustrate the reduction of the equation ; for which purpose, take the following example. 226. EXAMPLE 3. The vertical transverse section of the wall which supports the water in a reservoir, is 24 feet in perpendicular height ; what is the thickness at the base of the wall, supposing the section to be in the form of the frustum of an isosceles triangle, the slope or inclination on each side, being equal to 2 feet, and the specific gravity of the material If, that of water being expressed by unity ? Let the several numerical values here specified, be substituted instead of the respective symbols in the equation (170), and we shall obtain _24>O_ ___ nearly. The breadth of the section, or the thickness of the dyke at the bottom, being thus determined, the breadth or thickness at the top can easily be found, for we have 6.571 4 = 2.571 feet. 127. If the slope c should vanish; that is, if the side of the dyke on which the water presses be perpendicular to the horizon ; then, the equation (170), becomes (171). And if the opposite slope e becomes evanescent, while the slope c remains ; then we have * = 2^7 +ic< (172). But if both slopes vanish, or the section of the wall becomes rectangular; then, the equation (170) is (173). If therefore, the perpendicular altitude of the section, and the specific gravity of the materials of which the dyke is composed, remain as in the preceding example ; then we shall have b= 24X1 =4.571 feet. 2XfX 228. When the section of the wall assumes the form of a right angled triangle ; that is, when the slope c vanishes, and e becomes equal to the whole breadth b ; then we have 208 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. ds -W' (174). And exactly the same equation would arise, if the slope e remote from the fluid were to vanish, and the slope c adjacent to the fluid, become equal to b the whole breadth of the section; consequently, the thickness of a dyke in the case of a triangular section, whether the water presses on the perpendicular or hypothenuse of the tri- angle, is In all the preceding cases, it is supposed that the section of the dyke or embankment is of such dimensions, as to oppose an equipois- ing resistance to the pressure of the fluid which it supports ; but in the actual construction of all works of this nature, it becomes neces- sary, for the sake of safety, to enlarge the dimensions considerably beyond what theory assigns to them ; but it does not belong to this place to determine the limits of the enlargement. 2. OF THE PRESSURE OF FLUIDS AGAINST EMBANKMENTS OF LOOSE MATERIALS. 229. The theory which we have established above, supposes that a perfect connection obtains between all the parts of the dyke or embankment which is opposed to the pressure of the fluid, so that any one portion of it cannot be displaced or overthrown, unless the whole be overthrown at the same time; the formulge thence arising, are therefore, only applicable to dykes or embankments that are con- structed of masonry ; in those which are constructed of earth or other loose materials, and having the sides faced or fortified with stone, the same connection between the component portions of the wall does not exist, and consequently, although the several equations apply when the whole perpendicular height of the dyke is considered, yet the dyke will not resist equally at every part of the height, but is liable to be separated into horizontal sections. In order therefore, to adapt our prin- ciples to this case also, it becomes neces- sary to trace out the steps of another investigation ; for which purpose, Let ACD represent a vertical section of the dyke or embankment, whose summit at A is on a level with the surface of the fluid AE. OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 209 Take any point G in the line AGD, and through the point o thus assumed, draw the horizontal ordinate GB, cutting the vertical axis AC in the point B : now, it is required to determine the nature of the curve AGD such, that each portion of the dyke, or of its section, as AGE, estimated from the vertex, may be equally capable of resisting the horizontal pressure of the fluid exerted against AG ; or which is the same thing, that each portion may retain its stability and remain in equilibrio on its base GB; not separating from the lower portion GBCD, either by turning about the point c as a centre of motion, or by sliding in a horizontal direction along the base GB. Put x AE, the abscissa of the curve estimated from the vertex at A, y = EG, the horizontal ordinate corresponding to the ab- scissa x, s z= the specific gravity of the fluid, which endeavours to displace the dyke by pushing it along the lin BG, / zn the specific gravity of the materials of which the dyke is constituted, m m the momentum of the horizontal pressure, m' the momentum of the resistance offered by the dyke, and n :=z the number of times that the adhesion and friction of the dyke are equal to its weight. Then we have already seen, equation (145), that the momentum of the horizontal pressure of the fluid as referred to the point c, is from which, by substituting x s instead of d 3 , we obtain m which equation indicates the momentum of pressure at the point D. But the momentum of the resistance offered by the wall, that is, the momentum of the portion of the section represented by ABG, is m' = is'fy*x; and these momenta in the case of an equilibrium must be equal to one another ; hence we have from which, by taking the fluxion, we shall obtain or by suppressing the common factors, it becomes sx^ s'y*; by extracting the square root of both terms, we get x\/s y\/s'. Now, when x becomes equal to d, the whole perpendicular depth of the fluid, or the altitude of the section ; then y becomes equal to b, the thickness of the dyke, or the greatest breadth of the section ; VOL. i. p 210 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. consequently, if d and b be respectively substituted for x and y in the preceding equation, we shall have d^/7b^S (175). This equation involves the conditions necessary for preventing the dyke from turning about the point B, and if the equation be resolved into an analogy, we shall have b : d : : \/s : ^/sf. COROL. From which we infer, that the section is in the form of a rectilinear triangle, whose base is to the perpendicular height, as the square root of the specific gravity of the fluid, is to the square root of the specific gravity of the wall or dyke. 230. EXAMPLE. The perpendicular altitude of an embankment of earth is 20 feet ; what must be the breadth of its base, so that each portion of it estimated from the vertex, shall resist the effort of the fluid, to turn it round the remote extremity of the base, with equal intensity ; the water and the dyke having equal altitudes, and their specific gravities being 1 and 1 .5 respectively ? Here we have given rfrz 20 feet, szr 1, and s r zz 1 .5 ; consequently, by the preceding analogy, we have ^ 1.5 : ^ 1 : : 20 : 16.33 feet. 231. The conditions necessary for preventing the portion of the section ABG from sliding on its base, may be thus determined. We have seen (art. 220), that the momentum of the horizontal pressure, to urge the section along its base, is m Jse?% consequently, by substituting x 9 for d 2 , we have m Jsa: 2 , but the momentum of the section opposed to this, is m' zz ns'fyx; therefore in the case of an equilibrium, we have isx* = ns'fyx, from which, by taking the fluxion, we obtain sxx~ ns'yx, and by casting out the common factor, we get sx^nns'y. (176). From this equation, when converted into an analogy, we shall obtain x : y : : ntf : s. Which also indicates a rectilinear triangle, whose altitude is to the base, as n times the specific gravity of the embankment, is to the specific gravity of the fluid. If the water presses against the perpendicular side of the wall, the curve bounding the other side, so that the strength of the wall may be every where proportional to the pressure which it sustains, must be a semi-cubical parabola, whose vertex is at the surface of the fluid, and convex towards the pressure. OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 211 232. We are now arrived at that particular division of our subject, which comprehends some of the most interesting and important departments of hydrodynamical science ; it unfolds the principles of floatation, explains the method of weighing solid bodies in fluids, determines the relations of their specific gravities ; and moreover, it investigates the laws of equilibrium, and assigns the conditions neces- sary for a state of perfect or imperfect stability. Every term in this enumeration conveys the idea of mechanical action. Floating bodies, those which swim on the surface of a fluid, which is bulk for bulk heavier than the body afloat, are pressed downward by their own weight in a vertical line passing through their centre of gravity : and they are supported by the upward pressure of the fluid, which acts in a vertical line passing through the centre of gravity of the part which is under the water. When these lines are coin- cident, the equilibrium of floatation will be permanent. In the present instance we have merely to consider the principles of floatation as fluids exhibit the properties of the mechanical powers, as the lever or balance, the screw, &c. The pulley, in lowering a great weight or in lifting it up again, does no more than the ocean tide when it silently recedes and leaves dry, or majestically advances and without effort floats a stupendous ship. The lever or balance does no more than a canal lock effects, when it transfers from one level to another a heavy barge or vessel laden with ponderous commodities. And we behold too the ocean, like a vast screw or press, forcing down to its dark recesses vast masses, which in shipwrecks are sub- merged in its bosom, and which yet might be fashioned to be bulk for bulk much lighter than the devouring flood which has swallowed them up in its insatiable womb. The eternal and immutable laws of Nature, in all these cases, are most satisfactorily accounted for in the doctrine of fluid pressure and support j but this doctrine, like all the rudiments of human skill applied to natural phenomena, must depend on matters of fact, which can only be learned from observation and experi- ment, and which can generally and successfully be applied by the help of mathema- tical and philosophical investigations. This is the only scientific view we ought to take of all those truths that are denominated the phenomena of fluids, whose affections, from a series of concurring experiments, we undertake to expound j or assuming these as established pi-inciples that operate generally in the pressure and elasticity of fluids, we demonstrate them to be adequate to the production, not only of the particular effects adduced to prove their existence and power, but of all similar phenomena. This is the only method by which to make the results of practical men available in scientific discussions, and on the other hand render these discussions the handmaids of genius in constructive mechanics. This is the province of the mathematician; and we shall in the sequel follow it very closely, in expound- ing the doctrine of floatation and the specific gravities of bodies, the laws of equili- brium, and the conditions necessary for a state of perfect or imperfect stability, &c. P 2 CHAPTER IX. OF FLOATATION, AND THE DETERMINATION OF THE SPECIFIC GRAVITIES OF BODIES IMMERSED IN FLUIDS. OF the several particulars with which we concluded the last chapter, we shall speak in order, beginning with the theory of float- ation and the determination of the specific gravity of bodies, the leading principles of which are contained in the following proposition. PROPOSITION III. 233. When a body floats, or when it is in a state of buoyancy on the surface of a fluid of greater specific gravity than itself: It is pressed upwards by a force, whose intensity is equi- valent to the absolute weight of a quantity of the fluid, of which the magnitude is the same as that portion of the body below the plane of floatation.* Let ABC represent a vertical section of a solid body floating on a fluid, whose horizontal surface is DE, mn being the plane of floata- tion, and men the immersed portion of the floating body. Take any two points G and H on the surface of the solid, indefinitely near to each other, and through the points G and H thus arbitrarily as- sumed, draw the straight lines G F and HI, respectively parallel to DE the surface of the fluid, and meeting the opposite sides of the solid in the points r and i, so that each point in either of the intercepted portions GH and FI, may be considered as being at the same perpen- dicular depth h& or ZF below the horizontal surface of the fluid. At H and i erect the perpendiculars H r and i s, which produce to t and w, and through the points G and F, draw the straight lines G b and F/, respectively perpendicular to the surface of the solid in the * The Plane of Floatation is the imaginary plane, in which the floating solid is supposed to be intersected by the horizontal surface of the fluid. OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 213 points G and F ; make ob and F/each equal to /IG or ZF, the perpen- dicular depth of the points G and F below the surface DE; then, according to the principles which we have propounded and demon- strated in the first proposition and its subordinate inferences, the perpendicular pressures upon the indefinitely small portions of the body GH and FI, may be expressed as follows, viz. /> zr s XG H X G &, and p' m s X F i X F/, where s denotes the specific gravity of the fluid, and p, j/ the respec- tive pressures exerted by it perpendicularly to GH and FI, any indefinitely small portions of the floating body. But it is manifest from the resolution of forces, that the pressures of the fluid in the directions bo and y*F, may each be decomposed into two other pressures, the one vertical and the other horizontal ; for by completing the rectangular parallelograms oabc and Tdfe, it is obvious that the pressures in the directions G, CG and dr, ev are, when taken two and two, respectively equivalent to the pressures in the directions bo andy*F. Now, the horizontal pressures CD and CF, by construction are equal to one another, and they operate in contrary directions; consequently they destroy'each other's effects, and the upward vertical pressures on the solid at the points G and F, are respectively indicated by the straight lines G and C?F drawn into the specific gravity of the fluid; therefore, the whole vertical pressures on the indefinitely small por- tions GH and FI, are as follows, viz. j9 = 5XGHXG, and jy'zzrsXFiX^F, where p and p', instead of indicating the perpendicular pressures as formerly, are now considered in reference to the vertical pressures. Since the parallel straight lines GF and HI are indefinitely near to one another, the lines G n and FI may be assumed as nearly straight, and consequently, the elementary triangles Giir and FIS are respec- tively similar to the triangles GBO and F/W; therefore, by the pro- perty of similar triangles, we have G 6 : G a : : G H : G r, and $f: F d : : F i : F s ; and from these analogies, by equating the products of the extreme and mean^terms, we obtain G&XGnziGaXGH, an( j F y XFSZHFC? XFI. Let therefore, the products G&Xor and F/XFS be substituted instead of GiiXctG and FiXc?r in the above values of p and //, and we shall have /; s X G b X G r, and p m s X F/ X F s. 214 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. Now, these pressures are manifestly equal to the weights of the columns Gt and vu considered as fluid, and since the same may be demonstrated with respect to every other portion of the immersed surface, we therefore conclude, that the whole pressure upwards, is equal to the sum of the weights of all the columns Gt, FU, &c. ; that is, to the weight of a quantity of the fluid equal in magnitude to the immersed part of the body ; hence the truth of the proposition is manifest. COROL. From the principles demonstrated above, it follows, that when a solid body floating on the surface of a fluid is in a state of quiescence : The pressure downwards is equal to the buoyant effort ; that is, the weight of the floating body, is equal to the weight of a quantity of the fluid, whose magnitude is the same as that portion of the solid, which falls below the plane of floatation. PROBLEM XXXI. 234. A cylindrical vessel of a given diameter, is filled to a certain height with a fluid of known specific gravity, and a spherical body of a given magnitude and substance is placed in it : It is required to determine how high the fluid will rise in consequence of the immersion of the spherical segment which falls below the plane of floatation. Let ABCD represent a vertical section passing along the axis of a cylindrical vessel, filled with an incompressible and non-elastic fluid to the height ED, EF being the sur- face of the fluid before the sphere whose diameter is mn, is placed in it, and ab the surface after the immersion of the segment tnu, the liquid rising to the height A D. Then it is manifest from the nature of the problem, that the spherical segment tvnwu, together with the quantity of fluid in the vessel, must be equal to the capacity of the cylinder whose diameter is DC, and perpendicular altitude D; for the fluid rises in consequence of the immersion of the segment, and fills the spaces atvE and buw^s all around the vessel ; we have therefore to calculate OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 215 the spherical segment tvmuu, and the cylinders EFCD and a ben, for which purpose, Put d zz ED, the height to which the vessel is originally filled with the fluid, $ == DC, the diameter of the cylindrical vessel, r zz ct, or cv, the radius of the sphere, s = the specific grav ; ty of the fluid in the vessel, s' nr the specific gravity of the floating body, and x D, the height to which the fluid rises on the immersion of the spheric segment. Then, since by the principles of mensuration, the solid content or capacity of a sphere, is equal to two thirds of that of its circumscribing cylinder, it follows, that the capacity of the sphere mvnw, is ex- pressed by 3.1416r 2 x2rXf=: 4.1888^; but^we have elsewhere demonstrated, that the magnitudes of bodies are inversely as their specific gravities ; consequently, the magnitude of the part immersed, is determined by the following analogy, viz. .:.':-. 4.1888^: Now, as we have already observed, the quantity of fluid in the vessel at first, is .7854X^X^7854^, and the capacity of the cylinder formed by the fluid and the spherical segment, is consequently, by addition, we shall have .7854 tf x = .7854^-f H^^i . s and therefore, if all the terms of this equation be divided by the quantity .78543% we shall obtain 16rV ' (177). Or if the height to which the vessel is originally filled, be subtracted from both sides of the above expression, the increase of height in con- sequence of the immersion of the spheric segment, becomes , 16rV d=x 3?7' where &'=: a the increase of height. 216 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 235. Either of these equations will resolve the problem, but the latter form is the most convenient for a verbal enunciation, and the practical rule which it supplies is as follows. RULE. Multiply sixteen times the specific gravity of the sphere, by the cube or third power of its radius ; then, divide the product by three times the specific gravity of the fluid, drawn into the square of the cylinder's diameter, and the quotient will give the increase of height, in consequence of the immersion of the spheric segment. 236. EXAMPLE. A cylindrical vessel whose diameter is 8 inches, is filled with water to the height of 10 inches; how much higher will the water rise, and what will be its whole weight, when a globe of alder of 6 inches diameter is dropped into the vessel ; the specific gravity of alder being equal to .8, when that of water is expressed by unity ? Here, by operating according to the above rule, we get 16rV=16x3x3x3x. 8 z= 345.6, and in like manner we have 33^ = 3x8x8x1 192; consequently, by division, we obtain ar'zr - z= z= 1.8 inches, and the whole height is 11.8 inches. v oo s ly.2 237. If the specific gravity of the globe, and that of the fluid in which it is placed, are equal to one another, then equation (178) becomes , 33* ' (179). In this case it is manifest, that the sphere is wholly immersed in the fluid ; consequently, the increase of height wilt be equal to the altitude of a cylinder, whose diameter is 3, and whose capacity is equal to that of the immersed body ; hence, the method of computa- tion is obvious ; but the practical rule deduced from the equation for this purpose, may be expressed in the following manner. RULE. Divide sixteen times the cube or third power of the radius of the sphere, by three times the square of the cylin- der s diameter, and the quotient will give the increased height of the fluid. 238. EXAMPLE. A cylindrical vessel whose diameter is 12 inches, is filled with fluid to the height of 6 inches; to what height will OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 217 the fluid ascend when a sphere of 4 inches diameter is placed in it, the specific gravities of the fluid and the sphere being equal to one another ? lii this example there are given 5 ~ 6 inches and r zz 2 inches ; therefore, by operating as directed in the rule, we shall have 16r 3 =:16X2 3 128, the dividend, and in like manner for the divisor, we get 3S 2 = 3 X 12 2 432, the divisor ; consequently, by division, we obtain x f Hi= 0.2962 of an inch. Hence it appears, that the height of the fluid in the vessel, is increased by a quantity equal to 0.2962 of an inch, in consequence of the immersion, and the whole height to which it rises, is 6.2962 inches. PROBLEM XXXII. 239. A vessel in the form of a paraboloid, is placed with its vertex downwards and its base parallel to the horizon; now, supposing the vessel to be filled to the w th part of its capacity with a fluid of known specific gravity, and let a spherical body of a given size and substance be placed in it : It is required to ascertain the height to which the fluid will rise, in consequence of the immersion of the spherical segment. Let ABC represent a vertical section passing along the axis of the vessel, whose form is that of a paraboloid, generated by the revolution of the common parabola ; and suppose the vessel to be filled with an incompressible and non-elastic fluid to the height sc, DE being its horizontal surface when in a state of quiescence, before the sphere whose diameter is mn is placed in it; then will a b be the surface or the plane of floatation after the immersion of the segment rnw, the fluid rising to the height ts all around the spherical body. Now, it is obvious from the nature of the problem, that, the solidity of the spherical segment rnw, together with the quantity of fluid in the vessel, is equal to the magnitude of the paraboloid acb, whose 218 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. base is ab and axis cf ; therefore, in order to calculate the solidity of the segment, and that of the paraboloids DCE and acb, Put d n: me, the whole axis or height of the paraboloid, p zn the parameter or latus rectum of the axis, r zz: cr, cm or cw, the radius of the sphere, s the specific gravity of the fluid in the vessel, / zz the specific gravity of the floating body, and x zz tc, the whole height to which the fluid ascends, n being the part originally filled. By the principles of solid mensuration, the capacity or solidity of a sphere, is equivalent to two thirds of that of its circumscribing cylin- der ; consequently, the capacity of the floating sphere, is now, we have demonstrated in another place, that the magnitudes of bodies, are inversely as their respective gravities ; hence we have for that portion actually immersed, *:^: 4.1888,* : Again, by the principles of mensuration, the solidity of a paraboloid is equal to one half the solidity of its circumscribing cylinder, and by the property of the parabola, we have niA*~pd; therefore, the capacity of the paraboloidal vessel, is 3.1416X^^X^1-5708^^, and consequently, the quantity of fluid in it is expressed by But the capacity, or the solid content of the paraboloid a c b, whose axis is tc, becomes consequently, by addition and comparison, we have 4.1888rV \.570Spd* 1 .57(% X* zz -- -f - , s n and dividing all the terms by 1.5708, we get 8rV pd* **- = + - OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 219 and again, if all the terms be divided by p the parameter of the parabola, and the square root be extracted from both sides of the equation, we shall have But because the parameter of a parabola is a third proportional to any abscissa and its ordinate ; it follows, that if b denote the base A B of the paraboloid, of which the axis is d, we shall have let this value of the parameter be substituted instead of it in the above equation, and we shall obtain (180). 240. The following practical rule supplied by this equation, will serve to direct the reader to the method of its reduction. RULE. Multiply thirty -two times the axis of the paraboloid, by the cube or third power of the radius of the sphere drawn into its specific gravity ; then, divide the product by three times the square of the base of the vessel multiplied by the specific gravity of the Jluid, and to the quotient, add the square of the axis or depth of the vessel, divided by the number, which expresses what part of it is occupied by the Jluid ; then, the square root of the sum, will give the height to which the Jluid rises after the immersion of the spheric segment. 241. EXAMPLE. The axis of a vessel in the form of a paraboloid is 27 inches, and the diameter of its mouth is 18 inches ; now, supposing that the vessel is one fifth full of water, into which is dropped a sphere of hazel whose diameter is 8 inches ; to what point of the axis will the fluid ascend, the specific gravity of hazel being 0.6, when that of water is expressed by unity ? By proceeding according to the rule, we get 32X27X4X4X4X.6 = 33177.6, the dividend, and in like manner, for the divisor, we have 3X18X18X1 = 972, the divisor ; consequently, by division, we obtain 220 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. This is the value of the first term under the radical sign, in the expression for x equation (180), and the value of the second term, is 27 8 - = 145.8 ; therefore, by addition and evolution, we obtain a;=^/179.93 = 13.414 inches nearly. 242. If the specific gravity of the ball, and that of the fluid in which it is placed, be equal to one another, then equation (180) becomes _ /32^ ^ 'V -W~ + n' and by reducing the fractions under the radical sign or vinculum to a common denominator, we obtain a? = 243. The practical method of reducing the above equation, is expressed in words at full length in the following rule. RULE. Multiply the cube or third power of the- radius of the sphere, by thirty two times the number which indicates what part of the vessel is occupied by the fluid, and to the product add three times the axis of the vessel drawn into the square of its diameter ; then, divide the sum by three times the square of the vessel's diameter, drawn into the number which denotes what part of it the fluid occupies ; multiply the quotient by the axis of the vessel, and extract the square root of the product , for the height to which the fluid rises. 244. EXAMPLE. Let the dimensions of the vessel and the immersed body, remain as in the preceding example, the vessel containing also the same quantity of fluid ; to what height on the axis will the fluid ascend, supposing its specific gravity to be the same as that of the immersed body ? Here, by operating as directed by the rule, we get 32wr 3 = 32X5X4X4X4 = 320X32 = 10240, and 3b*d= 3X 18 X 18 X27 = 972 X27 = 26244 ; consequently, the sum of the parenthetical terms is 32r 3 4- 3b*d = 10240 + 26244 = 36484, and for the denominator of the fraction, we have 36 7 w = 3X18X18X5 = 324X15 = 4860; OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 221 consequently, by division, we obtain 32r+3ffd_ 36484 36 s n 4860 ~ therefore, by multiplication we shall have 7.507X27 = 202.689, and finally, by evolution, it is x= y 202.689= 14.23 inches. COROL. Hence it appears, that when the specific gravities of the fluid and the immersed body, are equal to one another, the fluid rises in the vessel to the height of 14.23 inches ; but when the specific gravities are to each other as 1 : 0.6, it rises only to 13.414; the reason of the difference, however, is manifest, for in the case of equal specific gravities, the spherical body is wholly immersed ; but when the specific gravities are unequal, only a part of the body falls below the plane of floatation. From the above we deduce the following inferences. 245. INFERENCE 1. If a homogeneous body be immersed in a fluid of the same density with itself: It will remain at rest> or in a state of quiescence, in all places and in all positions. Let ABCD represent a vessel, filled with an incompressible and non-elastic fluid to the height D, and let G be a homogeneous body, of the same density or specific gravity as the fluid. Now, it is manifest, that when the body G is put into the vessel and left to itself, it will by reason of its own weight, sink below ab the original surface, and raise the fluid to the height E D, where the body will be entirely under the fluid, and the whole mass in a state of equilibrium with the surface at EF. Then it is evident, that the body being of the same density as the fluid in which it is placed, it will press the fluid under it, just as much as the same quantity of the fluid would do if put in its stead, and conse- quently, the pressure exerted by the solid, together with that of the superincumbent fluid, presses downwards with the same energy, as if it were a column of fluid of equal depth. Therefore, the pressure of the body against the fluid at H, is equal to the pressure of the fluid against the body there; consequently, 222 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. these two pressures are equal and opposite to one another, and must therefore be in a state of equilibrium, in which case, the body will remain at rest. Hence, the truth of the inference is manifest with respect to a vertical pressure ; but it is equally true in reference to a motion horizontally and obliquely; for the horizontal pressures are obviously equal to one another, and they are in opposite directions; therefore, they are in equilibrio with one another, and no motion can take place. And again, with regard to the oblique pressure, it is evidently compounded of a vertical and horizontal one ; but we have just demonstrated that these are equal and opposite; consequently, the body can have no oblique motion, it must therefore remain at rest in any place and in any position. If the specific gravity of the immersed body be greater than that of the fluid, the pressure downwards will exceed the pressure upwards ; consequently, the weight of the body will overcome the resistance of the fluid under it, and it will therefore sink to the bottom. But if the specific gravity of the body be less than that of the fluid, the pressure upwards will exceed the pressure downwards ; therefore, the buoyant principle will overcome the weight of the solid, and it will rise to the surface of the fluid. 246. INF. 2. If a solid body be immersed in a fluid, and the whole mass be in a state of equilibrium : The pressure upwards against the base of the body, is equal to the weight of a quantity of fluid of equal magnitude, together with the weight of the superincumbent fluid. 247. INF. 3. If a solid body be placed in a fluid of greater or less specific gravity than itself : The difference between the pressures downwards and up- ivards is equal to the difference between the weight of the solid and that of an equal bulk of the fluid. 248. INF. 4. Heavy bodies when placed in fluids have a twofold gravity, the one true and absolute, the other apparent or relative. Absolute gravity is the force with which bodies tend downwards. By reason of this force, all sorts of fluid bodies gravitate in their proper places, and their several weights, when taken conjointly, compose the weight of the whole ; for the whole is possessed of weight, as may be experienced in vessels full of liquor. OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 223 Apparent or relative gravity, is the excess of the gravity of the body above that of the fluid in which it is placed. By this sort of gravity, fluids do not gravitate in their proper places ; that is, they do not preponderate ; but opposing one another's descent, they retain their positions as if they were possessed of no weight. 249. INF. 5. If a heavy irregular heterogeneous body descends in a fluid, or if it moves in any direction, and a straight line be drawn, connecting the centres of magnitude and gravity of the body : It will so dispose itself as Jo move in that line, the centre of gravity preceding the centre of magnitude . This is a manifest and a beautiful fact ; for the centre of gravity being surrounded by more matter and less surface than the centre of magnitude, it will meet with less resistance from the fluid ; conse- quently, the body will so arrange itself, as to move in the line of direction with its centre of gravity foremost. 250. What has been here adverted to, in regard to bodies of greater density or specific gravity sinking in a fluid, must only be understood to apply to such as are solid ; for if a body be hollow, it may swim in a fluid of less specific gravity than that which is due to the substance of which the body is composed ; but if the hollows or cavities are filled with the fluid, the body will then descend to the bottom. Again, if bodies of greater specific gravity than the fluid in which they are placed, be reduced to extremely small particles, they may also be suspended in the fluid ; but the principle or force by which this is effected, does not belong to hydrodynamics. PROPOSITION IV. 251. If a solid homogeneous body, be placed in a fluid of greater or less specific gravity than itself: It will ascend or descend with a force, which is equivalent to the difference between its own weight, and that of an equal bulk of the fluid. The principle announced in this proposition is almost self-evident, yet nevertheless, it may be demonstrated in the following manner. Put m the common magnitude of the body and the fluid, w' the weight of the solid body, s' its specific gravity. 224 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. w zz the weight of an equal quantity of the fluid, s zz: its specific gravity, and / zz the force with which the body ascends or descends in the fluid. Then, because as we have elsewhere demonstrated, the absolute weights of bodies, are as their magnitudes and specific gravities ; it follows, that w zz: ms, and w' zz ms' ; but according to the third inference preceding, the difference between the pressures downwards and upwards : Is equal to the difference between the weight of the solid body, and that of an equal bulk of the fluid. But the difference between the upward and downward pressures, is equivalent to the force of ascent and descent ; consequently, we have /zz: w^w'-mms^ms', and this, by collecting the terms, becomes /z=m(sv-s'). (182). If, therefore, the specific gravity of the solid be less than that of the fluid, the force of ascent will be /z=m(s-/); but when the specific gravity of the solid exceeds that of the fluid, the force of descent becomes /zz:m for the weights which the body loses in air and in water ; but we have deduced it as an inference from Proposition V., that when the same body is weighed in different fluids, it loses weights in proportion to the specific gravities of the fluids in which it is weighed ; consequently, we have x w' : x w : : s' is; therefore, by making the product of the mean terms equal to the product of the extremes, we have s (x w') s' (x w), and from this, by separating the terms, and transposing, we get (s s') x ~ s w' s'w; consequently, by division, we obtain __sw' s' w s s' (185). 268. The equation in its present form, supplies us with the follow- ing practical rule for its reduction. RULE. Divide the difference between the products of the alternate weights and specific gravities, by the difference of the specific gravities, and the quotient will be the real weight of the body. 269. EXAMPLE. A certain body when weighed in water and in air, is found to equiponderate 12 and 13.9975 Ibs. respectively; what is its real weight, the specific gravities of air and water being as 1 to .0012? Here, by operating as directed in the rule, we have 1X13.9975 .0012X12 x .0012 14 Ibs. From which it appears, that a body of 14 Ibs. avoirdupois, will completely fulfil the conditions of the question. OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES, 233 PROBLEM XXXV. 270. If the weights which a body indicates, when weighed in air and in water, are exactly ascertained : It is required from thence to determine the specific gravity of the body, the specific gravities of air and water being known. Here also, as in the case of the preceding Problem XXXIV., and Proposition V., the aid of a diagram is not required ; for it would be totally inconsistent with scientific precision, to denote the specific gravities of bodies by geometrical magnitudes, Put W i= the real weight of the solid body, w =z the weight when weighed in water, w' zz the weight when weighed in atmospheric air, s =. the specific gravity of water, expressed by unity, s' =. the specific gravity of air, and S = the required specific gravity of the solid body. Then, according to the principle announced and demonstrated in the 5th proposition, we have W w : W ::s : S ; where it is manifest, that W w; expresses the weight which the body loses by being weighed in water ; therefore, we have W (W -w) : W : : Ss : S; or by equating the products of the extremes and means, we get Sw = (Ss)Vf; and by proceeding in a similar manner when the body is weighed in air, we obtain Now, from the first of these equations, we have and from the second, it is W = (S=7) ; hence by comparison, we obtain Sw Sw' 234 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES, and by taking away the denominators, we get Sw s'w~Sw' s uf, from which, by transposing and collecting the terms, we obtain (10* w)S-=.sw' s'w, and finally, by division, we have sw' s' w '' w' w ' (186). 271. The practical rule or method of reducing this equation, may be expressed in words in the following manner. - RULE. Divide the difference between the products of the alternate weights and specific gravities, ly the difference of the weights when weighed in air and in water, and the quotient will express the specific gravity of the body. 272. EXAMPLE. A certain body when weighed in water indicates exactly 12 Ibs. avoirdupois ; but when the same body is weighed in air, it indicates 13.9975 Ibs.; required the specific gravity of the body, the specific gravities of water and air being as in the preceding problem, or as 1 to .0012 ? The process performed according to the directions given in the rule, or after the manner indicated in equation (186), will stand as follows. 1X13.9975 .0012X12 13.9975 12 Therefore, a body whose specific gravity is seven times the specific gravity of water, will fulfil the conditions of the question. - PROBLEM XXXVI. 273. If the weights which a solid body indicates, when weighed in air and in water, together with its specific gravity and real weight, are exactly ascertained : It is required from thence, to determine the magnitude of the body, on the supposition that it is globular. If the specific gravity of the body and its real weight were unknown, the solution of the present problem would include that of the two pre- ceding ones ; but in order to abbreviate the investigation, we have supposed the specific gravity and the real weight of the body to be given ; the process of the solution is therefore as follows. OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 235 Put W zz the real weight of the globular body, S zz its specific gravity, w zz the weight which the body indicates when weighed in water, s zz the specific gravity of water, vf zz the weight which the body indicates when weighed in air, s' zz the specific gravity of air, and d zz the required diameter of the solid body. Then, according to the principles of mensuration, the solidity of a globe is expressed by the cube or third power of its diameter, multi- plied by the constant decimal .5236 ; therefore, we have dx dxd* .5236 zz .5236d 8 ; but it has been stated in a former part of this work, that the absolute weight of any body, is expressed by its magnitude drawn into the specific gravity ; hence we have Wz=.5236Sd 3 ; consequently, by division, we obtain and from this, by extracting the cube root, we get .52365* (187). 274. The equation in its present form, expresses the diameter of the body in terms of its absolute weight and specific gravity ; this is certainly the simplest and only mode of determining the magnitude of any body or quantity of matter, when the weight and specific gravity are known a priori ; but when this is not the case, we must have recourse to other methods ; and a very elegant and simple one, consists in weighing the body in water and in air, as implied in the problem, and then proceeding as follows. By equation (185), Problem XXXIV., it appears, that the real or absolute weight of the solid, expressed in terms of its relative weights, and the specific gravities of the fluids in which it is weighed, viz. water and air, is !j /-V^ s s 1 ' and by equation (186), Problem XXXV., the specific gravity of the solid expressed in terms of the same quantities, is 236 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. _ sw* s'w w' w But the real or absolute weight of any body, is expressed by its magnitude drawn into the specific gravity ; consequently, we have s w> s/ w _ W - W let this value of the real weight be compared with that above, and we shall have .5236 d 3 (s w' s'w) _ sw' s' w w w s s' If the expression (sw' s'w) be suppressed on both sides of the above equation, we shall obtain .5236rf_ 1 w' w s s' ' and again, by suppressing the denominators, we get .5236(5 t')d* = u/ w; therefore, dividing by . 5236 (s 5'), we have ".5236(5 57 and finally, by extracting the cube root, we obtain .5236(5 5') (188). 275. Now, the methods of reducing the equations (187) and (188), or the practical rules derived from them, may be expressed as follows. 1. When the absolute weight and specific gravity are given. RULE. Divide the absolute weight of the body, by .5236 times the specific gravity, and the cube root of the quotient will be the diameter of the solid sought. 2. When the weights indicated by the body in water and in air are given. RULE. Divide the difference between the weights, as obtained from weighing the body in air and in water, by .5236 times the difference between the specific gravities of water and air ; then, the cube root of the quotient will be the diameter of the solid sought. 276. EXAMPLE 1. The absolute weight of a globular body is 14 Ibs.* and its specific gravity 7 ; what is its diameter ? OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 237 This example corresponds to equation (187), and must therefore be resolved by the first case of the foregoing rule, observing to bring the numerator into the same denomination with the denominator, that is, reducing Ibs. avoirdupois into ounces; or thus, 14Xl6:zrW, the absolute weight, from which we get Hence it appears, that a globular body, whose specific gravity is seven times greater than that of water, will weigh 14 Ibs. when its diameter is 3.9313 inches, which corresponds very nearly with a globe of cast iron. 277. EXAMPLE 2. A globular body whose specific gravity and absolute weight are unknown, indicates 12 Ibs. avoirdupois when weighed in water, and 13.9975 Ibs. when weighed in air; what is its magnitude, the specific gravity of water and air being to one another as the numbers 1 and .0012 ? This second example corresponds to the conditions represented in equation (188), and must therefore be resolved by the second case of the foregoing rule, the numerator being brought into the same deno- mination with the denominator, or the Ibs. avoirdupois being turned into ounces, as (13.9975 12) 16, from which we obtain l - 3 -^- 7 ^= r =^^ 4 ^^9ieg nearly 4 inches, the same $**, .5236(1 .0012) as above. From the principles established in the foregoing Proposition (V), and the problems derived from it, we deduce the following inferences. 278. INF. 1. When bodies of equal weights, but of different magni- tudes, are immersed in the same fluid : The weights which they lose, are reciprocally proportional to their specific gravities, or directly proportional to their 279. INF. 2. When a solid body is weighed in air, or in any other fluid whatever : The difference between its absolute weight, and the weight exhibited in the fluid, is the same as the weight of an equal bulk of the fluid. 280. INF. 3. If two solid bodies of different magnitudes, when weighed in the same fluid indicate equal weights : The greater body will preponderate when they are brought into a rarer medium. 238 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 281. INF. 4. If two solid bodies of different magnitudes, indicate equal weights when weighed in the same fluid : The lesser body will preponderate when they are placed in a denser medium. 282. INF. 5. If two or more solid bodies, when placed in the same fluid, sustain equal diminutions of weight : The magnitudes of the several bodies are equal among themselves. This is manifest, for the losses of weights are as the weights of the quantities of fluid displaced ; and these are as the magnitudes of the bodies which displace them. PROBLEM XXXVII. 283. If two bodies of equal weights, but different specific gravities, be exactly equipoised in air, and then immersed in a fluid of greater specific gravity, the smaller body will prevail : It is therefore required to determine, what weight must be added on the part of the greater body, to restore the equilibrium. Put s zz: the specific gravity of the fluid, in which the bodies are immersed, after being equipoised in air, / zz: the specific gravity of the greater body, s" zz: the specific gravity of the smaller body, w the common weight of each, w' zz: the weight lost by the greater body, by reason of the immersion, w" zz: the weight lost by the lesser body, and x zz: the weight which must be added to restore the equi- librium. Then, because by the preceding proposition, when a body is im- mersed in a fluid, the weight which it loses, is to its whole weight, as the specific gravity of the fluid is to that of the body ; it follows, that s r : s : : w : w' ', and this, by reducing the proportion, gives w' zz: > S Again, for the weight lost by the lesser body, we have s" : s : : w : w" : OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 239 which by reduction gives ws =-F- Now, it is manifest, that the weight required to restore the equili- brium, must be equal to the difference between the results of the above analogies ; therefore, we obtain H __ w s ws ~~7 7 r ' which, by a little farther reduction, becomes X = 77' ' (189). 284. The practical rule which this equation affords, may be ex- pressed in words at length in the following manner. RULE. Multiply the difference between the specific gravities of the bodies, by their common weight in air, drawn into the specific gravity of the fluid in which they are immersed, and divide the result by the product of the specific gravities of the bodies, for the weight to be added in order to restore the It may be proper here to observe, that the weight determined by this rule must not be immersed in the fluid, it must only be attached to that side of the balance on which the greatest weight is lost. 285. EXAMPLE. Suppose that 84 Ibs. of brass, whose specific gravity is 8.1 times greater than that of water, is equipoised in air by a piece of copper, whose specific gravity is 9 times greater than that of water ; how much weight must be applied to the ascending arm of the balance to restore the equilibrium, the same being destroyed by immersing the bodies in water, of which the specific gravity is expressed by unity ? Here, by attending to the directions in the rule, we get 84X1X(9 8.1) .. *= 8.1X9 =l-0371bB. Hence it appears, that if a mass of brass and of copper, each equal to 84 Ibs. when weighed in air, be immersed in a vessel of water, the copper will preponderate, in consequence of its greater specific gravity; and in order that the equilibrium may be again restored, a weight of 1 .037 Ibs. must be attached to the ascending arm of the balance, or that from which the brass is suspended . 240 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES* PROBLEM XXXVIII. 286. If two bodies of different, but known specific gravities, equiponderate in a fluid of given density : It is required to determine the ratio of the quantities of matter which they contain. Put 5 the specific gravity of the fluid, in which the bodies are found to equiponderate, m zz the magnitude of the greater body, s' zz its specific gravity, m'zz the magnitude of the lesser body, s" zz its specific gravity, w zz the weight of the greater body in the fluid, and w/zz the weight of the lesser body under the same circum- stances. Then, by Proposition V., when a solid body is immersed in a fluid of different specific gravity, the weight which it loses, is to its whole weight, as the specific gravity of the fluid, is to the specific gravity of the solid ; it therefore follows, that / : s : : m s' : m s zz the weight lost by the greater body ; but the weight of the body in the fluid, is manifestly equal to the difference between its absolute weight, and that which it loses in consequence of the immersion ; hence we have w zz m s' m s zz m (s' s) ; and by a similar mode of procedure, we obtain s" : s : : m 1 s" : m' s zz the weight lost by the lesser body; consequently, the weight which it possesses in the fluid, is w' zz m 1 s" m' s zz m' (s" s). Now, according to the conditions of the problem, these are in equilibrio with one another; therefore by comparison, we have m (s' s) zz m'(s" s), and by converting this equation into an analogy, it is m : m' : : (s" s) : (s' s); and finally, if we multiply the first and third terms by s', and the second and fourth by s", we shall have ms' : m's 11 : : s'(s" s} : s"(s's). 287. EXAMPLE. Twenty ounces of brass, whose specific gravity is eight times greater than that of water, and a piece of copper whose OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 241 specific gravity is nine times greater, are in equilibrio with one another in a fluid whose specific gravity is unity ; required the weight of the copper ? Here we have given m s' = 20 ounces ; / = 8 ; s" 9, and s \ ; consequently, by substitution, the above analogy becomes 20 : m's" : : 8(91) : 9(81); and by equating the products of the extremes and means, we get 64mVzzl260, and dividing by 64, we have m 1 s" in =19-f^- ounces. It therefore appears, that 20 ounces of brass and 19|^ ounces of copper, are in equilibrio with each other, when immersed in a fluid whose specific gravity is unity ; but if put into a fluid of greater density, the copper will prevail. PROBLEM XXXIX, 288. Suppose a cylinder and cone, of the same altitude, base, and specific gravity, to balance each other at the extremities of a straight lever, when immersed in a fluid of given density ; the cone being suspended at the vertex, and the cylinder at the extremity of the axis. Now, suppose a cone equal to the one proposed, to be abstracted from the cylinder, and its place sup- plied by another of the same magnitude and half the specific gravity ; it is manifest that in this state, the cone will prepon- derate : It is therefore required to determine, how much must be taken from the cone, in order that the equilibrium may be again restored. Let AB be a straight inflexible lever, supported upon and easily moveable about the fulcrum F, and let the cone CDE and the cylinder GHIK, (equal in altitude, base, and specific gravity,) be suspended from the extremities at A and B. Then it is manifest, that in conse- quence of the equality of the bases ,y. and altitudes, the magnitude of the cylinder is equal to three times the magnitude of the cone ; and since the specific gravities of the VOL. i. B 242 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLED BODIES. bodies are the same, it follows also, that the weight of the cylinder is equal to three times the weight of the cone ; consequently, by the principles of the lever, the length of the arm AF, is three times the length of the arm BF; for it is a well known property in the doctrine of mechanics, that when two bodies of different weights are in equi- librio on the opposite arms of a straight lever : The lengths of the arms are to each other, reciprocally as the weights of the suspended bodies. Now, suppose the cone LKI, which is obviously equal in magnitude to CDE, to be abstracted from the cylinder, and to have its place sup- plied by another cone of half the specific gravity as the former ; then it is evident, that if the cone CDE is suffered to retain its magnitude, it will preponderate and cause the cylinder to ascend ; it is therefore necessary, in order that the equilibrium shall'not be disturbed, to diminish the magnitude, and consequently the weight of the equili- brating cone; and for the purpose of assigning the quantity of diminution, Put m m the magnitude of the conical body CDE, m'~ the magnitude of the cylindrical body GHIK, m"-=. the magnitude of the remaining portion cab, w 1=1 the weight which the cone loses in the fluid, w' =. the weight lost by the cylinder, w" the weight lost by the remaining cone cab, s =n the specific gravity of the fluid, and s' =i the specific gravity of the cone and cylinder. Then, since the weight which a body loses by being immersed in a fluid, is to its whole weight, as the specific gravity of the fluid is to the specific gravity of the body, we have s' : s : : m s' : w ; therefore, by equating the products of the extremes and means, it is wmmsi=. the weight lost by the cone ; but according to the principles of mensuration, the magnitude of a cylinder is equal to three times the magnitude of a cone of the same base and altitude ; consequently, we have m' nr 3m, and for the weight lost by the cylinder, we get s' : s : : 3ms 1 : w' ; Horn which, by equating the product of the extremes and means, we obtain w/zr 3ms~m's, OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 243 and in like manner, the weight lost by the cone cab, is found to be w"m"s. But the weights which the several bodies possess in the fluid, are manifestly equal to the difference between the absolute weights and the weights lost ; and the absolute weights are equal to the magni- tudes drawn into the specific gravities ; therefore, we have ms' ms zz m (s' 5) zz the weight of the cone in the fluid, 3ms' 3ms3m(s' s)zzthe weight of the cylinder, m"s' w" szzm" (s r s) zz the weight of the remaining cone. Now, if from the weight which the cylinder possesses in the fluid, we subtract the corresponding weight of the cone, and to the remainder add the weight of another cone of equal magnitude and half the specific gravity ; then, the reduced weight of the cylinder in the fluid becomes 2ws' 4- J*w*' 3mszzwi(2Js' 3s). But according to the conditions of the problem, this weight is to be in equilibrio with the weight of the remaining cone; therefore, by the property of the lever, we have ro(2js' 3s) : m"(s' s) : : 3 : 1 ; and from this, by equating the products of the extremes and means, we get 3m" (s' s)=m (2 Is' 3s), in which equation ra" is unknown ; in order ^therefore to determine its value, divide both sides of the equation by 3(s' 5), and it becomes 6(s' s) ' (190), But this that we have determined, is the magnitude of the part which remains, whereas the problem requires the magnitude of the part to be cut off; now, the magnitude of the whole cone is m ; con- sequently, by subtraction, we have __ m(5s' 65) ms' - 6 (s--s) : - 6(7=0' 09')- 289. The practical rule for reducing this equation is very simple, it may be expressed in the following manner. RULE. Multiply the magnitude of the cone by its specific gravity, and divide the product by six times the difference between the specific gravity of the cone and cylinder, and that of the fluid, and the quotient will give the magnitude of the part to be cut off, in order to restore the equilibrium^ R 2 244 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES, 290. EXAMPLE. Suppose that a cone and cylinder of copper, whose specific gravity is nine times greater than that of water, are immersed in a fluid, whose specific gravity is 1.85, and placed under the con- ditions specified in the problem ; how much must be cut from the lower part of the cone to restore the equilibrium, the diameter of the bases and the altitude of the cone and cylinder being respectively 2 and 5 inches ? Here, by operating according to the rule, we shall have 5.236* X 9 47.124 = 6(9=L85)= : 1^ the part to be cut off from the cone, in order that the remainder may equipoise the cylinder; or we may calculate the magnitude of the equipoising cone by equation (190), in the following manner. 5.236(5X9 6X1.85) m" ^ ~ i4.135 cubic inches; o(y l.oo) which, by subtraction, gives 1.098 cubic inches for the part to be cut off. Put d =. the diameter of the base of the cone and cylinder, and h ~ the common height or altitude. Then, the equations (190) and (191) will become transformed, in terms of the dimensions, into those that follow, viz. ,,__.2618 s' (TV" w), and finally, by division, we shall obtain __ ?v' s'(w" TV) ~ w + rv'-w" ' (199). 305. The practical rule by which the reduction of the above equa- tion is effected, may be expressed in words at length in the following, manner. RULE. From the weight of the vessel with the solid in it, when filled up with water -, subtract the weight of the vessel when full of water only ; then multiply the remainder by the specific gravity of the air at the time of observation, and subtract the product from the iveight of the solid in air for a first number. To the vjeight of the vessel when full of water, add the weight of the solid when weighed in air, and from the sum, subtract the weight of the vessel with the solid in it, when filled up with water, and the remainder mill be a second number. Divide the first number by the second, and the quotient will give the specific gravity of the solid. 306. EXAMPLE. The weight of a vessel when full of water is 68 Ibs. avoirdupois, and the weight of a solid body when weighed in air of a medium temperature, is 34 Ibs. ; now, when the solid is placed in the vessel, its bulk of water is expelled, and the vessel being then weighed, is found to indicate 86 Ibs. ; required the specific gravity of the solid body? When the air is of a medium temperature, its specific gravity is very nearly expressed by the fraction 0.0012, that of water being unity; therefore, by proceeding according to the rule, we have _ 34 -.0012(86- 68) _ 68 + 3486 which, by referring to a table of specific gravities, is found to correspond very nearly with the opal stone, a silicious material of very great value, for the senator Nonius preferred banishment to parting with his favourite opal, which was coveted by Antony. I CHAPTER XL OF THE EQUILIBRIUM OF FLOATATION. BY the equilibrium of floatation is generally meant the position of a floating body^ when its centre of gravity is in the same vertical line with the centre of gravity of the displaced fluid. When the lower surface of the floating body is spherical or cylindrical, the centre will coincide with the centre of the figure ; as, in all circum- stances, the height of this point, as well as the form of the volume of fluid displaced, must remain invariable. In the next proposition, we shall prove, that the place of the centre is determined by the doctrine of forces combined with the elementary principles of hydrostatics, by considering the form and extent of the surface of the displaced portion of the fluid, compared with its bulk, and with the situation of its centre of gravity. Our inquiry will be found to embrace also rectangular figures ; solids in the form of paraboloids and cylinders, together with the equilibrium of floatation of solids immersed in fluids of different specific gravities ; the theory of construction of the aerometer ; the hydrostatic balance, and the method of weighing solid bodies in fluids. The reader will therefore consider this syllabus as the paraphrase of a definition for the term equilibrium of floatation. PROPOSITION VI. 307. When a body floats in a state of equilibrium on the surface of an incompressible and non-elastic fluid : The centre of gravity of the whole body, and that of the part immersed ; or which is the same thing, that of the fluid displaced, must occur in the same vertical line* Let ADBH be a vertical section passing through G and g, the centres of gravity of the whole body AHBD; and of the immersed part A DB, which falls below EF the plane of floatation. * The vertical line, which passes through the centres of gravity of the whole body, and the part immersed beneath the plane of floatation, is generally denomi- nated the line of support. 256 OF THE EQUILIBRIUM OF FLOATATION, For since g is the centre of gravity of the immersed part of the body, or of the fluid ADB, it is also the centre of all the forces or weights of the particles of fluid in ADB, tending downwards; but because the body is at rest, the same point g is also the centre of all the pressures of the fluid tending upwards, by which the weight of the body ADB H is sustained in a state of equilibrium. Now, it is manifest, that the sum of all the downward forces, is equal and opposite to the sum of all the upward forces, otherwise the body could not be in a state of rest ; but the direction in which the weight of the body tends downwards from c, is perpendicular to the horizon; consequently, the line CD which passes through G and g, the centres of gravity of the whole body and of the part immersed, must also be perpendicular to the horizon ; for if it is not, the body must have a rotatory motion, but according to the hypothesis, the body is at rest; therefore, the line CD is perpendicular to the horizon. From Propositions III. and VI., it is obvious, that for a floating body to remain at rest, or in a state of equilibrium, two conditions must obtain, and these are, The weight of the floating body, and that of the fluid dis- placed, must be equal to one another. This is manifest from the inference under the third Proposition, and the second condition to which we have alluded forms the substance of Proposition VI., viz. The centre of gravity of the whole body and that of the part immersed, or of the fluid displaced, occur in the same vertical line. It is extremely obvious, from the nature of the subject, that both the above conditions must have place ; for if the first do not obtain, the body will ascend or descend in a direction which is perpendicular to the horizon ; and if the second fail, the body will turn about its centre of gravity, until the centre and that of the fluid displaced occur in the same vertical line ; and if both these conditions fail together, the body will partake of a progressive and a rotatory motion at one and the same time. From the Proposition which we have demonstrated above, two or three very useful inferences may be deduced, as follows. OF THE EQUILIBRIUM OF FLOATATION. 257 308. INF. 1. If any homogeneous plane figure be divided symme- trically by its vertical axis, and placed in a fluid of greater specific gravity than itself : It will remain in equilibria with its bisecting axis vertical. 309. INF. 2. If any homogeneous solid, generated by the revolu- tion of a curve round its vertical axis, be placed in a fluid of greater specific gravity than itself: It will remain in equilibria in that position, that is, with its axis vertical. 310. INF. 3. If in any homogeneous prismatic body, whose axis is horizontal, the centre of gravity of the section made through its middle parallel to its base, be in the same vertical line with the centre of gravity of that part of the solid which falls below the plane of float- ation : The body will remain in equilibria in that position, if placed in a fluid of greater specific gravity than itself. This is manifest, for the centres of gravity of the whole prism, and of the part immersed, may be conceived to lie in those points, and consequently, the prismatic body is in a state of equilibrium. PROPOSITION VII. 311. When a solid body floats upon a fluid of greater specific gravity than itself, and has attained a state of equilibrium : The magnitude of the body, is to that of the part immersed below the plane of floatation, as the specific gravity of the fluid is to that of the floating body. For by the inference to the third proposition, when the body floats in a state of equilibrium": The weight of the floating body, is equal to the weight of a quantity of the fluid, whose magnitude is the same as that portion of the solid which falls below the plane of floatation. And according to this principle, the truth of the above Proposition is demonstrated ; for, Put m = the magnitude of the whole floating body, m' the magnitude of the part immersed, s := the specific gravity of the floating body, w its weight, VOL. I. S 258 OF THE EQUILIBRIUM OF FLOATATION. s' n: the specific gravity of the fluid, and w/iz: the weight of a quantity of the fluid, of the same magni- tude as that part of the body which falls below the plane of floatation ; then, according to the above in- ference just stated, we get w~w. But because the weight of any body is expressed by the product of its magnitude drawn into its specific gravity ; it follows, that w ms, and w' m's', consequently, by comparison, we have ms = m's'. (200). Therefore, if this equation be converted into an analogy, the truth of the Proposition will become manifest ; for m : m' : : s' : s, being precisely the principle which the Proposition implies. From the principle demonstrated above, various curious and inte- resting questions may be resolved, and by selecting a few which point directly to practical subjects, the information afforded by their reso- lution will sufficiently repay the labour of an attentive perusal. 312. EXAMPLE I. A cubical block of fir, whose specific gravity is 0.55, floats in equilibrio on the surface of a fluid whose specific gravity is 1.026 ; how much of the block is above, and how much below the plane of floatation, the entire magnitude being equal to 512 cubic inches ? Here, by the Proposition, we have 512 : m' : : 1.026 : 0.55, and from this, by equating the products of the extreme and mean terms, we get 1.026m' zz 281.6, and finally, dividing by 1.026, we obtain m'zz ' m 274. 464 cubic inches. It therefore appears, that the quantity of the solid immersed below the plane of floatation, is 274.464 cubic inches ; consequently, the part extant is 512 274.464 = 237.536 cubic inches, being less than half the magnitude of the body, by 18.464 cubic inches. 313. EXAMPLE 2. Let the specific gravities and the magnitude of the body remain as in the last example ; what weight must be added OF THE EQUILIBRIUM OF FLOATATION. 259 to the body, in order that its upper surface may be made to coincide with that of the fluid ? Put x nz the weight which must be added to the solid, in order that it may sink to a level with the surface of the water ; then, we have m: m + x :: 0.55 : 1.026, and by equating the products of the extremes and means, we get 0.55 (m + x) = 1.026m; therefore, by transposition, we obtain 0.55* = 0.476m; but according to the question, mzr:512 cubic inches, hence we get 0.55*:= 243.712, and finally, by division, we have 243 712 x zn - - 443.1 13 cubic inches; but a cubic inch of fir of the given specific gravity, weighs 0.0198 Ibs. avoirdupois, very nearly ; consequently, the weight to be added for the purpose of making the solid sink to the same level as the surface of the fluid, is .0198X443.113 8.774 Ibs. nearly. 314. But to determine generally, the magnitude which must be added to the original solid, in order that its surface maybe coincident with that of the fluid :- Let zziithe weight to be added; then, by the Proposition, we have m -{- x : m : : s : s, from which, by equating the products of the extremes and means, we get s (m -\- x) s' m, and by separating the terms, it becomes ^m -\- sx^ns'm, and finally, by transposition and division, we obtain _m(s' s) s (201). Therefore, the practical method of reducing this equation, may be expressed in words in the following manner. RULE. From the specific gravity of the fluid, subtract that of the solid ; then, multiply the remainder by the magnitude of the solid, and divide the product by its specific gravity for the weight to be added. s2 260 OF THE EQUILIBRIUM OF FLOATATION. 315. EXAMPLE 3. A cubic mass of oak, whose specific "gravity is 0.872, is placed in a cistern of water, and when it has attained a state of equilibrium with its sides vertical, it stands 20 inches above the surface of the fluid; what is the magnitude of the solid, the specific gravity of the water being represented by unity ? In order to resolve this example, we shall first investigate a general formula, which will apply to every case of a similar nature, when the specific gravity of the fluid and that of the solid are given ; for which purpose, Put x nr the length of one side of the solid ; then is x a* the length of that portion which is below the plane of floatation. But by the principles of mensuration, the magnitude of the whole body is razz a? 8 , and that of the part immersed, is m'= (x a) Xs'zz a: 8 ax* ; therefore, by the Proposition, we obtain m ' m! \ '. s' i s ; and this, by substitution, becomes a; 8 : x* a a? 2 : : s' : s ; from which by equating the products of the extremes and means, we get sz'zz s'O 8 as 2 ); and by separating the terms, we obtain sx 9 zz 5' x 3 as' x 9 ; consequently, by transposition and division, it is as' ~7^7)' (202). And the practical rule supplied by this equation, may be expressed in words at length in the following manner. RULE. Multiply the specific gravity of the fluid by the height of the body above its surface, and divide the product by the difference between the specific gravity of the fluid and that of the solid, and the quotient will give the side of the cube required. * The quantity a, is here put to denote the height of the body abovelhe fluid. OF THE EQUILIBRIUM OF FLOATATION. 261 Taking, therefore, the data as proposed in the foregoing example, and we shall obtain 20X1 #~ , Q S7 , =r 156.29 inches; consequently, the magnitude, or cubical contents of the body, is 156.29X 156.29X 156.29 = 3815627.7 inches. In addition to the foregoing examples, which might very appro- priately have been ranked under the head of problems, the seventh proposition affords the following inferences. 316. INF. 1. If two bodies floating on the same fluid, be in a state of equilibrium : The specific gravities of those bodies, will be to one another directly as the parts below the plane of floatation , and in- versely as the whole magnitudes of the bodies. 317. lNF.^2. If the same body float upon two different fluids, and be in a state of equilibrium on each : The specific gravities of the fluids, will be to one another, inversely as the parts of the body below the plane of floatation. 318. INF. 3. If different bodies float in equilibrio on the surfaces of different fluids, and if the parts below the planes of floatation be equal among themselves : The specific gravities of the fluids, will be to one another directly as the weights of the bodies, or directly as the magni- tudes of the bodies drawn into their specific gravities. 319. INF. 4. If any number of bodies of the same weight, but of different specific gravities, float in equilibrio on the surface of the same fluid : The magnitudes of the parts below the plane of floatation, are equal to one another. 320. INF. 5. If a body float in equilibrio on the surface of a given fluid, and if the part below the plane of floatation be increased or diminished by a given quantity : The absolute weight of the body, (in order that the equili- brium may still obtain,) must be increased or decreased by a weight, which is equal to the weight of the portion of the fluid that is more or less displaced, in consequence of increasing or diminishing the immersed part of the body, or that which falls below the plane of floatation. 262 OF THE EQUILIBRIUM OF FLOATATION. 321. The principle announced in the last inference, may be demon- strated in the following manner. Put ra the magnitude of the body at first, when in a state of equilibrium, ra':=r the part originally below the plane of floatation, m"=: the part by which it is increased or diminished, 5 zn the specific gravity of the body, s' the specific gravity of the fluid, and w ~ the weight by which the body is increased or diminished, in consequence of the increase or decrease of the im- mersed part. Then, because the quantity of fluid displaced, is equal to the mag- nitude of the body which displaces it, it follows, that the weight of the displaced fluid is expressed by (m'-f- w")s', and the weight of the whole solid with which it is in equilibrio, is (ms + w} ; consequently, we have (rn 1 -|- m H )s' i=.ms-\-w. Now it is manifest, that in the case of the first equilibrium, m V rr m s ; it therefore follows, that t m"s' = w. That is, the weight by which that of the body must be increased or diminished, to restore the equilibrium : Is equal to the weight of that quantity of the fluid which is more or less displaced, in consequence of the increase or decrease of the part below the plane of floatation. PROBLEM XLVI. 322. If a solid body in the form of a paraboloid, be in a state of quiescence on the surface of a fluid, whose specific gravity bears any relation to that of the body : It is required to determine how much of the solid will sink beneath the plane of floatation. Let GACBH be a vertical section passing along the axis of the solid, and cutting the plane of floatation in the line AB ; CD being the axis OF THE EQUILIBRIUM OF FLOATATION. 263 of the solid, GH the diameter of its base, and ABHG the portion which falls below EF the surface of the fluid. Put a :=: CD, the axis of the paraboloid, d =. G H, the diameter of its base, m the magnitude of the entire solid GACBH, w'mthe magnitude of the immersed part, s the specific gravity of the body, s' =z the specific gravity of the fluid, and x nrci, the axis of that portion of the body, which in a state of equilibrium, remains above the plane of floatation. Consequently, by the seventh proposition, we have m : m' : : s' : s. But by the principles of mensuration, the solidity of a paraboloid is equal to one half the solidity of its circumscribing cylinder ; there- fore, we get m .3927ad 2 , and similarly, we obtain solidity ACB = 1.5708^ a; 2 , where p is the parameter of the generating parabola. Now, the writers on conic sections have demonstrated, that accord- ing to the property of the generating curve, 4ap z= d* ; let this value of d? be substituted instead of it in the preceding value of m, and we shall obtain consequently, by subtraction, we get therefore, by substituting these values of m and m' in the above analogy, it is a 2 : a 2 re 2 : : / : s ; and from this, by equating the products of the extreme and mean terms, we obtain s'a? s'# 2 n=$a 2 , and by transposition we have s'x 1 z=a 2 (s' s) ; therefore, by division and evolution, we obtain 264 OF THE EQUILIBRIUM OF FLOATATION. IHCl A/ f _ V ' and finally, by subtraction, we have (203). 323. The practical rule for effecting the reduction of the above equation, may be expressed in words at length in the following manner. RULE. Divide the difference between the specific gravity of the fluid and that of the solid, by the specific gravity of the fluid; then, from unity subtract the square root of the quotient, and multiply the remainder by the axis of the parabola, and the result will give the height of the frustum that falls below the plane of floatation. 324. EXAMPLE. The axis of a paraboloid which floats in equilibrio on the surface of a fluid, is 29 inches ; what part of the axis is im- mersed below the plane of floatation, supposing the body *to be of oak, whose specific gravity is 0.76, that of water being unity ? Here, by proceeding as directed in the rule, we get v 1 0.76 ) zz 14.79 inches very nearly. If the vertex of the figure be downwards, as in the annexed dia- gram, then the part of the axis which falls below the plane of floatation will be greater than it is in the preceding case ; for it is manifest, that since the same magnitude or part of the body must be immersed in both cases, it will require a greater portion of the axis, towards the vertex of the figure, to constitute that magnitude, than it would require towards the base. Therefore, by retaining the foregoing notation, we have, by the principles of mensuration, GACBII mzz 1.5708pa% and ACBZZIW'ZZ \.57Q8px?; consequently, by the seventh proposition, we obtain 1.5708pa 2 : 1.5708;?* 2 : : s' : s ; therefore, by suppressing the common factors 1.5708p, and equating the products of the extreme and mean terms, we get s' x" m s a 2 , OF THE EQUILIBRIUM OF FLOATATION. 265 and this, by division, becomes *=%> and finally, by extracting the square root, we get (204). 325. The practical rule deduced from this equation is very simple ; it may be expressed in words at length in the following manner. RULE. Divide the specific gravity of the solid, by that of the fluid on which it floats ; then, multiply the square root of the quotient by the axis of the body, and the product will give the height of the part below the plane of floatation. Therefore, by retaining the data of the foregoing example, we shall obtain as under, c i x = 29 V0.76 = 25.251 inches; being a difference of 10.46 inches, in the depths of immersion, for the two cases of the question. PROBLEM XLVII. 326. When a body floats in equilibrio on the surface of a homogeneous fluid, it is a necessary condition, that the centre of gravity of the body, and that of the fluid displaced, shall be in the same vertical line. Supposing, therefore, that the equili- brium is disturbed by the addition or subtraction of a certain weight : It is required to determine, how far the body will be depressed or elevated in consequence of the extraneous weight? The above problem will obviously admit of two cases, for a weight may be added to a body, and it may be subtracted from it ; in the one case the body will descend, and in the other it will ascend ; the following general solution, however, wiH answer both cases. Let AFBC represent a vertical section of the solid body, floating in a state of equilibrium on a fluid of which the hori- zontal surface is K L ; and suppose, that when the body is acted on by its own weight only, the straight line D E is coin- 266 OF THE EQUILIBRIUM OF FLOATATION. cident with the surface of the fluid ; but in consequence of the addi- tional weight abed, the body descends through the space GH, where it again attains a state of quiescence, and the plane of floatation mounts to AB. Now, it is manifest, that when the body is acted on by means of its own weight only, (in which case, DE is coincident with the surface of the fluid,) the weight of the whole body is equivalent to that of a quantity of the fluid, whose magnitude is DCE; but when the weight abed is applied, the compound weight is equivalent to that of a quantity of the fluid, whose magnitude is ACB; consequently, the subsidiary weight abed, and the weight of a quantity of the fluid, whose magnitude is ABED, are equal to one another. Draw the -straight line ef parallel and indefinitely near to DE; then is DE/e, the small elementary increase of the immersed portion of the body, corresponding to any indefinitely small increase of the weight abed. Put a =. the area of the horizontal section passing through AB, determinable from the position and the figure of the body, before the weight abed is applied, n :z: the weight abed, rv nr the fluxion or small elementary variation of w ; x G H, the distance through which the body passes in con- sequence of the weight n being applied. x zz the fluxion or elementary variation of x, corresponding to w, and s =: the specific gravity of the fluid. Then, because the line ef is supposed to be indefinitely near to D E, it follows, that the portion of the body whose section is D E/e, may be considered as uniform in area between its bases, and consequently, its magnitude is expressed by ax ; but DE/C, is equal to the quantity of fluid displaced by reason of the elementary weight TV, and it is a well attested principle in hydrostatics, that the weight of the quantity of fluid displaced, and that of the body which displaces it, are equal to one another ; therefore we have and the aggregate of the small elementary weights, or the whole weight added, is ?y = /as*. (205). This is the general form of the equation of equilibrium ; but it admits of various modifications, according to the conditions of the OF THE EQUILIBRIUM OF FLOATATION. 267 question and the nature of the body. For instance, if the body be a solid of revolution, and r the radius of the section coincident with the plane of floatation ; then, the above equation becomes w^nrsfax, (206). where the symbol TT denotes the number 3. 1416, or four times the area of a circle whose diameter is expressed by unity. 327. The solution of the problem, however, may be effected inde- pendently of the fluxional analysis, especially in all cases where the floating body is symmetrical with respect to its axis ; for if it be in the form of a right cylinder with its axis vertical, as in the annexed diagram; then, the solution becomes an object of the greatest simplicity ; for since the area of the horizontal section is constant, the space through which the body moves will be the same, whether the weight be added to it or subtracted from it. Let A B c D be a vertical section of a cylinder, floating in equilibrio on a fluid whose surface is GH, the axis mn being perpendicular to the horizon, and sup- pose the weight n to be placed on the upper end of the cylinder ; it is obvious that the equilibrium will then be de- stroyed, and the body will continue to descend, until it has displaced a quan- tity of the fluid, whose weight is equal to that of the compound mass, consisting of the cylinder, together with the applied body whose weight is TV; or it will continue to descend, until the weight of the fluid displaced by the space IKFE is equal to n the weight of the applied body ; in which case, the equilibrium will again obtain, and the plane of floatation, which originally cut the cylinder in E F, will now be transferred to i K. Again, on the other hand, if a,weight rv be subtracted from the cylinder, supposed to be in a state of equilibrium with the plane of floatation passing through EF, the body will then ascend, until the weight of the fluid which rushes into its place becomes equal to the weight subtracted, in which case the solid will again be in a state of quiescence with the plane of floatation passing through a b. Put r ~ id or nd, the radius of the horizontal section, m the magnitude of the space IKFE or EF#a, x de or ec, the space through which the body is depressed or elevated in consequence of the extraneous weight, 268 OF THE EQUILIBRIUM OF FLOATATION. w zr the weight which is added to or subtracted from the cylinder, and 5 z= the specific gravity of the fluid. Then, by the principles of mensuration, the solidity of the cylinder, of which the section is IKFE or EF&O, becomes and the weight of an equal magnitude of the fluid, is m s = 3.l4l6r*sx', but this, by the nature of equilibrium, is equal to the disturbing weight ; hence we have w=3.Ul6r*sx, and from this, by division, we obtain ~3.1416rV (207). 328. The practical rule for reducing this equation is very simple ; it may be expressed in words at length in the following manner. RULE. Divide the given disturbing weight, whether it be added to or subtracted from the cylinder, by the area of the horizontal section, drawn into the speci/ic gravity of the fluid, and the quotient mill express the quantity of descent or ascent accordingly. 329. EXAMPLE. A cylinder of wood, whose diameter is 24 inches, floats in equilibrio with its axis vertical, on the surface of a fluid whose specific gravity is expressed by unity ; now, supposing the equilibrium to be destroyed by the addition or subtraction of another body, of which the weight is 56 Ibs. ; through what space will the body move before the equilibrium be restored ? Here, by proceeding as directed in the rule, we have _ 56 a '~~3.1416xl2 2 X.03617*~~ 3 ' 42 Again, if the body should be in the form of a paraboloid, floating in equi- librio on the surface of a fluid with its ~~ vertex downwards, as represented in the xj annexed diagram, where ACB is a ver- tical section passing along the axis CD, and GH the surface of the fluid on * The decimal fraction 0.03617 expresses the weight in Ibs. of a cubic inch of water. OF THE EQUILIBRIUM OF FLOATATION. 269 which the body floats, with the plane of floatation originally passing through EF, but which, on the addition or subtraction of the weight rv, ascends to i K or descends to a b. Put p zr the parameter of the generating curve, S ~ ec, the distance of the vertex below the surface of the fluid at first, m r the magnitude or solidity of the paraboloidal frustum, of which IKFE is a section, #/:=r the magnitude of the frustum whose section is EF^a ; x rn de, the descent of the body in consequence of the addi- tion of the weight rv, x' rr ec, the corresponding ascent in the case of subtraction, and s rzi the specific gravity of the fluid. Then we have crf = 5 -j- x, and c c rr 8 otf, and according to the writers on mensuration, we have and in a similar manner, we obtain m' = 1.5708p(28* f *'2) ; and since the weight of any body is expressed by its magnitude drawn into its specific gravity, it follows, that the weight of a quantity of fluid equal respectively to m and m 1 , are THSIZ: 1.5708;?s(23;r + x 2 ), andm's 1.5708ps(fcc' x' 3 ). Now, these according to the conditions of the problem, are respec- tively equal to the disturbing weight ; hence we have in the case of addition, m = 1 .57Q8ps(Z$x + * 2 ), (208). and in the case of subtraction, it is tv= 1.5708^5(2^' *' 8 ). (209). 330. The equations which we have just obtained, are precisely the same as would arise, by taking the fluent of the expression in equation (206) ; it therefore appears, that although the fluxional notation is the most convenient for expressing the general result, yet in point of sim- plicity as regards symmetrical bodies, there is little advantage to be derived from its adoption. Suppose that in the first instance, the equilibrium is destroyed by the addition of the weight rv, and let it be required to determine how far the body will descend in consequence of the addition. Equation (208) involves this condition ; consequently, if both sides be divided by the expression 1 .5708^5, we shall obtain 270 OF THE EQUILIBRIUM OF FLOATATION. which being reduced, gives x 1/8"H 8- /oim Y 1.5708j9s (210). 331. And the practical rule for reducing this equation, may be expressed in words at length, as follows. RULE. Divide the weight which disturbs the equilibrium, by 1.5708 times the parameter of the generating parabola, drawn into the specific gravity of the fluid, and to the quotient add the square of the distance between the vertex of the figure and the plane of floatation in the first position of equilibrium ; then, from the square root of the sum, subtract the said dis- tance, and the remainder will express the quantity of descent. 332. EXAMPLE. A solid body in the form of a paraboloid, floats on a vessel of water in a state of equilibrium with its vertex downwards, when 12 inches of the axis are immersed below the plane of floata- tion ; how much farther will the body sink, supposing a weight of 28 Ibs. to be laid on its base, the parameter of the generating parabola being 16 inches? Here, by pursuing the directions of the rule, we get, 90 12 2 H 12 =z 1.22 inches. 1.5708X16X. 03617 333. If the weight w should be subtracted from the paraboloid instead of being added to it, the quantity of ascent will then be deter- mined by equation (209), where we have divide both sides of this equation by the quantity 1.5708p5, and we shall obtain 1.5708ps J which, by transposing the terms, becomes By completing the square, we get Or THE EQUILIBRIUM OF FLOATATION. 271 and finally, by extracting the square root and transposing, it is ' = a V a '- (211). 334. And the practical rule by which this equation is reduced, may be expressed in words at length, in the following manner. RULE. Divide the weight which is proposed to be subtracted from the paraboloid, by 1.5708 times the parameter of the generating parabola drawn into the specific gravity of the fluid, and subtract the quotient from the square of the dis- tance of the vertex below the plane of floatation in theflrst position of equilibrium ; then, from the said distance , subtract the square root of the remainder for the quantity of ascent required. 335. If we take the data of the foregoing example, and proceed according to the directions of the rule, we shall obtain x 1 = 12 4 / 144 __ _ ?? _ 1.35 inches, y 1.5708X16X.03617 exceeding the descent in the former example, by 0.13 of an inch. Numerous other examples akin to the above, respecting bodies of various forms, and placed in different positions, might here be pro- posed ; but since they are all resolvable by the general formula, equation (206), we have thought proper to omit them. PROBLEM XLVIII. 336. If a body which is symmetrical with respect to its verti- cal axis, floats upon a fluid in a state of equilibrium : It is required to determine what weight must be placed upon the body, so that it shall sink to a level with the surface of the fluid, the specific gravities of the solid and the fluid, together with the magnitude of the solid, being given. In the second example to the seventh proposition, we have advanced principles of nearly a similar import to those required for the solution of the present problem, yet nevertheless, we think a separate solution in the present instance is necessary, since it can be somewhat differ- ently represented ; for which purpose, Put m ~ the entire magnitude of the floating body, w its weight before the external body or weight is super- added, 272 OF THE EQUILIBRIUM OF FLOATATION. rv' rr the added weight, n>" the weight of the fluid displaced, s =z the specific gravity of the fluid, and s' = the specific gravity of the floating solid. Then, because the absolute weight of any body, is expressed by its magnitude drawn into its specific gravity ; it follows, that the weight of the floating solid, is w ms', and in like manner, the weight of the displaced fluid, is n" m s ; now, it is manifest, from the nature of the problem, that the weight of the displaced fluid is equal to the weight of the floating body, together with the superadded weight; consequently, we have / -J- rv ~w' -f- w/wis ; from which, by transposition, we obtain w'=ffi(* '). .. (212). 337. The practical rule for the reduction of this equation is very simple : it may be expressed as follows. RULE. Multiply the difference between the specific gravities of the fluid and the floating solid, by the whole magnitude of the floating body t and the product will express the value of the added iveight. 338. EXAMPLE. A mass of oak, whose specific gravity is .872, that of water being unity, floats in equilibrio on the surface of a fluid whose specific gravity is 1.038 ; what weight applied externally to the floating body, will depress it to the level of the fluid surface, sup- posing the magnitude of the body to be equal to 8 cubic feet ? Here, by operating as the rule directs, we shall have w' 8(1.038 .872) 1.328 cubic feet of water ; but it is a well known fact, that one cubic foot of water weighs 62 Ibs. avoirdupois, very nearly ; consequently, we have wf=1.328X62J = 831bs. PROPOSITION VIII. 339. If a solid body, which is specifically heavier than one of two fluids which do not mix, and specifically lighter than the other, be immersed in the fluids : / UNIVERSITY V Of .yS OF TIIR EQUILIBRIUM OF FLOATATION. 273 It will float in equilibria between them, when the weight of the fiuids respectively displaced, are together equal to the weight of the solid body which causes the displacement ; the specific gravities of the fluid being supposed known. Let A BCD be a vertical section passing through the centre of gravity of the floating body, and suppose that IK is the common surface of the two fluids, in which the solid is quiescent, GH being the surface of the lighter fluid. Now, it is manifest, that since the body ABCD is specifically heavier than one of the fluids, and specifically lighter than the other, it cannot be entirely at rest in either, but must rest between them in such a position, that the sum of the weights of the fluids displaced shall be equal to the whole weight of the solid. Let EFD be perpendicular to i K, the common surface of the fluids in which the body floats ; then it is evident, that the pressure down- ward on any point of the base D, is equal to the weight of the incum- bent line of solid particles, whose altitude is BD the thickness of the body, together with the weight of EB the superincumbent column of trie lighter fluid ; and again, the pressure upwards on the same point D, is equal to the weight of a column of the heavier fluid whose alti- tude FD, together with the weight of a column of the lighter fluid, whose altitude is EF. Put d EB, the depth of the body below the upper surface of the lighter fluid, d' EF, the whole depth of the lighter fluid, or the depth of the common surface, 3 FD, the depth of the body below the common surface, or the surface 1 of the heavier fluid, $ = BD, the whole thickness of the solid body, 5 = the specific gravity of the lighter fluid, s' zz the specific gravity of the heavier fluid, s" = the specific gravity of the solid body, p rzr the downward pressure, and p' ~ the upward pressure. Then, because the weight of any body, whether it be fluid or solid, is expressed by the product of its magnitude drawn into its specific gravity, it follows that the downward pressure on the point D, is VOL. I. T 274 OF THE EQUILIBRIUM OF FLOATATION. p i= 2' s" -|- ds, and in like manner, the pressure upwards, is p' = Ss' -\-d's. But when the body floats in a state of equilibrium, the upward and the downward pressures are equal to one another ; hence we have from which, by transposing and collecting the terms, we get SY'zzas'-Kd' d)s. Now, it is obvious, that what we have demonstrated above with respect to the point D, may also be demonstrated to hold with respect to every other point of the surface whose section is ADC; consequently, by taking the aggregate of the upward and downward pressures, we obtain &c.)"=(3 + S + 3+ &c.)s' + {(d' + ef+ d'-f&c.) Put w zz (o ; -f- S' + 2' -j- &c.) to infinity, equal to the magnitude of the entire floating body, m (3 + I 4- 3 4- &C to infinity, equal to the part immersed in the heavier fluid, and m"= {(d' 4- d' 4- d' 4- &c.) (d -f d 4- d + &c.)} to infinity, equal to the part immersed in the lighter fluid ; conse- quently, by substitution, we get ms" = m's' + m"s. (213). This equation involves the principle announced in the Proposition, and its application to practical cases will be exemplified in the reso- lution of the following Problems. PROBLEM XLIX. 340. Suppose that a solid body in the form of a regular cube, is observed to float in equilibrio between two unmixable fluids of different specific gravities : It is required to determine, how much of each fluid is dis- placed by the body, the specific gravities of the body and the fluid being given. Let A BCD be a cubical body, floating in equilibrio between the two unmixable fluids, whose upper horizontal surfaces are respec- OF THE EQUILIBRIUM OF FLOATATION. 275 tively IK and GH, GH being the surface of contact of the two fluids. Put m =: A BCD, the magnitude of the whole body, x m ABCFE, the magnitude of the part immersed in the lighter fluid, and m a; rr EFC D, the magnitude of the part immersed in the heavier fluid. Therefore, if the specific gravities of the body and the fluids be respectively denoted by s", s and s', as in the Proposition; then, we shall have ms"=:xs-\- (m x)s', from which, by transposition, we obtain x (s' 5) zz m(s' s"), and finally, by division, it becomes _ '' 341. An equation of an extremely simple and convenient form, from which we deduce the following practical rule. RULE. Multiply the magnitude of the immersed solid by the difference between its specific gravity and that of the heavier Jluid; then divide the product by the difference between the specific gravities of the fluids, and the quotient will give the magnitude of the part immersed in the lighter Jluid. 342. EXAMPLE. A cubical piece of oak containing 2000 inches, and whose specific gravity is 0,872, that of water being unity, floats in equilibrio between two fluids, whose specific gravities are respec- tively 1.24 and 0.716 ; what portion of the solid is immersed in each of the fluids, supposing them to be altogether unmixable ? The operation being performed according to the rule, will stand as below. 2000(1.24 0.872) (1.^4-0.716)- 14 4 ' 58 ublC mches ' This result expresses the solid contents of that part of the body which is immersed in the lighter fluid ; consequently, the part which is immersed in the heavier fluid, is 2000 1404.58 = 595,42 cubic inches. T 2 276 OF THE EQUILIBRIUM OF FLOATATION. 343. In this example, a greater portion of the body is immersed in the lighter fluid than what is immersed in the heavier ; but this cir- cumstance manifestly depends upon the nature of the immersed body, and the relation of the specific gravities ; for an instance may readily be adduced, in which exactly the reverse conditions will obtain : Thus, let the magnitude of the body and the specific gravities of the fluids remain as above, and suppose the specific gravity of the body to be 1.17 instead of 0.872 ; what then are the parts immersed in the respective fluids ? The numerical process is represented as below. 2000(1.241.17) * = - (1.24- 0.716) = 267.175 cub.c. nc hes. Here then, we have 267.175 cubic inches for the portion which is immersed in the lighter fluid, while that immersed in the heavier is 2000-267.175~1732.825 cubic inches; this agrees with the case represented in the diagram, for there the body displaces a greater quantity of the heavier than it does of the lighter fluid. PROBLEM L. 344. Having given the specific gravities of two unmixable fluids, and the magnitude of a solid body which floats in equili- brio between them : It is required to determine the specific gravity of the solid, so that any proposed part of it may be immersed in the lighter fluid. Put m zr the magnitude of the immersed solid, the same as in the preceding Problem, s ~ the specific gravity of the lighter fluid, s' the specific gravity of the heavier fluid, x = the specific gravity of the solid body, being the required quantity, and n n= the denominator of the fraction which expresses the part of the body immersed in the lighter fluid. Then, according to the principle demonstrated in Proposition VIII., we shall obtain ms , (m m)s' mx= \-- , n n and from this, by a little farther reduction, we get mnxmmns' m(s r s) ; OF THE EQUILIBRIUM OF FLOATATION. 277 consequently, by division, we obtain _mns' m(s' s) * mn (215). 345. And from this equation we deduce the following rule. RULE. Multiply together, the magnitude of the body, the number which expresses what part of it is immersed in the lighter, and the specific gravity of the heavier fluid ; then, from the product subtract the difference between the specific gravities of the fluid drawn into the magnitude of the solid body, and divide the remainder by the magnitude of the body t multiplied by the number which expresses what part of it is immersed in the lighter fluid ; then shall the quotient express the specific gravity of the body. 346. EXAMPLE. The specific gravities of two unrnixable fluids are respectively 1.24 and 0.716, that of water being unity ; now, sup- posing that when these fluids are poured into the same vessel, a body of 2000 cubic inches which is in equilibrio between them, has one seventh part of its magnitude immersed in the lighter fluid ; what is the specific gravity of the body ? Here, by proceeding according to the rule, we have mns / = 2000x7Xl.24 17360 7M(5's):= 2000 (1.24 0.716) zz 1048 subtract difference zr 163 12; consequently, by division, we shall obtain mns' m(s' s) 16312 x s = r- - zz: 1.17 nearly. mn 2000X7 J 347. From what has been done above, it is easy to ascertain what will be the specific gravity of the body, when equal portions of it are immersed in the lighter and in the heavier fluids ; for in that case, we have n equal to 2, which being substituted in equation (215), gives *~s' J(*' s). (216). And the practical rule for reducing this equation may be expressed in words at length, in the following manner. * This equation is susceptible of a simpler form, for by casting out the common factor m, it is 278 OF THE EQUILIBRIUM OF FLOATATION. RULE. From the specific gravity of the heavier fluid, subtract half the difference between the given specific gravities, and the remainder will be the specific gravity of the solid body. 348. EXAMPLE. Let the specific gravities of the fluids remain as above ; what must be the specific gravity of the body, so, that when it is in a state of equilibrium, one half of it may be immersed in each solid ? One half the difference of the given specific gravities, is (1.24 0.716) =0.262; consequently, by subtraction, we have a- =1.24 0.262 = 0.978, and with this specific gravity, a body, whatever may be its magnitude, will be equally immersed in the two unmixable fluids. 349. If the specific gravity of the lighter fluid vanish, or become equal to nothing; then equation (215) becomes and by converting this equation into an analogy, we get m : m' : : s 1 : s". This analogy expresses the identical principle, which is announced and demonstrated in Proposition VII. preceding ; it is therefore pre- sumed, that the examples already given will be found sufficient to illustrate the application of this very elegant and important property. Since the magnitude of the whole floating body is equal to the sum of its constituent parts, it follows, that according to our notation, m = m' 4~ m " '* consequently, by substitution, equation (215) becomes (m 4- m") s" = m' s' 4- m" s, or by transposing and collecting the terms, we get m ("-) = '(*-*"), and by converting this equation into an analogy, we obtain m : m" ::( s " s) : (s's). By comparing the terms of the proportion as they now stand, it will readily appear, that if the specific gravity of the lighter fluid be increased, the term (s" s) is diminished, while (s' s") remains the same ; consequently, the first term ni will be diminished with respect to the second term m" ; which implies, that the part of the body in the lighter fluid will be increased ; hence arises the following very curious property, that OF THE EQUILIBRIUM OF FLOATATION. 279 If any body float upon the surface of a fluid in vacuo, and air be admitted, the body will ascend higher above the surface, and consequently, the proportion of the immersed part to the whole will be diminished. PROBLEM LI. 350. Suppose a solid body to float in equilibrio on the surface of water, both in air and in a vacuum : It is required to determine the ratio of the parts immersed in the water in both cases. Put m zz: the magnitude of the whole floating body, m' zr the magnitude of the part immersed below the surface of the water, when the incumbent fluid is air, m" the portion immersed when the body floats on water in vacuo, s zz: the specific gravity of air, s' zz: the specific gravity of water, and s" zz: the specific gravity of the floating body. Then we have m m', for the part above the surface of the water, when the incumbent fluid is air, and m m" for the extant part when the body floats in vacuo ; consequently, by equation (213), we have, when the body floats in air, ms" zz: m' s' -\-(m ra') s, from which, by a little reduction, we obtain m' (s' s) zz m (s" s), and finally, by division, it becomes ,_*(*"-*) - (s'-s) ' (217). and again, when the body floats in vacuo, we have but in this case, s vanishes, hence we get ms" = m"s, and by division, it is _ms" (218). Let the equations (217) and (218) be compared with one another in the terms of an analogy, and we shall have ' - m ( s '' s ) . . a . ms " . ' (S' 5) S' 280 OF THE EQUILIBRIUM OF FLOATATION. therefore, by equating the products of the extreme and mean terms and casting out the common quantity m, we obtain m"(s"s) _ m's" s' sY~ : ~7"' by clearing the equation of fractions, we get m"s'(s" s) = nf8 l! (S s), and finally, by division, we have '(" ) (219). 351. Now, it is manifest, that in order to determine from this equation, what part of the body is immersed in the water when it floats in vacuo, it is necessary in the first place, to ascertain how much of it is immersed when the floatation occurs in air : Equation (217) determines this, and the practical rule deduced from the equa- tions (217) and (219) may be expressed in words at length in the following manner. RULE. From the specific gravity of the floating body, sub- tract the specific gravity of air ; multiply the remainder by the magnitude of the body, and divide the product by the difference between the specific gravities of water and air, for the part which is immersed in water, when the incumbent fluid is air. Again. Multiply the difference between the specific gravities of water and air by the specific gravity of the floating body ; divide the product by the difference between the specific gra- vities of the solid body and air, drawn into the specific gravity of water ; then, multiply the quotient by the magnitude of the part immersed in water when the body floats in air, and the product will be the magnitude of the part immersed in water, when the body floats in vacuo. 352. EXAMPLE. A mass of oak whose specific gravity is 0.925, con- tains 185 cubic inches; what quantity of it exists below the plane of floatation, supposing it to float on water in vacuo, the specific gravity of the air being 0.0012 at the instant of observation ? By operating according to the directions given in the first clause of the rule, the quantity below the plane of floatation when the incumbent fluid is air, becomes 185(0.925 0.0012) m =. : - 171.108 cubic inches; (10.0012) OF THE EQUILIBRIUM OF FLOATATION. 281 therefore, according to the second clause of the rule, the part immersed when the body floats in vacuo, becomes 0.925(1 0.0012) = X171 - 108: = 353. If we refer to the equation (218) preceding, it will readily appear, that the above result may be determined with much less labour and greater simplicity ; for the magnitude of the immersed part, when the body floats in vacuo, is there expressed in terms of the weight of the body and the specific gravity of water, and the practical rule for reducing the equation, may be expressed in words at length, as follows. RULE. Multiply the magnitude of the body by its specific gravity ; then divide the product by the specific gravity of water, and the quotient will express the magnitude of the immersed part when the body floats in vacuo. Therefore, by taking the data as proposed in the above example, the magnitude of the immersed part becomes 185X0.925 m'~ - - - ^ = 171.125 cubic inches; being precisely the same quantity as we obtained by the foregoing prolix operation. If we compare the computed values of m' and m" with one another, we- shall find that the latter exceeds the former by a very small quan- tity, that is, 171.125 171. 108 =z 0.017, which verifies the concluding inference under Problem L. 354. On the principles which we have explained and illustrated in the foregoing problems, depends the construction and application of the Hydrometer, an instrument which is generally employed for detect- ing and measuring the properties and effects of water and other fluids, such as their density, gravity, force, and velocity. When the hydrometer is employed to determine the specific gravity of water, it is sometimes denominated an aerometer or water-poise ; and being an instrument of very general utility in numerous philoso- phical experiments, we think it will not be amiss in this place, to discuss its nature and properties a little in detail ; and we may here observe, that the following problems and remarks are quite sufficient to establish and exemplify its most important applications. The hydrometer, or aerometer, in general consists of a long cylin- drical stem of glass, or other metal, connected with two hollow balls, 282 OF THE EQUILIBRIUM OF FLOATATION. into the lower of which is introduced a small quantity of mercury or leaden shot, for the purpose of preventing the instrument from over- turning, and causing it to float steadily in a vertical position, or per- pendicularly to the surface of the fluid in which it is immersed. Numerous schemes have been promulgated by different ingenious and experienced philosophers for the improvement of this instrument ; but however much the forms which have been recommended may differ among themselves, yet the general principle is the same in all. The following is a list of the principal writers who have registered their improvements in the annals of science, viz. Desaguliers, Guyton, Nicholson, Speer, Adie, Charles, Atkins, Clark, Dicas, Brewster, Deparcieux, Fahrenheit, Jone Quin, Sikes, and Wilson. 355. It would be quite superfluous to detail the various alterations and improvements suggested by these authors; suffice it to say, that in all there is something different and in all there is something common ; but that which merits the greatest share of our attention, by reason of the extreme delicacy of its indications -and the simplicity of its construction, is the hydrometer of Deparcieux, which was presented to the Academy of Sciences in the year 1766. This instrument, which was intended by its inventor to measure the specific gravities of different kinds of water, is represented in the annexed figure, where AC is a glass phial about seven or eight inches in length, loaded with mercury or leaden shot, to prevent it from overturning; and in order that no air may lodge below it, when it is immersed in the fluid, the lower part is rounded off into the form of a spheric segment. In the cork of the phial at A, is fixed a brass wire of one twelfth of an inch in diameter, and from thirty to thirty six inches long, or of any other length which may be found convenient for the purpose, but such, that when the phial is loaded and immersed in spring water of a medium temperature, the entire phial and about one inch of the wire should be below the graduated scale DH, which is fixed upon the side of the tin vessel DEFG ; to the other end, or summit of the wire, is attached a small box B, intended for containing the minute weights which it may be found necessary to apply, in order to cause the instrument to sink to a certain fixed point in the OF THE EQUILIBRIUM OF FLOATATION. 283 different kinds of water whose specific gravities are required to be found.* The white iron vessel DEFG, is used for holding the fluid on which the experiment is to be performed ; it is generally about three feet in length, and from three to four inches in diameter, according to the circumstances under which it may happen to be employed. The small scale DH, is attached for the purpose of measuring the different depths to which the instrument sinks when differently loaded, or when it is immersed in fluids of different specific gravities. The indications of this instrument are so extremely delicate, that if a small quantity of alcohol, or a little common salt, be added to the fluid, the phial will ascend or descend through a very sensible dis- tance, which circumstance greatly enhances the value of the aerometer; for in proportion to its sensibility and the delicacy of its indications, are its importance and utility to be appreciated. We come now to consider the theory of this instrument, and we shall just remark in passing, that the same principles, under very slight and obvious modifications, will apply to any other hydrometric instrument, of a similar, or nearly similar nature and construction, to that which forms the subject of our present discussion. PROBLEM LII. 356. Having given the capacity or volume of the phial, toge- ther with the dimensions of the immersed wire, and the entire weight of the aerometer : It is required to determine the specific gravity of the fluid, in which the instrument settles in a state of equilibrium. Now, because the weight of any body when floating in equilibrio, whatever may be its form and the substance of which it is composed, is equal to the weight of the fluid which it displaces ; it follows, that if we put c =: the capacity or volume of the phial immersed in the fluid, I zz the length of the immersed wire, r nr the radius of its transverse section, TTZZ: 3.1416, the number which expresses the circumference of a circle whose diameter is equal to unity, * To the bottom of the box B we have affixed the arm a b, from one extremity of which is suspended the wire cd carrying the index i, the whole being truly balanced by the small ball b attached to the other extremity of the horizontal arm ab. In all other respects the instrument is that of Deparcieux. 284 OF THE EQUILIBRIUM OF FLOATATION. s m the specific gravity of the fluid sought, and w=. the entire weight of the aerometer, always known. Then, it is manifest, that the capacity or volume of the phial, together with the magnitude of the immersed wire, is equal to the quantity of fluid displaced ; and the weight of this quantity of fluid is equal to the weight of the aerometer ; but by the principles of men- suration, the magnitude of the immersed wire is expressed by Trr*l; consequently, the quantity of fluid displaced is c-f-7rr 2 /, and the magnitude of any body multiplied by its specific gravity is equal to its weight ; hence we have t0 = (c + **/)*; (220). therefore, by division, we obtain w (221). 357. Here follows the practical rule for reducing the equation. RULE. Divide the entire given weight of the aerometer, by the capacity or volume of the phial, increased by the quantity of wire immersed, and the quotient will give the specific gra- vity of the fluid. 358. EXAMPLE. The whole weight of an aerometer, when so loaded as to have the attached wire depressed 15 inches below the surface of the fluid, is 23 ounces; required the specific gravity of the fluid, supposing the diameter of the wire to be one twelfth of an inch, and the capacity of the phial 40 inches ? Here, by the mensuration of solids, the magnitude of the wire is IGX^TXTT^ - 082 of a cubic inch, very nearly; therefore, the whole quantity of fluid displaced, is 40 4- 0.082 zz: 40.082 cubic inches ; therefore, by the rule, we obtain The number 0.5738, which we have obtained from the above calcu- lation, expresses the weight of one cubic inch of the fluid in ounces ; but since it is customary to express the specific gravity of bodies in ounces per cubic foot, it becomes necessary, for the sake of compari- son, to reduce the above result to that standard ; hence we have s zn 0.5738 X 1728 z= 991 .5264 ounces per cubic foot for the specific gravity of the fluid on which the experiment was tried. OF THE EQUILIBRIUM OF FLOATATION. 285 PROBLEM LIII. 359. Having given the capacity or volume of the phial, the whole weight of the aerometer, the specific gravity of the fluid, and the radius of the wire : It is from thence required to determine, how much of the stem or wire is immersed below the surface of the fluid when the instrument rests in a state of equilibrium. By recurring to the equation marked (220), and separating the terms, we obtain Trr*sl w cs; from which, by division, we get w cs T7T' (222). 360. The practical rule for reducing this equation, may be expressed in words at length, in the following manner. RULE. From the entire weight of the hydrometer, subtract the capacity of the phial drawn into the specific gravity of the fluid ; then, divide the remainder by the area of a trans- verse section of the wire, drawn into the specific gravity of the fluid, and the quotient will express how far the wire is immersed below the upper surface of the fluid, when the instrument floats in a state of equilibrium. 361. EXAMPLE. The entire weight of an aerometer, when so adjusted as to remain at rest in a fluid whose specific gravity is 0.5738*nr23 ounces ; what length of the stem or upright wire falls below the sur- face of the fluid, supposing its diameter to be one twelfth of an inch, and the magnitude of the immersed phial 40 inches ? Here, by the foregoing rule, we have 23 40x0.5738 '= 3^1416X^7x07673-8 = 15 ' 33 mcheS ** 362. If the entire weight of the aerometer be multiplied by 1728, the number of cubic inches in one cubic foot, the formulas (221) and (222) become transformed into * The number 0.5738, by which the specific gravity is here expressed, is the weight in ounces of one cubic inch, which being reduced to the standard of one cubic foot, gives srr 0.57 38x1728 =98 1.5624 oz. 286 OF THE EQUILIBRIUM OF FLOATATION. d ,__1728?<; cs c from the first of which the standard specific gravity is obtained, and in the second, the specific gravity as calculated from the first must be employed. By comparing the quantities in equation (222) with each other, it will readily be perceived, that a very small variation in w the weight of the instrument, or in s the specific gravity of the fluid, will produce a very considerable variation in Z, the immersed portion of the stem or wire ; for it is manifest, that the numerator of the fraction w cs, expresses the weight of the fluid displaced by the wire or upright stem of the instrument, and consequently, since r the radius of the stem is a very small quantity, it follows, that the weight of the fluid which it displaces must also be very small. PROBLEM LIV. 363. Suppose that a small variation takes place in the density of the fluid in which the instrument is immersed : It is required to determine the corresponding variation that takes place in the depth to which it sinks before the equili- brium is restored. Let the notation for the first position of equilibrium remain as in Problem LII., and let I' denote the immersed length of the stem or wire, corresponding to the specific gravity s' ; then, by equation (222), we have _ w cs' 7T?'V ' consequently, by subtraction, the variation in length becomes w cs w cs' and this, by a little farther reduction, gives (223). 364. The practical rule for reducing this equation, may be expressed in words as follows. RULE. Multiply the whole weight of the aerometer by the variation in the specific gravity ; then, divide the product by the area of the transverse section of the upright stem or wire, OF THE EQUILIBRIUM OF FLOATATION. 287 drawn into the greater and lesser specific gravities of the fluid, and the product will express the required variation in the position of the instrument. 365. EXAMPLE. Suppose the specific gravity of the fluid to vary from 0.5738 to 0.5926 ounces per cubic inch during the time of the experiment, what is the corresponding variation in the depth of the instrument, its whole weight being 23 ounces, and the diameter of the upright stem equal to one twelfth of an inch ? Here, by attending to the directions in the rule, we obtain . ., 23(0.5926 0.5738) Hence it appears, that by a difference of 0.0325 in the absolute specific gravity of the fluid, there arises a difference of 233 inches in the position of the instrument ; this seems a very great difference, and is in reality far beyond the bounds prescribed for the whole apparatus to occupy ; it serves, however, to exemplify the extreme delicacy of the principle, and when the changes in the specific gravity are very minute, the corresponding changes in depth will nevertheless be suffi- ciently distinct to admit of an accurate measurement. 366. By diminishing the diameter of the upright stem, or increasing the entire weight of the instrument, which is equivalent to an increase in the weight of the fluid displaced, the sensibility of the aerometer may be greatly increased. This is manifest, for by inference 5, equation (202), it will readily appear, that if the specific gravity remains the same, the quantity by which the instrument sinks in the fluid on the addition of a small weight w r , varies directly as the mag- nitude of the weight added, and inversely as the square of the radius of the upright stem. Let us suppose, that by the addition of the small weight w', the length of the part of the stem Z, which is originally immersed, becomes equal to /' ; then, by the principles of mensuration, the increased magnitude of the immersed stem is 7rr*(l' I); but the weight of a body is equal to its magnitude multiplied by its specific gravity; hence we have *r\l' l)s = w'; and this, by division, becomes r-r=+-. irr^s Now it is obvious, that by the supposition of a constant specific gravity, the quantity ITS is also constant : it therefore follows, that 288 OF THE EQUILIBRIUM OF FLOATATION. w' V I varies as . r* In the above investigation, we have supposed the specific gravity of the fluid to remain constant ; but admitting it to vary, so that s may become equal to s' ; then, in order that the upright stem may rest at the same depth of immersion, w must become equal to (w -f- w') ; if, therefore, we substitute s and (w -)- w'), for s and w in equation (223), we shall obtain _ w -\-w' cs' I , , TrrV an equation from which we find the value of s' to be w w' and by a similar reduction, equation (223) gives w consequently, by analogy, and suppressing the common denominator, we get s' : s : : w -}- w' : w. From this analogy, the difference between the specific gravities in the two cases can very easily be ascertained, for by the division of ratios, we have s' s : s : : w -\- w' w : w, which, by reduction, becomes __ w' s ~ 17* (224), 367. This is a very simple equation for expressing the difference of the specific gravities ; it may be reduced by the following practical rule. RULE. Multiply the added weight by the lesser specific gravity ; then, divide the product by the lesser weighty and the quotient will be the difference between the specific gravities sought, 368. EXAMPLE. An aerometer, whose absolute weight is equal to 23 ounces, is quiescent in a fluid whose specific gravity is 0.5738 ounces, as referred to a cubic inch ; but on being put into a denser fluid, it requires the addition of 0.7536 of an ounce, to cause the instrument to sink to the same depth ; what is the specific gravity of the denser fluid? OF THE EQUILIBRIUM OF FLOATATION. 289 Here then we have given w' = 0.7536 of an ounce, and s = 0.5738 ; consequently, by the above rule, we have consequently, the specific gravity of the heavier fluid, is s' = 0.5738 -f 0.0188 = 0.5926 ; and this, when reduced to the standard of one cubic foot, becomes 0.5926x1728 = 1024.0128, which, on being referred to a table of specific gravities, will be found to correspond with sea water at a medium temperature. In the above operation, we have taken the specific gravity as re- ferred to one cubic inch of the fluid only, but the well informed reader will readily perceive, that the same result would obtain if the specific gravity should be estimated by the cubic foot ; for in that case, we should have w 1 0.7536 of an ounce, and 5 = 991.5264, conse- quently, by the rule, we have 23 therefore by transposition, the specific gravity of the denser fluid, is s' = 991.5264 4- 32.4864 = 1024.0128, being precisely the same result as that which we obtained on the former supposition. 369. The diagram which we have employed to illustrate the general principle of the aerometer, is at the best but a very rude and imper- fect representation, and in its present state, it is altogether unfitted for ascertaining the specific gravities of fluids with any degree of pre- cision ; it is therefore requisite, in cases where extreme accuracy is required, to have recourse to some other method of indicating the precise measure of density, and for this purpose, the hydrometer or aerometer, is very advantageously replaced by the HYDROSTATIC BALANCE, an instrument which determines the specific gravities of bodies with the greatest correctness, and which, on account of its simplicity and cheapness, is rendered available for almost every purpose in which the specific gravity of bodies forms the subject of inquiry. The Hydrostatical Balance, so called, is nothing more than a common balance, furnished with some additional apparatus for ena- bling it to measure the specific gravities of bodies with accuracy and expedition, whether the bodies be in a solid or a fluid state. The description of the instrument is as follows. VOL. i. u 290 OF THE EQUILIBRIUM OF FLOATATION. Let AB be the beam of a balance very nicely equipoised upon its centre of motion at c, and suspended from the fixed object represented at F, the centering being so delicately exe- cuted, that the equilibrium of the instru- ment is disturbed by the smallest por- tion of a grain being added to or sub- tracted from either arm of the beam. D and E are two scales, which, together with their appendages are also balanced with the greatest exactness ; one of them as E having a hook in the middle of its bottom surface, to which the weight w is suspended by means of a horse hair, or any other flexible substance of such extreme levity, as to have no sensible effect upon the equilibrium. p is an upright pillar placed directly under the centre of motion, and carrying the circular arc mmm, which serves to prevent a too great vibration on either side, and also, by means of the index i, which is fixed on the beam immediately under the fulcrum, it indicates the exact position of equilibrium ; for it is manifest, that when the beam is horizontal, the pointer must be directly over the middle of the arc. The pieces in the scale D, denote the weight of the body when weighed in air ; but when the body is immersed in water, as repre- sented by the figure abed, the scale D with its accompanying weights, must evidently preponderate, and for the purpose of restoring the equilibrium, small weights must be placed in the opposite scale at E ; and since the weights thus added, indicate the weight of a quantity of water of equal bulk with the immersed body, it follows, that the specific gravity of the body can from thence be determined. The hydrostatical balance, like the hydrometer or aerometer pre- viously explained, has undergone various alterations and improve- ments, according to the ideas of the different individuals who have had occasion to apply it in their inquiries respecting the specific gravities of bodies ; but since the general principle is the same in all, under whatever form the instrument may appear, it would lead to nothing useful to enter into a detailed description of the various improvements which it has received, and the numerous changes that have been made upon it ; we shall therefore refrain from farther dis- cussion on the nature of its construction, and proceed to exemplify the manner in which it is applied to the determination of specific gravities. OF THE EQUILIBRIUM OF FLOATATION. 291 PROBLEM LV. 370. Having given the specific gravity of distilled water, equal to 1000 ounces per cubic foot : It is required to determine the specific gravity of a solid body that is wholly immersed in it. It is manifestly implied by the total immersion of the body, that its specific gravity exceeds the specific gravity of the fluid in which it is immersed ; therefore, attach the body to the hook in the bottom of the scale E by a very fine and light thread, and balance it exactly by weights put into the other scale at D ; then, immerse the body in the water, and find what weight is required to restore the equilibrium, the weight thus required will measure the specific gravity of the body. Put w z= the weight of the body when weighed in water, w' ~ the weight when weighed in atmospheric air, s zn the specific gravity of water, and s' the specific gravity of the body sought. Then is w w' equal to the weight which must be put into the scale E to restore the equilibrium; consequently, by the fifth proposition, we have . w' w : w' : : s : s' ; from which, by reduction, we get ,'-_^i- ~w' w (225). 371. The following is the practical rule in words at length for reducing the above equation. RULE. Multiply the weight of the body when weighed in air, by the specific gravity of the fluid, and divide the product by the weight which it loses in water for the specific gravity of the body. This rule determines the specific gravity of the body when it exceeds that of the fluid in which it is weighed ; but when the body is specifically lighter than the fluid, the method of finding its specific gravity is shown in Problem XLIV., it is therefore unnecessary to repeat it here. 372. EXAMPLE. If a piece of stone weighs 20 Ibs. in air, but in water only 13J Ibs.; required its specific gravity, that of water being 1000 ? Here, by the rule, w=: 13J, w'= 20, s=z 1000, 20X1000 20000 therefore s = T^T ~ 3076.923 rz the specific gravity Z(j ~ Aug O.O of the mass when it is wholly immersed in water. u 2 CHAPTER XII. OF THE POSITIONS OF EQUILIBRIUM. PROBLEM LVI. 373. Suppose that a solid homogeneous triangular prism, floats upon the surface of a fluid of greater specific gravity than itself, with only one of its edges immersed : It is required to determine in what position it will rest, when it has attained a state of perfect equilibrium. Let ABC be a vertical transverse section, at right angles to the axis of the homogeneous prism, floating A in a state of equilibrium on the fluid whose horizontal surface is IK. Bisect AB, BC the sides of the tri- angle in the points r and n, and D E, D c in the points H and m ; draw the straight lines CF and AH, intersecting one another in the point G, and CH, EWI intersecting in g ; then is G the centre of gravity of the whole triangle ABC, and g the centre of gravity of the triangle DEC, which falls below DE the plane of floatation. Join the points G, g by the straight line G # ; then, according to the prin- ciple announced and demonstrated in the sixth proposition, the straight line gc, is perpendicular to DE the surface of the fluid. Draw FH, and because CF and CH the sides of the triangle CFH, are cut proportionally in the points G and g, it follows from the prin- ciples of geometry, that the straight lines Gg and F H are parallel to one another ; but we have shown that gG is perpendicular to the hori- OF THE POSITIONS OF EQUILIBRIUM. 293 zontal surface of the fluid, or the plane of floatation passing through DE; consequently, F H is also perpendicular to DE, and FD, FE are equal to one another. Put a A B, the unimmersed side of the triangular section, b nz BC, one of the sides which penetrate the fluid, c zz AC, the other penetrating side, d =2 CF, the distance between the vertex of the section, and the middle of the extant side, zz the angle ACF, contained between the side AC and the line CF, ^' zz the angle BCF, contained between the line CF and the side B, s zz the specific gravity of the solid body, s' zz the specific gravity of the fluid on which it floats, x zz CD, the immersed portion of the side AC, and y zz CE, the immersed portion of the side BC. Then, according to the principles of geometry, since the line CF is drawn from the vertex of the triangle at c, to the middle of the base or opposite side at F, it follows, that AC 2 -f BC 2 ZZ2(AF 2 -f CF 2 ), or by taking the symbolical representatives, we shall obtain from which, by reduction, we get dzziV2(^4-c 2 ) a*. (226). Since all straight lines drawn parallel to the axis of the prism are equal among themselves ; it follows, that the weight of the whole solid ABC, and that of the portion DEC below the plane of floatation, which corresponds to the magnitude of the fluid displaced, are very appro- priately represented by the areas drawn into the respective specific gravities of the solid and the fluid on which it floats. Now, the writers on the principles of mensuration have demon- strated, that the area of any right lined triangle : Is equal to the product of any two of its sides, drawn into half the natural sine of their contained angle. Therefore, if we put a' and a" to represent the areas of the triangles ABC and DEC respectively, we shall have for the area of the triangle ABC, 294 OF THE POSITIONS OF EQUILIBRIUM. and for the area of the triangle DEC, it is But according to the principle demonstrated in the third proposition preceding, the weight of a floating body : Is equal to the weight of the quantity of fluid displaced. Consequently, the weight of the solid prism whose section is ABC, is equal to the weight of the fluid prism, whose section is D EC ; that is, \bcs sin.ty 4- 4>') = \xys' sin.ty -f 0'), and from this, by suppressing the common quantities, we get bcs xys. (227). By the principles of Plane Trigonometry, it is F D 2 zz d* + x* 2c?a- cos.<, and F E 2 =z d* -|- y* 2dy cos.^' ; but these by construction are equal ; hence we have Let the value of d as expressed in equation (226), be substituted instead of it in the above equation, and we shall obtain a?* x cos.(j> V 2(c 2 -f 6 2 ) a 2 ?/ 2 y cos. f V 2(c 2 -f 6 2 ) a 2 . (228). Recurring to equation (227), by division, we have bcs , r , . , . 6Vs 2 ?/ zn . , the square of which is w 2 zn ; xs x*s 2 substitute these values of y and y 9 in equation (228), and it is bcscos.Q' . and multiplying by a? 2 we obtain, rcV bcs cos.0' 5 s from which, by transposition, we get a- 4 cos. / 2^+c 2 a 2 X x*-\ -- - 26 2 c 2 a 2 X x= (229). 374. The equation as we have now exhibited it, involves the several circumstances that accompany the equilibrium of a floating body, and its root determines the position in which the equilibrium obtains ; the general form of the expression, is however exceedingly complex, and rising as it does to the fourth order or degree, the resolution is neces- sarily attended with considerable difficulty, especially when the sides OF THE POSITIONS OF EQUILIBRIUM. 295 of the transverse section are represented by large numbers ; in parti- cular cases, the ultimate form will admit of being modified, and may in consequence, be rendered somewhat more simple ; but it must nevertheless be understood, that whenever the position of equilibrium is required by computation, we must inevitably perform a very irksome and laborious process. A geometrical construction may also be effected by the intersection of two hyperbolas ; but since this implies a knowledge of principles higher than elementary, we think proper to pass it over, and proceed to illustrate the application of the above equation by the resolution of a numerical example. 375. EXAMPLE. Suppose a triangular prism of Mar Forest fir, the sides of whose transverse section are respectively equal to 28, 26, and 18 inches, to float in equilibrio in a cistern or reservoir of water, having only one angle immersed ; it is required to determine the posi- tion of equilibrium, on the supposition that the two longest sides of the section penetrate the fluid, the specific gravity of the prism being to that of water as 686 to 1000 ? By recurring to equation (229), and comparing its several consti- tuent quantities with the parts of the diagram to which they respec- tively refer, it will readily appear, that x, cos.0 and cos.0' are the only terms whose values require to be calculated ; of which cos.0 and cos.0' are to be determined from the nature of the figure, and x from the resolution of the biquadratic equation in which its values are involved. The length of the straight line CF, which is drawn from the vertex of the section at c, to the middle of the opposite side at F, is accord- ing to equation (226), expressed by consequently, by substituting the numerical values of the sides, we obtain _ d= 1^2(28* + 26 2 ) 18 2 = 25.4754784 inches. Hence, in the triangles ACF and BCF respectively, we have given the three sides AC, AF, FC and BC, BF, FC to find COS.ACF and cos. BCF; for which purpose, we have the following equations as deduced from the elements of Plane Trigonometry, viz. In the triangle ACF, it is 4 c d a 2 and in the triangle BCF, it is 296 OF THE POSITIONS OF EQUILIBRIUM. Now, by substituting the numerical values of a, b and c, as given in the question, and the value of d as deduced from calculation, the absolute values of cos.0 and cos.0' will stand as below. Thus, for the absolute numerical value of cos.^>, we have 4(784-f649) 324 and for the absolute numerical value of cos.^', it is 649) 324 Let the numerical values of cos.^> and cos.^' as determined by the above computation, together with the numerical values of a, b, c, s, and s', as given in the question, be respectively substituted in equa- tion (229), and we shall obtain x 4 48.2S59* 3 4- 23894.7* =: 249408 ; but in order to simplify the resolution of this equation, it will suffice to take the co-efficients to the nearest integer, for the error thence arising will be of very little consequence in cases of practice, and the modification will very much abridge the labour of reduction ; the equation thus altered, will stand as below. x 4 4Sx s -f- 23895* = 249408. Therefore, if this equation be reduced by the method of approxima- tion, or otherwise, the value of or will come out a very small quantity less than 22 inches ; but taking it equal to 22, the result of the equa- tion is 22 4 48 X22 3 4- 23895X22 = 248842. By substituting the given values of b, c, s and s', with the com- puted value of x, in equation (227), we shall have 22000^ = 499408, from which, by division, we obtain 499408 22000 ' Consequently, from these computed dimensions, together with the sides of the section given in the question, the prism may be exhi- bited in the position which it assumes when floating in a state of equilibrium. 376. Construct the triangle ABC to represent the transverse section of the floating prism, and such that the sides AC, BC, and AB are respec- tively equal to 28, 26, and 18 inches; make CD and CE respectively OP THE POSITIONS OF EQUILIBRIUM. 297 equal to 22 and 22.7 inches, and through the points D and E, draw the straight line IK, which will coincide with the plane of floatation, or the sur- A face of the fluid on which the body floats. \ Jx^xT; -3 Bisect A B, the extant side of the sec- T \^ ^^1 tion in the point F, and join F D and F E ; then, the conditions of equilibrium ma- nifestly are, that the lines FD and FE are equal to one another, and that the area of the immersed triangle DCE, is to the area of the whole triangle ACB, as the specific gravity of the solid is to the specific gravity of the fluid. That the lines FD and FE are equal to one another, appears from an inspection and measurement of the figure ; but the following proof by calculation will be more satisfactory, inasmuch as numbers can be more correctly estimated than measured lines, which depend for their accuracy upon the delicacy of the instruments and the address of the operator. In the plane triangle DCF, we have given the two sides DC and CF, respectively equal to 22 and 25.4754784 inches, and the natural cosine of the contained angle DCF equal to 0.94769 ; consequently, the third side D F can easily be found ; for by the principles of Plane Trigono- metry, we know that DF 2 =:DC 2 4- FC 2 2DC.FCCOS.DCF; therefore, by substituting the respective numerical values, we obtain DF 2 = 484 + 649 2X22X25.4754784X0.94769 = 70.72; consequently, by extracting the square root, it is DFZZI V 7 0.72 = 8.4 inches. Again, in the plane triangle ECF, we have given the two sides EC and CF, respectively equal to 22.7 and 25.4754784 inches, and the natural cosine of the contained angle ECF equal to 0.93906; consequently, by Plane Trigonometry, we have EF 2 zrEC 9 -|-CF 2 2EC.CFCOS.ECFJ and substituting the respective numerical values, we obtain EF 2 =: 515.29 -f- 649 2 X22.7 X25.4754784 X 0.93906 =: 77.895 ; therefore, by extracting the square root, we shall have E F zz: V 777895 8 .82 inches. 298 OF THE POSITIONS OF EQUILIBRIUM. Hence/the length of the line DF is 8.4 inches, and the length of EF is 8.82 inches, giving a difference of 0.42, or something less than half an inch ; being as small a difference as could be expected, from the manner in which the co-efficients of the equation that furnished the value of x were modified, and also from the circumstance of x being determined only to the nearest integer, without considering the frac- tions with which it might be affected. 377. Upon the whole then, the position of equilibrium is sufficiently manifest, from the condition of equality between the straight lines D F and EF ; we shall therefore proceed to inquire if it be equally apparent, from the proportionality between the triangles ACB and DCE. Since cos. = 0.94769, and cos. f 0.93906, it follows that = l8 37' and ':=: 20 6'; consequently, by addition, the whole angle ACB becomes 04-f = 18 37'4-20 6' = 38 43'; therefore, in each of the triangles ACB and DCE, we have given the two sides AC, BC and DC, EC with the contained angle ACB common to both, to find the respective areas. Now, the writers on mensuration have demonstrated, that when two sides of a plane triangle, together with the contained angle are given : The area of the triangle is equal to the product of the two sides drawn into half the natural sine of their included angles 378. This is a principle which we have already stated in the investi- gation, and expressed analytically in deducing equation (227) ; we shall now employ it in determining the areas of the triangles according to the magnitudes of the sides and the contained angle, as given in the example and derived from computation. The natural sine of 38 43' is 0.62547, and the sides AC and BC are respectively 28 and 26 inches ; consequently, by the above principle, we have a' = 1(28X26X0.62547) 227.671 square inches. The natural sine of the contained angle remaining as above, the sides DC and EC as derived from computation, are equal respectively to 22 and 22.7 inches ; hence, from the same principle, we have a"= 1(22X22.7X0.62547) = 156.1798 square inches. Now, according to the conditions of the question, the specific gravity of the fluid is 1000, and that of the floating body is 686; consequently, we obtain 1000 : 227.671 : : 686 : 156.1823. OF THE POSITIONS OF EQUILIBRIUM. 299 In this case the error x is extremely small, amounting only to 156.1823 156.1798 = 0.0025 of a square inch; hence we con- clude, that the position of equilibrium under the given conditions, is very nearly the same as we have found it to be from the resolution of the equations (227) and (229). 379. The preceding solution, however, indicates only one position of equilibrium ; but it is manifest from the nature of the equation (229), that there may be more, for by transposition, we have bcscos.ti' / a i T -a 2 X x . = 0, B -,/E and it is demonstrated by the writers on algebra, that in every equa- tion of an even number of dimensions, having its last term negative, there are at least two real roots, the one positive and the other nega- tive ; consequently, the above equation has two of its roots real and determinable ; but the equation being of four dimensions has also four roots, hence, the other two roots may also be real, and in that case, there will be three values of x positive and the fourth negative; but for every positive value of x there may be a position of equilibrium, that is, there may be three positions, in which the body may float in equilibrio with the angle ACB downwards ; but there cannot be more. 380. If the sides b and c are equal to one another, as represented in the annexed dia- gram, then cos.< and cos.^' are also equal, and the general equation becomes *Xa s -f. ? 2 a* X x -JT = 0. (^30) . Now, it is manifest from the relation of the terms in this equation, that it is resolveable into the two quadratic factors b*s b z s a? -~r, and x* cos. and x 2 cc s and the positions of equilibrium are indicated by the number of real positive roots which these equations contain. 300 OF THE POSITIONS OF EQUILIBRIUM. By extracting the square root of both sides of the equation or*zz _ we shall obtain This expression exhibits two roots of the original equation (230), one positive and the other negative ; but the positive root only becomes available in determining the position of equilibrium, the negative one referring to a case whjch does not exist. It has already been shown in equation (227), that when a solid body floats in equilibrio on a fluid of greater specific gravity than itself; then we have xys' bcs, but according to the supposition, b and c are equal to one another ; hence we get from which, by division, we obtain b*s y = ^' or, by substituting the above value of x, it becomes . (232 , Hence it appears, that the values of x and y are each of them expressed by the same quantity; consequently, the triangle DCE is isosceles, and AB the extant side of the section, is parallel to DE the base of the immersed portion, both of them being parallel to the plane of floatation or the horizontal surface of the fluid. 381. The practical rule for the reduction of the equation (231) or (232), may be expressed in words at length, in the following manner. RULE. Divide the specific gravity of the solid body, by the specific gravity of the fluid on which it floats ; then, multiply the square root of the quotient, by the length of one of the equal sides of the section, and the product will give the portion of that side which is immersed below the plane of floatation, or that which is intercepted between the vertex of the section and the horizontal surface of the fluid. 382. EXAMPLE. A prism of wood, the sides of whose transverse section are respectively equal to 20, 28 and 28 inches, is placed with its vertex downwards in a cistern or reservoir of water whose OF THE POSITIONS OF EQUILIBRIUM. 301 surface is horizontal ; it is required to determine, what position the solid will assume when in a state of equilibrium, its specific gravity being to that of water as 686 to 1000 ? Here, by the rule, we have from which, by extracting the square root, we get ^0.6860.8282; and finally, by multiplication, we obtain x = 28 X 0.8282 = 23.1896 inches. But according to equation (232), y possesses the very same value ; consequently, if 23.1896 inches be set off from the vertex of the section upwards on each of its equal sides, the straight line which joins these points will coincide with the plane of floatation, or the horizontal surface of the fluid on which the body floats. 383. This is the most natural and obvious position of equilibrium, and such as must always obtain when the body is homogeneous, and symmetrical with respect to a vertical plane passing through the axis and bisecting the base ; but there may be other situations in which the body may float in a state of quiescence, and the circumstances under which they occur must be determined by the resolution of the following equation, viz. b*s Complete the square, and we shall have 2 o 8 X * -f ( 1 and by extracting the square root, we get x COS.0V a = : J COS 2 ^(46 3 a 2 ) -j hence, by transposition, we shall obtain, (233). / ^ a*^y J cos 2 .^(4^ a 2 ) -, The corresponding values of y are (234). y = \ cos.< V46 2 a?2\/ 302 OF THE POSITIONS OF EQUILIBRIUM. Expressions of this form, arising from the reduction of an adfected quadratic equation, are in general rather troublesome and difficult to render intelligible in words, and even when intelligibly expressed, they are to say the least of them, but very dull and uninviting guides, from which a tasteful reader turns with disgust ; we are therefore unwilling to crowd our pages with long and formal directions for the purpose of reducing equations, when it is probable after all, that nine out of every ten of our readers will pass them over, and proceed immediately to discover their object by the direct resolution of the original equation. 384. It is however necessary, in conformity to the plan of our work, to express the most important final equations in words at length, and since the preceding forms are of considerable utility in the doctrine of floatation, it would be a direct violation of systematic arrangement, to omit the verbal description, and leave the subject open only to algebraists ; we shall therefore, in order to render both parts of the operation intelligible, endeavour to express the method of reduction in as brief and comprehensive a manner as the nature of the subject will admit. 1 . To determine the value of x. RULE. From four times the square of one of the equal sides of the section, subtract the square of the base, or side opposite to the vertical angle ; multiply the square root of the remainder by one half the natural cosine of half the vertical angle, and call the product ?n. From four times the square of one of the equal sides of the section, subtract the square of the base, or side opposite to the vertical angle, and multiply the remainder by one fourth of the square of the natural cosine of half the vertical angle, or that which is immersed in the fluid ; then, from the product, subtract the quotient that arises, when the specific gravity of the solid, drawn into the square of one of the equal sides of the section, is divided by the specific gravity of the fluid, and call the square root of the remainder n. Finally, to and from the quantity denoted by m, add and subtract the quantity denoted by n ; then, the sum in the one case, and the difference in the other, will give the two values of*. 2. To determine the corresponding values of y. RULE. Calculate the values of m and n, precisely after the manner described above; then, from and to the quantity OF THE POSITIONS OF EQUILIBRIUM. 303 denoted by m, subtract and add the quantity denoted by n, and the difference in the one case, and the sum in the other, will give the values ofy corresponding to above values of x. 385. These are the rules by which the other positions of equilibrium are to be determined ; but it is necessary to remark, that beyond cer- tain limits no equilibrium can obtain. In the first place, in order that the body may float with only one of its angles immersed, it is mani- festly requisite, that the equal sides of the section should each be greater than m -j- n ; and in the second place, in order that x and y may be real positive quantities, the expression J cos 2 .^>(46 2 a 2 ) must . tfs s , cos 2 .d>(4 2 a ) exceed , or must be less than - r . ,, - . ' 2 reason of these limitations is obvious from the nature of the quadratic formula (233) and (234), but it will be more satisfactory to show, that unless the data of the question are so constituted as to fulfil these conditions, the rules will fail in determining the positions of equilibrium ; or in other words, there is no other position in which the body will float at rest, but that which is indicated by the equa- tions (231) and (232). 386. EXAMPLE. The data remaining as in the preceding example, let it be required to determine from thence, whether under the pro- posed conditions, the body can float at rest in any other position than that which we have already assigned for it, by the reduction of the equations (231) and (232), in which the extant side or base of the figure is parallel to the horizon. By the principles of Plane Trigonometry, we have \ cos.0 n= T V v 7 28 + 1 0) (28 1 0) = !(0.93406) = \ cos.20 55' 29" ; consequently, by proceeding according to the rule, we'get m~\ cos.(46 2 a s )*zz0.46703 ^4 X28 2 20*=: 24.429 very nearly. Again, to determine the value of w, it is i cos. 8 0(46 2 a 2 ) =. 0.46703 2 (4 X28 2 20 2 ) = 596.768, and for the value of the term, involving the specific gravities, we have consequently, by subtraction, we get 596.768 537.824 = 58.944. It therefore appears from the last result, that both the values of x and y are real positive quantities ; consequently, one of the limiting 304 OF THE POSITIONS OF EQUILIBRIUM. conditions is answered, and we shall shortly see, whether or not the data are sufficient to satisfy or fulfil the other condition. By extracting the square root of 58.944, we get w=-V/58.944 = 7.677 nearly; therefore, by addition and subtraction, the values of x, are a: = m + 72 = 24.429 +7.677 = 32. 106, and x m n 24.429 7.677 zz 16.752 inches. Now, we have seen by equation (234), that the corresponding values of y are expressed in the same terms, having the signs of the second member reversed ; hence we have y = 16.752, and y = 32.106 inches. But here we have m -\- n zn 32.106 inches, being greater than b the downward side of the transverse section, which by the question is only 28 inches ; it therefore follows, that with the proposed data and under the specified circumstances, there is only one position in which the body can float in a state of rest, and it is that which we have already determined, where the base of the section, or the extant side of the body, is parallel to the surface of the fluid. But we may here observe, that notwithstanding the values of a: and y, as we have just assigned them, do not satisfy the conditions of the question, yet they are not to be considered as being useless ; for they actually serve, with a slight modification of the body, to furnish posi- tions in which it will float at rest, although those positions do not agree with the case, in which only one angle of the figure falls below the plane of floatation. 387. The positions of equilibrium corresponding to the preceding values of a; and y, are as represented in the an- nexed diagrams, where E D is the horizontal sur- face of the fluid, ABC being the position which the body assumes when x is equal to 32.106 and y equal to 16.752 inches, and abc the correspond- ing position when y is equal to 32.106 and x equal to 16.752 inches ; these being the respective values as obtained by the above numerical process. OF THE POSITIONS OF EQUILIBRIUM. 305 That the positions here exhibited are those of equilibrium, is very easy to demonstrate, for produce the sides CA and cb to meet the surface of the fluid in the points E and D, and bisect AB and a b in the points F and/; then, if the straight lines FE, FH and/D, /i be drawn, they will be equal among themselves. This is one of the conditions of equilibrium, as we have already demonstrated in the construction of the original diagram, and the other condition is, that the areas of the immersed figures ECH and DCI, are respectively to the whole areas ABC and a be, as the specific gravity of the solid, is to the specific gravity of the fluid which sup- ports it. Now, if the first of these conditions obtain, that is, if the straight line FE be equal to FH, and/D equal to/i, then, by the principles of Plane Trigonometry, we shall have EC 2 -j-CF 2 2EC.CFCOS.ECFZTHC*-}- c ^ 2HC.CF COS.FCH ; but the angles ECF and FCH are equal to one another, and each of them equal to ; consequently, by substituting the literal representa- tives, we have x a -f- d* 2rf x cos.^> y 3 - -)- d* 2dy cos.0, or by expunging the common term e? 4 , we get x* 2dx cos.0 z= ?/* ^dy cos.0, and this, by transposing and collecting the terms, becomes # 2 ?/ 2 zz: 2e? cos.^(o? y) ; therefore, jf both sides of this equation be divided by the factor (x y), we shall obtain x -f- y zr: 2rf cos.^>. Now, by a previous calculation we found x to be equal to 32.106 inches, y 16.752 inches, d equal to \/28 8 10% and cos. equal to 0.93406 ; consequently, by substitution, we have 32.106 + 16.752 = 2x0.93406X6/19"; hence the equality of the lines FE and FH is manifest. What we have shown above with respect to the triangle ABC, may also be shown to obtain in the triangle a be, the one being equal and subcontrary to the other ; this being the case, it is needless to repeat the process; but we have yet to prove, that the area CEH, is to the whole area ABC, as the specific gravity of the floating body, is to that of the fluid on which it floats. therefore, by the principles of Plane Trigonometry, we get AC : AF : : rad. : sin.ACF, VOL. i. x 306 OF THE POSITIONS OF EQUILIBRIUM or numerically, we shall obtain 28 : 10 : : 1 : sin./>= 0.35714, and we have already found that cos.^ =: V (28 -f 10) (28 10) -f- 28 = 0.93406 ; but according to the arithmetic of sines, it is sin.20 2 sin.0 cos.0, and by substituting the above numerical values, we get \ sin.20 = 0.35714 X 0.93406 = 0.33359. Then in the triangle ECH, there are given the two sides EC and HC, respectively equal to 32.106 inches and 16.752 inches, together with half the natural sine of the contained angle ; to find the area of the triangle. Now, by the principles of mensuration, the area of any plane tri- angle is expressed by half the product of any two of its sides, drawn into the natural sine of the contained angle, hence we get 32. 106X 16.752 X0.33359 = 179.417 square inches. Again, in the isosceles triangle ABC, there are given the sides AC and BC, respectively equal to 28 inches, and half the natural sine of the contained angle ACB, equal to 0.33359 ; to find the area. Here, by the principles of mensuration, we have 28X28X0.33359 zz 261.53456 square inches; then, by the property of floatation, it is 1000 : 686 : : 261.53456 : 179.413 square inches. 388. Since this result agrees so very nearly with that derived from a direct computation of the triangular area, we may reasonably con- clude, that the positions exhibited in the diagram are those of equili- brium ; it is however necessary to remark, that since the weight of the body remains unaltered in what position soever it may be situated, it does not readily appear in what manner the adequate quantity of fluid is displaced, unless we conceive some physical plane, of sufficient breadth and totally destitute of weight, to be fixed on that edge of the solid which becomes immersed by reason of the change of position that the body is supposed to undergo. This plane, during the oscillation of the prism, will dislodge the fluid which occupies the space EAW or vbn, and the weight of this quantity of fluid added to that which is displaced by the quadrilateral figure cj mn or cbni, will be equal to the whole weight of the float- ing body OF THE POSITIONS OF EQUILIBRIUM. 307 389. The above modification, however, does not strictly accord with the conditions of the problem ; we must therefore inquire, whether the just principles of equilibrium do not depend upon some other element, such as the specific gravity. Now, we have already stated, that in order to have the values of x and y real positive quantities, it is neces- sary that s , cos s .0(46 a -a 9 ) should be less than - prs - s 4kb 9 and for a similar reason s . must be greater than And if the specific gravity of the fluid be denoted by unity, as is the case with water, then the specific gravity of the floating body must lie between the limits cos 9 ^(4& g a a ) and cos.ft V 46* a 9 4& 2 b The specific gravity of the floating body, as we have proposed it in the question, is 686, that of water being denoted by 1000; conse- quently, when the specific gravity of water is expressed by unity, that of the solid is 0.686; let us therefore try if this number lies between the above limits ; for which purpose, we must substitute 28 for b, 20 for a, and 0.93406 for cos.0 ; then we shall have as follows. 0.93406 2 (4X28 ? 20 2 ) For the greater limit we have s = - ;. OQa - - 0.761 4X ~ nearly. It is therefore manifest, that the specific gravity of the floating body, as we have employed it, is less than the greater limit, and con- sequently properly chosen with regard to it, and we have next to inquire if it exceeds the lesser limit ; for which purpose, it is 28 Here then it is obvious, that the lesser limit exceeds the given specific gravity ; and from this we infer, that without the modification specified above, the body will not fulfil the conditions of the problem in any other position than that in which its base is parallel to the surface of the fluid ; but if the specific gravity of the floating body fall between the numbers 0.761 and 0.745, all other things remaining, then the prism, besides the situation of equilibrium in which its base is parallel to the surface of the fluid, may have two others, in both of x2 308 OF THE POSITIONS OF EQUILIBRIUM. which the conditions of the question will be truly satisfied, for only one angle of the figure will fall below the plane of floatation. In order therefore to exhibit those positions, we shall suppose the specific gravity of the floating prism to be expressed by 0.753, which is the arithmetical mean between the limits above assigned ; then, by operating according to the rules under equations (233) and (234), we shall obtain _ tf 0.46703 ^4x28' 20 2 rz 24.429 as for- merly computed ; and after a similar manner, we have = a 9 ) = 0.46703 S (4X28 9 20*) - 1UUO =z2.528; consequently, by addition and subtraction, we shall get x = m -f n 24.429 -f 2.528 = 26.957 inches, and x = m n = 24.429 2.528 21.901 inches; and the corresponding values oft/, are 21.901 and 26.957 inches respectively. 390. The positions of equilibrium corresponding to the above values of a: and y, are as represented in the annexed diagrams, where it may be shown that the straight lines F E, F H and /D, fi are equal to one another, and also that the areas of the immersed spaces E c H and D ci are respectively to the whole areas ABC and abc, as the specific gravity of the solid, is to that of the fluid on which it floats, or as 0.753 to unity in the case of water. These conditions being satisfied, the body will float in equilibrio in the positions here exhibited; and it from hence appears, that the problem admits of a complete solution, by retaining the specific gra- vity of the solid within determinate limits. 391 . When the transverse section of the floating prism, is in the form of an equilateral triangle ; then a and b are equal to one another, and equation (230) becomes _ b s cos id) \/ 3 b^ s* r $ ' 5 / 9 > OF THE POSITIONS OF EQUILIBRIUM. 309 and if the value of s' be expressed by unity, as in the case of water, then we have 6V = 0. (235). Now, it is manifest that this equation is composed of the two quadratic factors x* b* srz:0, and .r'^-^cos.^V^Xa; 4- b*s=:&, whose roots give the positions of equilibrium. Since the sides a and b are equal to one another, and s' equal to unity ; then, the limits between which the value of s must be retained, are Jcos*.0 and cos.^~3l ; but in the case of the equilateral triangle, ^ zr 30 ; consequently, cos.^> J ^/ 3, and cos 2 .0 j ; therefore, by substitution, the above limits become T g s= 0.5625, and $ lrz:0.5, the arithmetical mean of which, is (0.5625 + 0.5) = 0.53125. Let this value of s be substituted instead of it, in each of the con- stituent quadratic factors, and the equations whose roots determine the positions of equilibrium, become respectively ** 0.531256% and x* bcos.(j>^/3^x = .531256*; but by the property of the equilateral triangle, =n 30, and consequently cos.0 \ : i#V3~ : : 0.53125 : 1* or by suppressing the common quantity J\/3, we have x 2 : 6 2 : : 0.53125 : 1 ; but x z ~ 416.5, and 2 ~ 784; therefore, by substitution, we obtain 416.5 :784 : : 0.53125 : 1. It is therefore evident, that by the above results, both the condi- tions of equilibrium are satisfied, and consequently, the body floats in a state of equilibrium when placed as represented in the diagram ; that is, with 20.4083 inches of its side immersed, and its base parallel to the plane of floatation. 393. The adfected quadratic equation a 2 42z:zz 416.5, has obviously two positive roots, each of them less than b the side of the section ; from which we infer, that besides the position of equilibrium above exhibited, the body may have other two, and these will be determined by the resolution of the equation, as follows. Complete the square, and we obtain ^ _ 42a; + 21 2 = 416 5 + 441 = 24.5, OF THE POSITIONS OF EQUILIBRIUM. 311 extract the square root of both sides, and we get x 21 = dby 7 2475 = =t 4.95 nearly; consequently, by transposition, we obtain x :r21 -f 4.95 == 25.95 inches, and x = 2l 4.95 16.05 inches; and the corresponding values of y are Now, the positions of equilibrium supplied by the above values of a; and y, are as exhi- bited in the subjoined diagrams, where LK is the surface of the water, ABC the position of the body corresponding to x equal 25.95 inches, and y equal 16.05 inches; abc being the position which the solid assumes, when the values of x and y reverse each other ; that is, when y equal 16.05 inches and y equal 25.95 inches. Bisect A B and a b in the points F and /, and draw the straight lines FE, FH andyi,yD to meet the surface of the water in the points E, H and i, D, the points in which the plane of floatation intersects the im- mersed sides of the solid ; then are the lines FE, FH andyi,yD equal among themselves, and the areas ECH, ICD, are respectively to the whole areas ABC, a b c as the number 0.53125 to unity. PROBLEM LVII. 394. Suppose that a solid homogeneous body, in the form of a triangular prism, floats upon the surface of a fluid of greater specific gravity than itself, in such a manner, that two of its edges shall fall below the plane of floatation : It is required to determine its position, when it has attained a state of perfect quiescence. Let ABC represent a section perpendicular to the axis of a solid homogeneous triangular prism, floating in a state of quiescence on a fluid whose horizontal surface is IK; A DEB and DCE being respec- tively the immersed and extant portions. 312 OF THE POSITIONS OF EQUILIBRIUM, Now, it is manifest, that since the whole section ABC, is divided by DE the line of floatation, into the two parts A DEB and DCE; it follows, that the centre of gravity of the section ABC, and the common centre of gravity of the two parts into which it is divided occur in the same point ; consequently, the centres of gravity of the triangular areas ABC and DEC, with that of the quadrilateral space A DEB, are situated in the same straight line. But by the principles of floatation we know^ that when the solid is in a state of quiescence, the centre of gravity of the whole section ABC, and that of the immersed portion A DEE occur in the same verti- cal line; that is, the vertical line passing through their centres of gravity is perpendicular to the horizontal surface of the fluid ; and for this reason, the vertical line passing through the centre of gravity of the whole section ABC, and that of the extant portion DEC, is also perpendicular to the horizon. Bisect the sides AB, BC and D E, EC in the points F, n and H, m and draw the straight lines CF, ATI and CH, Din intersecting two and two in the points G and g, which points are respectively the centres of gravity of the triangular spaces ABC and DEC. Draw the straight lines Gg and FH, and because CF and CH the sides of the triangle CFH, are cut proportionally in the points o and g, it follows, that Gg and FH are parallel to one another; but it has been demonstrated, that Gg is perpendicular to TK the horizontal surface of the fluid ; therefore, FH is perpendicular to DE the line of floatation; and since DE is bisected in H, it follows that FD and JE are equal to one another. Put a AB, the immersed side of the triangular section ABC, b ~ AC, one of the sides of the triangular section which penetrate the fluid; c :n BC, the other penetrating side of the figure, d m CF, the distance between the vertex of the section and the middle of the immersed side ; :rz ACF, the angle contained between the side AC and the bisecting line c F* OF THE POSITIONS OF EQUILIBRIUM. 313 ty' BCF, the angle contained between the bisecting line CF and the side B c, s zn the specific gravity of the floating solid, s' :zr the specific gravity of the supporting fluid, x :zz CD, the extant portion of the side AC, and y CE, the corresponding portion of the side BC. Then, since the area of any plane triangle, is expressed by the product of any two of its sides, drawn into half the natural sine of their included angle, it follows, that the area of the entire section ABC, is expressed as under, viz. and for the area of the extant triangle DEC, we have where the symbols a and a", denote the areas of the whole section and the extant portion respectively ; consequently, by subtraction, the area of the immersed part ADEB, is (a 1 a") = i3in.(0 + f)(bc xy). But by the principles of floatation, the area of the whole section ABC, is to the area of the immersed portion ADEB, as the specific gravity of the supporting fluid, is to the specific gravity of the floating solid ; that is !&csin.(4>4-<') : sin.( -\- 0') (be xy) : : s' : s ; from which, by casting out the common terms, we get be : (be xy) : : s' : s, and equating the products of the extremes and means, it is bcs bcs' xys' ; therefore, by transposing and collecting the terms, we obtain xys' = bc(s' s). (236). Since the line CF is drawn from the vertex of the triangle ABC, to the middle of the opposite side or base AB, it follows from the prin- ciples of geometry, that AC ? 4-BC 8 :=2(AF 2 -f CF 3 )* and this, by substituting the literal representatives, becomes 6 2 + c 2 =:2(K-f d 2 ); therefore, by transposition, we have 4d 2 zz2(6 2 + c 2 )-a 2 , and finally, by dividing and extracting the square root, we get d 4^2(^4- zz 2/ 2 Zdy cos.0' ; or by substituting the value of d, equation (237), we get x* cos.0 V 2(6 3 +c 2 ) a 3 X zzrz/ 9 cos.f V 2(^4-c 2 ) a 2 X y . (238). If both sides of equation (236) be divided by the expression #/, we shall obtain bc(s' s) y =^- consequently, by involution, we have y^ j> s ~ s - Let these values of y and y* be substituted instead of them in equa- tion (238), and we shall have ^bc(s' s)cos.' xs and multiplying all the terms by x , we get 6V(s' s) 2 bc(s -- -- and finally, by transposition, we have OF THE POSITIONS OF EQUILIBRIUM. 315 ) a 2 _6 2 cV-5) 2 "I 72 "" (239). 395. The above is the general equation, whose roots give the several positions in which the solid may float in a state of equilibrium ; it is similar to equation (229), having (5' s) instead of s, and (s' sf instead of s 2 ; the body may therefore have three positions of equili- brium, but it cannot have more, the very same as in the case, where it floated with only one of its edges below the surface of the fluid. The method of applying the general equation to the determination Of the positions of equilibrium, is to calculate the value of d, cos.0 and cos.^>' from the given dimensions of the section, and to substitute the several given and computed numbers instead of their symbolical equivalents ; this will give a numeral equation of the fourth degree, which may be reduced either by approximation or otherwise, accord- ing to the fancy of the operator. 396. EXAMPLE. Suppose a solid homogeneous triangular prism, the sides of whose transverse section are respectively equal to 28, 23 and 18 inches, to float in equilibrio on a cistern of water with two of its edges immersed ; it is required to determine the positions of equi- librium, on the supposition that the two longest sides of the section include the extant angle, the specific gravity of the prism being to that of water, as 565 to 1000 ? In order to resolve this question, we must first of all determine the length of the line cr, which is drawn from the extant angle at c to the middle of the opposite side AB ; for which purpose, let the dimensions of the section be respectively substituted according to the combination exhibited in equation (237), and we shall have d = \ V2(28 2 + 23 2 ) 1 8 2 \ v/ 2302 = 23.99 inches nearly. Consequently, in the triangles ACF and BCF respectively, we have given the three sides AC, AF, FC and BC, BF, FC to find cos. ACF and cos. BCF; for which purpose, the elements of Plane Trigonometry supply us with the following equations, viz. In the triangle ACF, it is and in the triangle BCF, it is 316 OF THE POSITIONS OF EQUILIBRIUM. Therefore, by substituting the respective values of a, b and c, as given in the question, and the value of d as computed above, we shall have the following values of cos.0 and cos.^'. Thus, for the absolute numerical value of cos.0, it is 4(28* + 23.99*) 18 2 _ A COS '* = -- 8X28X23.99 ~ ' 95166 ' and for the corresponding value of cos. 0', we have 4(23 2 -4-23.99 2 18 2 COS - = 8X23X23.99 =- 92747 ' Having ascertained the numerical values of cos. and cos.^>', let the respective quantities be substituted in equation (239), and it becomes 28X23X435X0.92747 4 0. 1000 28 2 X23 2 X435 2 1000 2 from which, by computing the several terms, we get * 4 45.66* 3 -f 12466* = 78478.36. The root of this equation will be most easily discovered by approxi- mation, and for this purpose, we shall adopt the method of trial and error, which Dr. Hutton has so successfully applied to the resolution of every form and order of equations, however complicated may be their arrangement. By a few simple trials, indeed it is almost self evident, that the value of x will be found between 15 and 16; consequently, by sub- stitution we obtain 15 4 45.66X15 8 4- 12466X15 78478.36 = 15113.64 too little. 16 4 45.66X16 3 + 12466X16 78478.36 = 509.72 too great. Here it is manifest that the errors are of different affections, the one being in defect and the other in excess ; hence we have 15113.64 + 509.72 : 16 15 :: 509.72 : 0.032; consequently, for the first approximation, we get x = 16 0.032 = 15.97 very nearly. Supposing therefore, that x lies between 15.9 and 16 ; by repeating the process, we shall have 16 4 45.66X 16 3 -f 12466X 16 78478.36 = 509.72 too great. 15.94 __ 45.66 X15.9 3 4- 12466 X 15.9 78478.36 zz!05.393too little. Here again, the e/rors are of different affections, the one being in excess and the other in defect; consequently, we have OF THE POSITIONS OF EQUILIBRIUM. 317 509.72-1-105.393 : 1615.9 : : 105.393 : 0.017 nearly; therefore, the second approximate value of x, is x = 15.9 + 0.017 = 15.917 inches. By again repeating the process, a nearer approximation to the true value of x would be obtained, but the above is sufficiently accurate for our present purpose ; therefore, let this value of x, together with the numerical values of b, c, s and s', be substituted in equation (236), and we shall obtain 159172/m 280140, and from this, by division, we get 280140 397. And the position of equilibrium corresponding to the above values of x and z/, is represented in the annexed diagram, where IK is the hori- zontal surface of the fluid, ABED the im- mersed part of the section, and DCE the extant part. Bisect AB in F, and draw the straight lines CF, FD and FE; then, as we have previously demonstrated, when the body floats in a state of equilibrium, the lines FD and FE are equal to one another. Now, in order to determine if this equality obtains, we must have recourse to equation (238), where we have c *) 'X x = t/ 2 cos. then, let the computed values of or, y, cos.0, cos.^', and the given values of a, b and c, be substituted instead of them in the above equa- tion, and we shall obtain 15.917 2 0.95166 v/2302 X 15.917 17.6 2 0.92747 /2302X 17.6, and this, by transposition and reduction, gives 726.77 = 56.43 783.20. 398. Another condition of equilibrium is, that the area of the im- mersed part ABED, is to the area of the whole section ABC, as the specific gravity of the solid is to that of the supporting fluid. This is a more necessary condition than the equality of the lines FD, FE; for such an equality may exist when no equilibrium obtains ; but it may be considered as a universal fact, that whenever the two conditions are satisfied at the same time, the body floats in a state of quiescence. 318 OF THE POSITIONS OF EQUILIBRIUM. We have already found that cos. 4>zz 0.95 166, and cos.0'= 0.92747 ; consequently, ^=17 53', and ' = 21 57' ; hence we have (< -f- 0') zr39 50', and by the principles of mensuration, we get a' = 1(28 X 23) sin.39 50' = 322 X 0.64056 = 206.26032, and the area of the extant part, is a" = J( 15 - 917 X 17.6) sin.39 50' = 140.0696 X 0.64056 = 89.72298 ; therefore, by subtraction, the area of the immersed part becomes (a' a") = 206.26032 89.72298 = 1 16.53734 ; consequently, by the principle of floatation, it is 206.26032 : 116.53734 : : 1000 : 565 nearly; from which it appears, that both the conditions of equilibrium are satisfied, and therefore the body as exhibited in the figure indicates a state of quiescence. 399. By finding the other roots of the equation, other situations of equilibrium may be assigned ; but since the one above given is that which would be adopted in practice, we consider that it would be a waste of both labour and time to search after the others ; we therefore leave the reduction of the resulting cubic equation for exercise to the reader, presuming that he will find his trouble and attention amply repaid, by the satisfaction to be derived from the confirmation of the principles by an actual construction. 400. When the triangle ABC becomes isosceles; that is, when the sides b and c are equal to one another ; then cos.0 and cos.0' are also equal, and the general equation (239), becomes transformed into b\s' s)cos.d> - . b\s' s) 2 2 2 or by transposing the absolute given quantity - - ^ - , we get s (240). Now, by carefully examining the nature of this equation, it will immediately appear to be composed of the two quadratic factors b\s's) _ W s} * 2 -- r =0, and* 2 cosV4 2 a 2 X*4--^ " ' where it is manifest, that each of these expressions involve two roots of the original equation, and the number of the real positive roots, indicates the number of positions in which the body may float in a OF THE POSITIONS OF EQUILIBRIUM. 319 state of quiescence, while the absolute values of the roots determine the positions themselves. 401. Let each of the above quadratic factors be transformed into an equation, by transposing the given term --^ -, and we shall obtain < _yy < ) 2 b\s' s) and when the value of s', or the specific gravity of the supporting fluid is expressed by unity, as is the case with water ; then, we have for the pure quadratic, x z 2 (1 s). (241). and for the adfected quadratic, it is ^2 ___ cos ^ ^ 4 #! _ a 2 x x __ #(i_ s ). (242). Let the square root of both sides of equation (241) be extracted, and we shall obtain x = b^/T^7; (243). but from equation (236), we have xy = b\l s), and this, by substituting the above value of x, becomes b /703~26.514, and similarly, for cos.^>, we get cos.0 = - V 4 X 28* 1 8 2 r= 0.94693 ; oo consequently, by substitution, we obtain zz 26.514X0.94693X2 ~ 50.212. Y 2 324 OP THE POSITIONS OF EQUILIBRIUM. 408. The expression for the area of the immersed figure ABED, is ^sin.2^>(6 2 xy}, and the expression for the area of the whole section ABC, is \tf sin. 2^ ; and by the principles of floatation, these are to one another, as the specific gravity of the floating solid, is to that of the fluid on which it floats ; hence we have xy) : i& 2 sin.20 : : 0.2013 : 1, and by suppressing the common term ^sin.2^, we get {tf xy} : 6 2 :: 0.2013 : 1, and from this, by putting the product of the extreme terms, equal to the product of the means, we obtain and finally, by substituting the numerical values, we have 27. 152X23.06 = 0.7987 X28 2 very nearly, which satisfies the other condition of equilibrium ; hence we infer, that the subcontrary positions represented above, are those which the body assumes when floating in a state of quiescence with two of its angles below the plane of floatation. 409. When a, b and c are equal to one another ; that is, when the triangular section is equilateral; then, the general equation (239), becomes b\s' *)* and from this equation, by transposing the given term -- pj -- , we get 8 '- s Now, it is manifest, that the equation in its present form, is com- posed of the two quadratic factors and these factors, by transposing the given term - , -- in each, become transformed into the following quadratic equations, viz. OF THE POSITIONS OF EQUILIBRIUM. 325 and supposing the specific gravity of the fluid, or the value of s' to be expressed by unity ; then, these equations become 'Resolving these equations by the rules which the writers on algebra have laid down for that purpose, we shall have for the pure quadratic, and ?/zz:6\/(l s}; and again, for the adfected form, it is (251). and y b {0.866 cos.^v/ 0.75 cos 8 .^ (1 r^ f ^ In the equations, (251), it is obvious that the values of a; and y are assignable, whatever may be the value of s, provided that it is less than unity ; and since x and y are each expressed by the same quantity, it follows that they are equal to one another, and consequently the body will float in equilibrio, when the immersed side or base of the section is parallel to the surface of the fluid. 410. EXAMPLE. If the floating prism be of fir from the forest of Mar, of which the specific gravity is 0.686, that of water being unity; then we have and if the value of b, or the side of the equilateral triangle be 28 inches, we get xi=y 0.56X28 = 15.69 inches; and the position of equilibrium corresponding to this common value of a; and y, is exhibited in the annexed diagram, where IK is the horizontal surface of the fluid, DCE being the extant portion of the floating body, and ABED the part immersed below the plane of floatation ; c D and c E being re- spectively equal to 15.69 inches. Bisect AB in F, and draw the straight lines FD and FE to inter- sect the surface of the fluid in the points D and E ; then, because the triangle ABC is equilateral, and CD equal to CE by the construction, it follows, that FD and FE are equal to one another; this satisfies one of the conditions of equilibrium, and we have now to inquire if the area of the immersed portion ABED, is to the area of the whole section ABC, as the fraction 0.686 is to unity. \|/ i==~=- 326 OF THE POSITIONS OF EQUILIBRIUM. Now, by the principles of mensuration, we know that the area of a plane triangle, of which the three sides are equal, is expressed by one fourth of the square of the side, drawn into the square root of the number 3 ; consequently, the area of the whole section ABC, is and the area of the extant part DEC, is therefore, the area of the immersed part ABED, is (a' a") = JftV 3" i*V~3 = 0-433 (fi x 2 ) ; hence, by the principles of floatation, we get 0.433 (& *') : 0.4336 2 : : 0.686 : 1, and by equating the products of the extremes and means, it is x* = b*(\ 0.686) 0.3146 2 . But b is 28 and x 15.69 inches ; therefore, if these values of b and x be substituted instead of them in the preceding equation, we shall have 15.69* nr0.314x28 2 =r 246.176. In this case also, one of the conditions of equilibrium is satisfied ; hence we conclude, that the position which we have represented above is the true one, since both the conditions upon which the equilibrium depends, have been fulfilled by the results as obtained from the reduc- tion of the formula. The value of x and y, as exhibited in equations (252), will indi- cate two other positions of equilibrium, subcontrary to each other ; but in order that those positions may be coiisistent with the conditions of the problem, it becomes necessary to assign the limits of s, or the specific gravity of the floating body ; for it is manifest, that beyond certain limits, the conditions specified in the problem cannot obtain. 411. Now, in the case of the isosceles triangle, it has been shown, that the greater limit of the specific gravity, is and consequently, when the triangle is equilateral, s=i~ -ft = 0.5; and moreover, it has also been shown, that when the triangle is isosceles, the lesser limit of the specific gravity, is OF THE POSITIONS OF EQUILIBRIUM. which, when the triangle is equilateral, becomes 7 s = =0.4375, 16 1 and the arithmetical mean of these, from equation (247), is 327 Let therefore this value of s be substituted instead of it in the expressions, class (252), and we shall obtain x = 25.95, and x = 16.05 inches, the corresponding values of y being y =z 16.05, and y =. 25.95 inches. 412. The positions of equilibrium, as indicated by these values of x and?/, are as represented in the annexed diagrams, where IK is the horizontal surface of the fluid, ABED, abed the immersed, and DEC, dec the extant portions of the section corresponding to the positions ABC and abc, in which CD and ce are each equal to 25.95 inches, and CE, cd equal to 16.05 inches, being the respective values of x and y, as determined from equation (252). Bisect A B and a b in the points F and f, and draw the straight lines FD, FE and fd, fe intersecting the horizontal surface of the fluid in the points D, E and d, e ; then, when the body floats in a state of equi- librium, the lines FD, FE,/C? and/e are equal among themselves. This is very easily proved, for since the triangle ABC is equilateral, the angle ACB is equal to sixty degrees, and consequently its half, or the angles ACF and BCF are each of them equal to thirty degrees; therefore, by the principles of Plane Trigonometry, we have DF 8 =CD 2 -4-CF ? 2CD.CFCOS.30, and similarly, by the same principles, we get F E 2 HZ C E 4 -4- C F 2 2C E.C F COS. 30 ; 328 OF THE POSITIONS OF EQUILIBRIUM. but according to the conditions of equilibrium, these are equal, hence we have CD 2 2C D.C F COS.30 = C E 2 2CE.CF COS.30 ; therefore, by substituting the analytical expressions, and transposing, we get x 3 z/ 2 = 2d cos.30(a? y\ and dividing both sides by (x y), we shall have 2dcos.30 = a:-f y. By Plane Trigonometry cos. 30 nr sin. 60, and by the property of the equilateral triangle, we have e? = b sin. 60 ; consequently, by sub- stitution, we get 2&sin 2 .60~ x + y; or numerically, we obtain 2X28x1 = 25.954-16.05 42. 413. Hence it appears, that in so far as the equilibrium of floatation depends upon the equality of the lines FD and FE, the condition is completely satisfied, and the same may be said respecting the lines fd and/e ; but it is manifest, that another condition must be fulfilled before the body attains a state of perfect quiescence, and that is, that the area of the immersed part ABED, is to the area of the whole section ABC, as the specific gravity of the solid body, is to that of the fluid on which it floats, or as 0.46875 to unity : now, this condition is evidently satisfied, when x y = P(l 0.46875), therefore, numerically we obtain 25.95X16.05 28 2 X0.53125i=41.65. Here then, both the conditions of equilibrium are satisfied, and from this we infer, that the positions exhibited in the diagram are the true ones, the downward pressure of the body in that state, being perfectly equipoised by the upward pressure of the fluid. 414. What we have hitherto done respecting the positions of equi- librium, has reference only to a solid homogeneous triangular prism, floating on the surface of a fluid with its axis of motion * horizontal ; * When a solid homogeneous body, in a state of equilibrium on the surface of a fluid is disturbed by the application of an external force, it will endeavour to restore itself by turning round a horizontal line passing through its centre of gravity, and th^s line on which the body revolves, is called the axis of motion. OF THE POSITIONS OF EQUILIBRIUM. 329 and but there are various other forms, which are not less frequent in the practice of naval architecture, nor less important as subjects of theo- retical inquiry : some of these we now proceed to investigate. PROBLEM LVIII. 415. Suppose that a solid homogeneous body in the form of a rectangular prism, floats upon the surface of a fluid of greater specific gravity than itself, in such a manner, that only one of its edges falls below the plane of floatation : It is required to determine what position the body assumes, when it has attained a state of perfect quiescence. Let ABCD be a vertical section, at right angles to the horizontal axis passing through the centre of gravity of the rectangular prism, and let I K be the surface of the fluid, on which the body floats in a state of equilibrium, being the extant portion mvn the part which falls below the plane of floatation. Bisect mn in F and vn in H, and draw the straight lines DF and mil, intersecting each other in g the centre of gravity of the immersed triangle mvn. Join the points A, c and B, D by the diagonals AC and BD, intersecting in o the centre of gravity of the rectangular section ABCD, and draw og. Then, because the body floats upon the surface of the fluid in a state of equilibrium according to the conditions of the problem ; it follows from the laws of floatation, that the straight line Gy is perpen- dicular to IK. Through F the point of bisection of mn the base of the immersed triangle, and parallel to go, draw FP meeting the diagonal BD in the point p, and join PTW, PW; therefore, because the straight line ga is perpendicular to mn the plane of floatation, it is evident that FP is also perpendicular to mn, and consequently, pm and PW are equal to one another. This is a condition of equilibrium which holds universally, and another is, that the area of the immersed triangle m D n, is to the area of the whole section ABCD, as the specific gravity of the solid, is to 330 OF THE POSITIONS OF EQUILIBRIUM. the specific gravity of the fluid on which it floats ; when both these conditions obtain, the body will float permanently in a state of equi- librium. Put a zz: A D or BC, one of the sides of the section that contain the immersed angle, b zz: DC or AB, the other containing side; s zz the specific gravity of the floating solid, s' zz the specific gravity of the supporting fluid, or that on which the body floats, x zz Dm, the part of the AD which is immersed under mn the plane of floatation, y zz DW, the corresponding portion of the side DC ; a' zz the area of the whole rectangular section A BCD, and o"zz the area of the immersed portion mvn. Then, since the section of the solid is considered to be uniform, with respect to the axis of motion, throughout the whole of its length, we have a"s' = a's. (253). But by the principles of mensuration, the area of the whole rectan- gular section A BCD, is expressed by the product of its two sides; that is, a' zz a b, and the area of the immersed triangle mow, is Let these values of a' and a" be substituted instead of them in the equation (253), and we shall have abs=\xys'. (254). By the property of the right angled triangle, it is B D 2 zz A D* 4- A B? > or by putting d to denote the diagonal BD, we get from which, by extracting the square root, we obtain and by the principles of Plane Trigonometry, it is \/a 8 + 6 2 : a : : rad. : COS.ADB ; and similarly, by Trigonometry, we have 1^/0? -j- & 2 : b : : rad. : cos. BDC ; OF THE POSITIONS OF EQUILIBRIUM. 331 therefore, by working out the above analogies, and putting radius equal to unity, we shall have COS.ADBIZZ , and cos. BD cur . Since gG and FP are parallel, and g? equal to one third of DF ; it follows, that GP is equal to one third of DP; or which is the same thing, DP is equal to three fourths of BD ; that is DPz=|Va 2 -f b\ When two sides of a plane triangle are given, together with the angle of their inclination, as is the case in the triangles WIDP and WDP; then, the writers on Trigonometry have demonstrated, that 7WP 2 ZZDW 2 -f- DP 2 - 2D97Z.DPCOS.ADB, and WP 2 DW 2 -|- DP 2 - 2DW.DPCOS.BDC ; and these, by the principles of floatation, are equal, hence we get Dm 2 - SDTW.DPCOS.ADBUZDW 2 - 2DW.D P COS.B DC. Let the analytical expressions of the several quantities Dm, on, DP, cos. ADB and COS.B DC, be substituted in the above equation, and we shall obtain *-!:=,- ( 255 ). If both sides of the equation (254), be divided by the expression Jars', we shall obtain as follows, viz. the square of which, is y ~ : Now, if these values of?/ and z/ 2 , be respectively substituted instead of them in equation (255), we shall obtain 3ab*s 2 __ " : : and finally, by reduction and transposition, we get (256). And if we consider the value of s', or the specific gravity of the fluid, to be expressed by unity, as is the case with water ; then the above general equation becomes 332 OF THE POSITIONS OF EQUILIBRIUM. 416. When the specific gravity of the solid body is so related to that of the fluid, as to fulfil the conditions of the problem, the roots of the above equation will determine the positions of equilibrium ; but since there cannot be more than three real positive values of x in the equation, it follows, that there cannot be more than three positions in which the prism will float in a state of rest, with only one of its edges below the surface of the fluid. 417. If a and b are equal to one another ; that is, if the transverse section of the floating body be a square at right angles to the axis of motion ; then, equation (257) becomes O j x 4 ' -- and from this, by transposition, we obtain a; 4 X* 3 -f 3b*sx 46V = 0. (258). Now, it is obvious, that this equation is composed of the two fol- lowing quadratic factors, O I a? 2 26 2 s, and x 9 X a; -f 2& 2 *; which being converted into equations, gives x^ Ws, (259). and similarly, from the adfected factor, we obtain x 9 ^Xx= 26 2 s. (260). Since these two quadratic equations are deduced from the factors which constitute the particular biquadratic (258), it follows, that the real positive roots which they contain, must indicate the positions of equilibrium according to their number. If we extract the square root of both sides of the equation (259), we shall obtain x = b^; (261). but by equation (254), we have lxy = b z s\ consequently, by division, we get OA2 (262). OF THE POSITIONS OF EQUILIBRIUM. 333 Here then it is manifest, that the values of x and y are each expressed by the same quantity ; hence we infer, that the body floats with one diagonal of its vertical section perpendicular to the surface of the fluid, and the other parallel to it. 418. The practical rule afforded by the equations (261 and 262), may be expressed in words at length as follows. RULE. Multiply the square root of twice the specific gravity of the solid, by the side of the square section, and the product will give the length of the immersed part , when the body is in a state of rest. 419. EXAMPLE. Suppose a square parallelopipedon, whose side is equal to 18 inches, to be placed upon a fluid with one of its angles immersed, and one of its diagonals vertical ; how much of the body will fall below the plane of floatation, supposing its specific gravity to be 0.326, that of the supporting fluid being equal to unity? Here, by operating according to the rule, we get ar= 18V 2 X0.326 zz 14.526 inches, and for the corresponding value of y, we have , = ^ = ^6^, Consequently, the position of equilibrium thus indicated, is as represented in the annexed diagram; where IK is the surface of the fluid, AC the horizontal and B D the ver- tical diagonal ; Dm and i>n being respectively equal to 14.526 inches, as deter- mined by the foregoing arithmetical process. 420. Take DP equal to three fourths of BD, and draw pm and pn meeting the surface of the fluid in the points m and n ; then are pm and pn equal to one another ; this is one of the conditions necessary to a state of equilibrium, when neither of the diagonals is vertical ; but in the present instance, the condition of equality will obtain wherever the point P may be taken, and consequently, the equilibrium is not in- fluenced by the position of that point. 334 OF THE POSITIONS OF EQUILIBRIUM. 421. The only condition, therefore, which establishes the equilibrium in this case, is, that the area of the immersed triangle mi>n, is to the area of the whole section A BCD, as the specific gravity of the solid is to that of the supporting fluid. 422. If the specific gravity of the solid be equal to one half that of the fluid on which it floats; then, AC will coincide with IK, and in this state the specific gravity attains its maximum value; for if it exceeds this limit, more than one angle of the solid will become immersed, and this is contrary to the conditions of the problem. 423. When the specific gravity of the floating solid is properly limited, the equation (260), has two real positive roots; hence we infer, that there are two other positions in which the body may float in a state of equilibrium, and these will be determined by the resolu- tion of the equation. Therefore, complete the square, and we get and by extracting the square root, it is 3b b x =4= -V (9 32s) ; consequently, by transposition, we have b C 4 - (263) . and the corresponding values of y, are (264). 424. Now, by attentively examining these equations, it will appear, that in order to have the values of x and y real quantities, the value of s y or the specific gravity of the solid body, must be such, that thirty two times that quantity shall not exceed the number 9 ; and moreover, in order that the greatest value of x and y may be less than b the side of the square section, it is necessary that thirty two times the specific gravity of the solid shall not be less than the number 8. 425. When the value of s is taken such, that 32s zz 9; then we have \/9 325 ~ ; in which case the values of x and y are each of them equal to three fourths of b ; but when the value of s is such, that 325 8 ; then we have <\/9 32s = =t 1 , and consequently, the two values of x are b and \b respectively, the corresponding values of y being \b and b; and the positions of equilibrium corresponding to OF THE POSITIONS OF EQUILIBRIUM. 335 the above values of x and y, are as represented in the annexed diagrams, where i K is the horizontal surface of the fluid ; No. 1 the position corresponding to x and y, when they are respectively equal to three fourths of b ; No. 2 the position indicated by x zn b and y=\b, and No. 3 that which corresponds to the reverse values of x and y, viz. when x is equal to \b, and y equal to b ; in both of which cases, one angle of the figure is under the plane of floatation, and another coincident with it; but this is scarcely consistent with the conditions of the problem, which distinctly intimates, that only one edge or angle of the floating body shall be immersed in the fluid, and this implies, that all the other edges or angles shall be wholly extant, or in other words, that the greatest values of x and y, shall be less than the side of the square section. In order, therefore, that this condition may obtain, the specific gravity of the body must be less than T 9 T , which gives the position in No. 1 ; and greater than 3- 8 T , which gives the positions in Nos. 2 and 3 ; consequently, by taking the arithmetical mean between these limits, we shall have s 0.265625, and the equations (263 and 264) become the corresponding values of y, being But the square root of 9 8.5 is 0.7071 very nearly ; therefore, if b be equal to 18 inches, as in the preceding example, we shall have 18X3.7071 the corresponding values of y being y = 10.318, and y = 16.682 inches. Now, it is manifest, that none of the positions represented above, resemble that which is indicated by the values of x and y just deter- mined ; but the true positions which these values furnish, are such as correspond to a state of equilibrium, and they are exhibited in the 336 OF THE POSITIONS OF EQUILIBRIUM, subjoined figures, whereas in all the previous cases, IK is the horizontal surface of the fluid ; men and mo A Bra being the areas of the immersed and extant portions of the body, corresponding to # 16.682 inches, and 2/z=:10.318 inches ; the subcontrary figures odp and oabcp being the respective areas when #=: 10.318 inches, and y zn 16.682 inches. Bisect mn in F, and through the point F draw FP at right angles to mn, meeting the diagonal AC in the point p, and join pm and PW; then it is manifest, that the straight lines pm and PW are equal to one another, as ought to be the case when the solid floats in a state of equilibrium; and moreover, the area of the immersed portions men, and odp, are to the area of the entire sections A BCD and abed, as the specific gravity of the floating solid, is to that of the supporting fluid. 426. If the conditions of the problem should be reversed, that is, if three angles of the figure be immersed beneath the plane of floata- tion, and one extant above it; then, by a similar mode of investiga- tion, it may be shown, that \xy s' &(' s) and furthermore, that .. __ ~ ' 2 ~ 2 Now, these being similar equations to those which correspond to the case of one angle being immersed beneath the surface of the fluid ; it follows, that all the other steps of the investigation would also be similar, and consequently they need not be repeated. PROBLEM LIX. 427. Suppose that a solid homogeneous prismatic figure, whose transverse section is rectangular, is found to float in a state of equilibrium on the surface of a fluid with its two edges immersed : OF THE POSITIONS OF EQUILIBRIUM. 337 It is required to determine the positions assumed by the solid, when it is in a state of quiescence. The solution of this problem is attended with greater difficulty than either of the preceding ones respecting the positions of equilibrium ; the superior difficulty in this case, arises from the situation of the lines, whose equality constitutes the second condition of equilibrium ; in the foregoing cases, this equality was determined by the resolution of the simple problem in Plane Trigonometry, where two sides and the contained angle are given, and it is required to find the third side, or that which subtends the given angle ; in the present instance, how- ever, this mode of comparison does not take place, and the equality of the lines alluded to, or rather the condition of equilibrium depend- ing on such an equality, can only be established by a series of com- plicated analogies, arising from the similarity of triangles determined by the construction. Let A BCD represent a transverse section perpendicular to the axis of a homogeneous rectangular prism, which floats in equilibrio on the surface of a fluid of greater specific gravity than itself, and in such a manner, that two of its angles are wholly immersed be- neath the plane of floatation re- presented by HE; IK being the horizontal surface of the fluid, HECD the immersed portion of the section, and ABEH the extant portion. Let G and g be the centres of gravity of the whole section ABCD, and the immersed part HECD; join G#, then, if the position which the body has assumed be that of equilibrium, the line Gg is perpen- dicular to HE the plane of floatation, and the area of the immersed part HECD, is to the area of the whole section ABCD, as the specific gravity of the solid is to that of the supporting fluid. Through the point c, the most elevated of the immersed angles of the figure, draw cb perpendicular to IK, and through the points o and g draw GC and ge perpendicular to cb; then, if the position which the body has assumed be that of equilibrium, the straight lines GC and g e are equal to one another. The conditions under which the body floats in a state of quiescence, therefore are, VOL. I. 7. 338 OF THE POSITIONS OF EQUILIBRIUM. 1. That the area of the immersed part, and that of the whole section, are to one another as the specific gravities of the solid and the fluid. 2. That the horizontal lines, intercepted between the centres of gravity, and the vertical line passing through the most elevated of the immersed angles, are equal to one another. Through the points o and g, draw the straight lines oa and gv perpendicular to BC the side of the section; and through the points a and v, draw ab and vd parallel to the horizon, and am,vn perpen- diculars to GC and ge\ and finally, through E and g, and parallel to CD and CB, draw the straight lines E and sr, and the construction is finished. Then it is manifest, that by means of the parallel and perpendicular lines employed in the construction, we can form a series of similar triangles, which will lead us by separate and independent analogies, to the comparison of the lines GC and ge, on whose equality the equi- librium of floatation depends. Put a =z AD or b c, the longest side of the transverse section, b =: AB or DC, the shortest side, d ~ gv, the perpendicular distance between the centre of gravity of the immersed part, and the side of the section B c ; a' the area of the whole section ABC D, a"~ the area of the immersed part HECD ; x zz DH, the distance between the lowest immersed angle, and the corresponding extremity of the line of floatation, y CE, the distance between the highest immersed angle and the other extremity ; and as heretofore, let s denote the specific gravity of the solid body, and s' the specific gravity of the fluid on which it floats ; then, by the principles of floatation, we have a" :a'::s: s', and from this, by equating the products of the extreme and mean terms, we get a's = a"s f . Now, by the principles of mensuration, the area of the rectangular section ABCD, is expressed by the product of its two containing side* AB and BC; hence we have OF THE POSITIONS OF EQUILIBRIUM. 339 and moreover, the area of the immersed part HE CD, is expressed by half the sum of the parallel sides drawn into the perpendicular dis- tance between them ; consequently, we obtain abs= \bs\x + y), and finally, by multiplication and division, it is 2s s'O + y)- (265). The equation which we have just investigated, involves one of the conditions of equilibrium, viz. that in which the area of the immersed part, and that of the whole section, are to each other, as the specific gravity of the solid is to that of the fluid ; but in order to discover the equation which involves the other condition, we must have recourse to a separate construction, as follows. Let IK be the surface of the fluid, and HECD the immersed portion of the section, as in the general dia- gram preceding. Bisect D H and c E, ~ \'- " the parallel sides of the figure, in the points t and z, and draw tz;* then, by the principles of mechanics, the l =^^-^ straight line tz passes through the centre of gravity of the figure HECD, and divides it into two parts such, that gz : gt : : SDH-J-CE : DH + 2cE. Through the point E draw E* parallel to DC, and bisect DC and HE in the points i and h; draw ih, bisecting E the side of the triangle HE* in the point q, and join th and uq, intersecting one another in the point o ; then is the point o thus determined, the centre of gravity of the triangular space H E t. Draw the diagonal D E, bisecting q i in p the centre of gravity of the rectangular space ECD t, and join po intersecting tz in g ; then is g the centre of gravity of the quadrilateral space HECD which falls below the plane of floatation. Through the point g thus determined, draw the straight line gv parallel to DC, the lowest immersed side of the section, and meeting CE in the point v; then is gv the quantity to be assigned by the construction. From the point z and parallel to CD, draw zf meeting DH perpen- * It is a circumstance entirely accidental, that the lines tz and or by supposing s' equal to unity, as is the case with water, we have t/:n2as x. OF THE POSITIONS OF EQUILIBRIUM. 343 Let this value of y be substituted instead of it, wherever it occurs in the above equation, and we shall obtain which being reduced and thrown into a simpler form, becomes 2x 8 6asx 2 -f (12aV 6a 2 s -f 6 2 )a;:=: 8aV-|- a 2 s 6aV. (268). Now, according to the nature of the generation of equations, it is manifest that the above expression is composed of one simple and one quadratic factor; but as #zzO, is obviously one of the members from which the equation is derived, for in that case, the whole vanishes, or which is the same thing, when all the terms of the equation are arranged on one side with their proper signs, the sum total is equal to nothing. Granting therefore, that as a:~0, is one of the constituent factors, then we shall have x~ as, and by referring to equation (265), we shall obtain therefore, by transposition, it is Consequently, the position of equilibrium assumed by the solid in this instance, is when x and y are equal to one another ; that is, when the side of the body is parallel to the horizon, the depth to which it sinks being determined by the measure of its specific gravity. Let all the terms of the equation (268) be transposed to one side, and let their aggregate be divided by (as #), and there will arise 2x 2 4asx+$a 2 s* 6a 2 s-j-6 2 0, and from this, by transposition and division, we obtain x 2 2a sx = a*s (3 4*) j#. From this equation it may be inferred, that if the roots or values of x be both real and positive quantities, and each of them less than a the upward side of the section ; then the body may have two other positions of equilibrium, which will be determined by reducing the equation. Complete the square, and we obtain V 3aV(l *) J# ; 344 OF THE POSITIONS OF EQUILIBRIUM. therefore, by evolution, it becomes x asi= 3a z s(l s) i and by transposition, we have x = as+^/3a?s(l s) %b* ; and the corresponding values of y f are (269). 3a 8 s(l s) \tf. (270). It would be superfluous in this place, to give a numerical example to illustrate the reduction of equations (269 and 270) ; we shall therefore drop the discussion of the oblong rectangular section, and proceed to inquire, what are the circumstances which combine to establish the equilibrium in a square. 430. Therefore, when a and b are equal to one another, that is, when the transverse section is a square ; then the general equation (268), becomes 2x* 6sa; 2 -f 6 8 (12s 8 6s -f 1 ) x = ^(Ss 8 6s*-fs); (271). but one of the constituent factors of this equation is, bs x ; consequently, by transposition and division, the other factor becomes 2a 8 4bsx 4- &X8s 8 65 -f 1) = ; and from this, by transposing and dividing by 2, we shall get a? 8 2bsx b\3s 4s 8 J). (272). Now, it is manifest, that when the section is a square, as we have assumed it to be in the present instance, the factor bs arzzO, gives x = bs, and from equation (265), we obtain y ' 2bs x ~%bs b$~ bs. Hence it appears, that the body will float in a state of quiescence, when any of its sides is horizontal, and in this case, the problem is reduced to the determination of the depth to which the body sinks, and this is entirely dependent on the A measure of its specific gravity. 431. If the specific gravity of the floating solid, be to that of the fluid j= on which it floats in the ratio of I to 2 ; j then the body will sink to one half its f depth, as represented in the annexed diagram, where IK is the horizontal OF THE POSITIONS OF EQUILIBRIUM. 345 surface of the fluid; EF the water line, or line of floatation; EFCD the immersed portion of the section, and ABFE the part that is extant, and these in the present case, are equal to one another, since the specific gravity of the fluid is double the specific gravity of the float- ing body. It is also obvious from the figure in this case, that if the body were to revolve about its axis of motion, till one of the diagonals assumed a vertical position, it would then float in equilibrio with one half the section immersed, the horizontal diagonal in that case coinciding with the surface of the fluid. If we resolve the quadratic equation (272), we shall obtain twa other positions, in which the body will float in equilibrio, provided that the specific gravity be retained within proper limits ; for it is on this limitation solely, that the equilibrium of floatation depends. Complete the square, and we shall have a* %bsx + 6V= b\3s 3s 2 j), extract the square root, and we obtain x bs=^b^3s(l s) , therefore, by transposition, we have x = b(s+)/3s(l s) J. (273). But by equation (265), we have 2bs = x -\-y ; consequently, by transposition, we obtain y 2bs x; therefore, by substitution, we get y b(s =pV 35(1 *) J. (274). Now, with regard to the limits of the specific gravity, it is easy to perceive, that if the quantity ^3s(\ s) be greater than s 9 the least values of x and y will be negative ; and if the expression * -|--V/3s(l s) J be greater than unity, the greatest values of x and y will be greater than b ; consequently, neither of them satisfies the conditions of the problem. But it is further manifest, that in order to have the values of x and y real quantities, the expression 3s(l s) must exceed the fraction ^ ; now the least value of s that will fulfil this condition, is s^z j, in which case we have 35(1 S)ZZ3X|XJ = T Q 6, from which subtracting | or T 8 F , we get 346 OF THE POSITIONS OF EQUILIBRIUM. the square root of which is J, hence it is 432. The position of equilibrium indicated by these values of a; and y, is represented in the annexed diagram, where IK is the hori- zontal surface of the fluid; AE the line of floatation ; A E c D the immersed, and ABE the extant por- tion of the section. Here it is obvious, that since the plane of floatation passes through the angle A, and bisects the oppo- site side in the point E; the immersed part AECD, is equal to three fourths of the entire section ABCD, as it ought to be, in consequence of the specific gravity of the body, being assumed equal to three fourths of the specific gravity of the fluid. It may also be readily shown, that the centre of gravity of the whole section, and that of the immersed part occur in the same vertical line ; but this is not necessary in the present instance, as we are only endeavouring to discover the limits of the specific gravity. 433. The position of equilibrium corresponding to the value of x\b, and y zn b, is similar and subcontrary to the position represented in the preceding diagram, and this being the case, it is unnecessary to exhibit it ; we shall therefore proceed to determine the greatest limit of the specific gravity that will fulfil the conditions of the problem ; for which purpose, we have &(1 -.) = !, from which, by separating the terms, we get 3 S 3s 2 J; therefore, by transposition and division, it becomes Complete the square, and we obtain s 2 s + imi 1 = ^, hence, by extracting the square root, we get (275). and finally, by transposition, we have 434. From what has been done above, it is manifest, that the least limit of the specific gravity is f , and the greatest is (3 -f- ^ 3) ; the OF THE POSITIONS OF EQUILIBRIUM. 347 former giving the position represented in the preceding diagram, and the latter that which is exhibited in the marginal figure ; where the body floats with one of its flat surfaces horizontal, IK being the surface of the fluid ; E F the water line, or line of floatation; EFCD being the im- mersed part of the section, and A B F E the part which is extant, the immersed part being to the whole section, as 0.788675 to 1 ; that is ED: AD:: i(3 4-^/3) ! Since (3 y/'S is also a root of the equation (275), it follows, that the body will float in equilibrio with one of its flat surfaces horizontal, as in the annexed figure, when the spe- cific gravity is equal to the above quantity ; for in that case the radical expression ^ 3s(l s) J in equa- tions (273 and 274) vanishes, and x and y become each equal to , and the immersed part of the section is to the whole, as 0.211 to 1 ; that is ED : AD : : i(3 V*3) : 1. 435. Having established the limits between which the solid floats in equilibrio with a flat surface upwards, but inclined to the horizon in various angles depending on the specific gravity ; we must now return to the equations (273 and 274), in which the conditions are indicated, that have enabled us to assign the above limits to the relative weight of the floating body. Taking the arithmetical mean between the limits above determined, we shall have s = |(0.75 + 0.788675) = 0.7693375 ; consequently, if the side of the square section be equal to 20 inches, the values of x and y will be determined by the following operation. 436. Let the mean calculated value of s the specific gravity of the floating body, and the given value of b the side of its square section, be respectively substituted in equation (273), and we shall have, for the greatest value of a-, x =20(0.7693375+ V 3x0.7693375 x 0.2306625-0.5) =18.96inches, 348 OF THE POSITIONS OF EQUILIBRIUM. and for the least value of x, it is a:zz20(0.7693375 v/ 3 X 0.7693375 X 0.2306625 0.5)zzl 1 .8 inches, the corresponding values of y as found from equation (274), are y 11.8 inches, and y 18.96 inches. Now, the positions of equilibrium corresponding to the above values of x and ?/, are as represented in the annexed diagrams, where 1 IK is the hori- zontal surface of the fluid ; E F and ef the lines of floatation ABCD the position corresponding to am 18.96, and y rz 11.8 inches ; the position abed, being that arising from the reversed values of x and?/; that is, a; 11.8 and y 18.96 inches; EFCD being the immersed part of the section in the one case, and efcd in the other. 437. If the specific gravity be taken equal to the complement of the above mean, we shall obtain two other positions of equilibrium in which the body will float, corresponding precisely to the above figures inverted, af- ter the man- A^ ^^o ner exhibited in the mar- ginal dia- gram ; where ABCD is the position cor- responding to 20 18.96zz 1.04 inches, and abed the position correspond- ing to 20 11.8:^:8.2 inches; the immersed portions EFCD and efcd in the one case, being equal to ABFE and abfe, the extant portions in the other. It is moreover manifest, that the centres of gravity of the immersed and extant portions of the section are situated in the same vertical ; for they are connected by a straight line which passes through the centre of gravity of the entire figure ABCD; but in the case of an equilibrium, the centres of gravity of the whole, and the immersed part, are situated in the same vertical line ; therefore also, the centres K OF THE POSITIONS OF EQUILIBRIUM. 349 of gravity of the immersed and extant parts occur in the same vertical. 438, We have now to inquire if the second condition of equilibrium be satisfied ; that is, if the area of the whole section and that of the immersed part, are to one another as the specific gravity of the fluid, is to that of the solid. Now, the area of the whole section, is 20x20 = 400, and that of the immersed portion, is (1 8.96 -f 11.8)10 = 307.6, or 92.4; and the mean specific gravity is, 0.7693375 or 0.2306625 ; therefore we have 400 : 307.6 : : 1000 : 769, and 400 : 92.4 : : 1000 : 231 ; consequently, the positions of equilibrium are as exhibited above. PROBLEM LX. 439. A solid homogeneous body, having the section which cuts the axis of motion perpendicularly, in the form of a common or Apollonian parabola, is supposed to float upon a fluid of greater specific gravity than itself: It is required to determine the position it assumes when in a state of equilibrium, supposing its base or extreme ordinate to be entirely above the surface of the fluid. In the resolution of this problem, we shall have occasion to advert to several properties of the common parabola, a curve which, by reason of its easy construction, and the simplicity of its equation, has been very extensively introduced into mechanical science; and from the frequency of its occurrence, it is presumed, that its chief properties are familiar to and clearly understood by the greatest part of our readers ; so that in tracing the positions of equilibrium, it will not be requisite to demonstrate any of the properties to which we refer, and which, by the nature of the investigation, we are constrained to employ. A Let AD B be a common para- bola, representing a transverse section of a solid uniform body> floating at rest upon a fluid whose surface is IK, and let T___ J[j DC be the axis, AB the base or extreme ordinate, (which, by "1 the conditions of the problem, 350 OF THE POSITIONS OF EQUILIBRIUM. is entirely above the surface of the fluid), and FII the line of floatation. Bisect FH in w, and through n draw nm parallel to DC the axis of the parabola, and meeting the curve in the point m ; then is mn when produced to r a diameter of the curve, whose vertex is in the point m. Through the point m, draw mt parallel to AB the base of the parabola, and meeting the axis DC in the point K ; produce CD to E, making BE equal to DK, and join EW, then by the property of the parabola, Em is a tangent to the curve in the point m, and it is parallel to FH the line of floatation, or the double ordinate to the diameter mr. Let P be the place of the focus; join ?WP, and through H the extreme point of the line of floatation, draw uv parallel to AB the base of the figure, and meeting the diameter mr perpendicularly in the point v; then are the triangles KETW and vnii similar to one another. Take DG equal to three fifths of DC, and mg equal to three fifths of mn; then are G and g respectively the centres of gravity of the whole parabola ADB and of the part FDH; join og, then by the principles of floatation, the straight lines Gg and FH are perpendicular to one another, and consequently, the triangles KEW and wng are similar. Put a m DC, the axis of the parabola or section of the floating body, 26zz AB, the base or double ordinate corresponding to the axis DC, a' zz the area of the whole parabola ADB, a"z= the area of the immersed part FDH, x ~ mn, an abscissa of the diameter mr, y zz: H n, the corresponding ordinate, z zz Km, the ordinate passing through m the point of contact, p zz the parameter or latus rectum to the axis, s zz the specific gravity of the floating solid, and s' zz the specific gravity of the fluid on which it floats. Now, supposing that all the sections which are perpendicular to the axis of motion, are equal to one another; then, according to the principles of floatation, we have a'szza'Y. (276). OF THE POSITIONS OF EQUILIBRIUM. 351 But the writers on mensuration have demonstrated, that the area of the common parabola, is equal to two thirds of its circumscribing rectangle, or equal to four thirds of the rectangle described upon the axis and the ordinate ; according to this principle therefore, we have a'zzi^DcXAC, and a"~%vnXmn. By the equation to the curve, it is therefore, by division, we obtain & DKrr ; P but according to the construction, we have P and by the property of the right angled triangle, it is K E 2 -f K m> E m 3 ; that is, E w 2 = therefore, by extracting the square root, we shall have P and from the similar triangles KETW. and vrm, we get Ts.m : Km : : nn : VH; and this, by substituting the analytical equivalents, becomes z : z : : y : v H ; P ' consequently, by working out the analogy, we have py VHT Hence then, by substituting the respective literal representatives, for the quantities DC, AC and VH, mn, the preceding values of d and a", become a'z 4 b, and a"= Therefore, let these values of a' and a" be substituted instead of them in the equation (276), and we shall obtain = If we suppose the axis of the parabola to be vertical, and its base or double ordinate horizontal ; then the points m and D coincide with 352 OF THE POSITIONS OF EQUILIBRIUM. one another, and KTWZHZ vanishes; consequently, in that case, equa- tion (277), becomes a b s x y s' ; but by the property of the parabola, we have y ^/px; and similarly, we obtain b zz ^p a ; therefore, by substitution, we get a s \/pa zz x s \/p x ; by squaring both sides, it is s'V 5 s a 8 , (278). and finally, by division and evolution, we have 440. The practical rule supplied by this equation, may be expressed in words at length in the following manner. RULE. Divide the square of the specific gravity of the floating solid, by the square of the specific gravity of the fluid on which it floats, then multiply the cube root of the quotient by the axis of the parabola, and the product will give the portion of the axis, which falls below the plane of floatation, or the surface of the fluid. 441. EXAMPLE. A solid body whose transverse section is in the form of a parabola, floats in equilibrio on the surface of a fluid with its vertex downwards, and its base or double ordinate horizontal ; it is required to determine how deep the body sinks, supposing the vertical axis to be equal to 40 inches, the specific gravity of the body and that of its supporting fluid, being to one another, as 686 to 1000. Here, by operating as directed in the rule, we shall have s 2 =686 2 z=: 470596; s' 2 =1000 2 =: 1000000; from which, by division, we obtain the cube root of which, is 0.470596 = 0.7778 ; OF THE POSITIONS OF EQUILIBRIUM. 353 consequently, by multiplication, we finally obtain a; =: 40X0.7778 = 31. 112 inches. 442. Therefore, the position of equilibrium corresponding to the above value of a?, is as represented in the annexed diagram, where AB is the base or double ordinate of the parabolic section, DC its axis; FH the water line, or double ordinate of the immersed portion FDII, DE the corresponding ab- scissa, and IK the horizontal sur- face of the fluid. That the condition is satisfied, in which the centres of gravity of the whole and the immersed part are situated in the same vertical, is manifest from the circumstances of the case ; and that the other con- dition is satisfied, in which the areas of the whole and the immersed part, are to each other, as the specific gravities of the fluid and the solid, will appear from the following calculation. Since the parabolas ADB and FDII, are similar to one another, having the same parameter and being situated about the same axis ; it follows, that a^ a : x \/ x : : a : a" ; but by the question, a is equal to 40 inches, and by the foregoing computation, we have found that & = 3 1.1 12 inches; therefore, we get SOv/To": 31.112t/3nT2 : : 1000 : 686, which satisfies the other condition of equilibrium, from which we infer, that if the specific gravity of the solid be taken within proper limits, the preceding diagram exhibits a position of floating. 443. The equation (278) was obtained on the supposition, that the axis of the parabola is vertical and the points D and m coincident, in which case the quantity z vanishes entirely from the figure ; but the same result will obtain whether we consider the points D and m to be coincident or not, as will appear from what follows. By the property of the parabola, that the distance of any point of the curve from the focus, is equal to the perpendicular distance between that point and the directrix, it follows, that mp (see Jig. art. 439) is equal to the sum of DK and DP taken jointly ; that is, VOL. I. 354 OF THE POSITIONS OF EQUILIBRIUM. Z 2 now, we have already seen that D K is expressed by , and by the nature of the curve, we have DP =r \p ; therefore, by addition, it is But since by the property of the parabola, the parameter of any diameter, is equal to four times the distance between the vertex of that diameter and the focus, we have P and by the equation to the curve, it is consequently, by extracting the square root, we obtain P Let this value of y be substituted instead of it in equation (277), and we shall obtain VP but by the equation to the curve, we have therefore, by substitution, we shall get VP and multiplying both sides by ^/p, we obtain from which, by squaring both sides, we get which is the identical expression, obtained on the supposition of a coincidence between the points D and m ; consequently, the value of x must be the same in both cases, and the position of floating depend- ing upon the specific gravity must also be the same. Now, by the construction we have seen, that the triangles KEW and wng are similar to one another; hence we get Em : EK : : gn : wn, OF THE POSITIONS OF EQUILIBRIUM. 355 or by taking the analytical equivalents, it becomes - , and from this, by working out the proportion, we get 4xz hence, by subtraction, we have ,.,'< j ws ns w;w; but because Emus is a parallelogram, it follows, that ws E?n; therefore, it is Again, the triangles wgn and WGS are similar to one another; but we have shown above, that KEW is similar to wng; therefore, KEW is similar to WGS, and we have EK : EWZ : : ws : SG ; taking therefore the analytical values, we obtain 2: ~ P P consequently, by reduction, we have 2p 5 But by the nature of the figure, SG is manifestly equal to EG ES ; z 2 now ESZZ mn^nx, and EG ^ZDG -J- DE; that is, EG fa-j consequently, we have s G f- a -| # ; let these two values of SG be compared with each other, and we get ~~Tp "IfIf" 1 '"*' (279). and finally, by reduction, we obtain ~~ 10 Now, the value of a;, as we have determined it from equation (278), is 2A2 356 OF THE POSITIONS OF EQUILIBRIUM. therefore, by substitution, we have (280). Here then, we have obtained a pure quadratic equation, which gives two subcontrary positions of equilibrium, provided that the specific gravity be taken within proper limits. Extract the square root of both sides of equation (280), and we shall obtain and if the specific gravity of the fluid be expressed by unity, as is the case when the fluid is water, then we shall have ( 2 81). But in order to have the value of z a real positive quantity, it is necessary that 6a should be greater than 5p -f- 6a^s 2 ; in order there- fore, to find the greatest value of * that will satisfy this condition, we must put these two quantities equal to one another, arid in that case we shall obtain transpose, and we obtain 6a$7 2 =6a 5p; divide by 6a, and it becomes 5p therefore, by involution, we get '=<'->'.. '"'..,.. and finally, by evolution, it is ,1, =(>-!)* Here then it is manifest, that in order that the positions determined by the equation may be those of equilibrium, it is necessary that the specific gravity of the floating body shall be less than ( 1 ^ > 6a/ 444. EXAMPLE. A solid body whose transverse section is in the form of a parabola, is placed in a cistern of water with its vertex OF THE POSITIONS OF EQUILIBRIUM. 357 downwards, in such a manner, that its base or extreme ordinate is entirely above the surface ; it is required to determine the position of the body when in a state of equilibrium, the parameter of the parabolic section being 16 inches, the axis 40 inches, and the specific gravity of the floating solid, to that of the supporting fluid as 1 to 2 ? In this example there are given, p 16 inches, am 40 inches, and s 0.5, the specific gravity of water being unity ; therefore, by sub- stitution, we get from equation (281) * ^ 1.6(6X40 5X16 6x40/0125) 3.75 inches. And the positions of equilibrium corresponding to this value of g, are as represented in the subjoined diagrams, and the following is the method of construction. With the parameter or latus rectum equal to 16 inches, and the subcontrary axes DC and dc each equal to 40 inches, describe the parabolas ADB and adb ; from c the middle of the base and towards the depressed part of the figure, set off cr equal to 3.75 inches, the computed value of z ; through the point r, draw rm parallel to the axis CD, and meeting the curve in the point m ; draw the tangent WE, and on the diameter mr, set off mn equal to 25.19 inches, the value of x as obtained by the reduction of equation (278) ; then through the point n, and parallel to the tangent ?WE, draw the straight line IK, which will coincide with the surface of the fluid, and cut the parabolas ADB and adb in F, H and/", h the extremities of the lines of floata- tion, corresponding to the positions of equilibrium which we have exhibited in the diagrams. We must now endeavour to prove, that the positions in which we have represented the body are those of equilibrium; and for this purpose, we must inquire if the equation (279) is satisfied by the sub- stitution of the computed values of x and z ; for when that is the case, the line which joins the centres of gravity of the whole section and the immersed part of it, is perpendicular to the surface of the fluid. 358 OF THE POSITIONS OF EQUILIBRIUM. Now, the values of x and z as we have determined them by calcu- lation, are respectively equal to 25.19 and 3.75 inches ; therefore, by substitution, equation (279) becomes 16 2 -j- 4X3.75* 2X25.19 __ 3X40 3.75 2 2X16 ~5~~ ~~5~~ ~16" from which, by transposition, we have 16 2 -f 4X3.75* 2X25.19 3x40 3.75 2 2 X 16 445. Here then, it is manifest, that one of the conditions of equi- librium is satisfied, viz. that in which the line which passes through the centres of gravity of the whole section and the immersed part of it, is perpendicular to the surface of the fluid; we have therefore in the next place, to inquire if the areas of the whole section and the im- mersed part, are to one another, as the specific gravity of the fluid is to that of the solid. Now, we have seen, equation (277), that vy+42 2 but by the nature of the parabola, b zn \/ ap ; hence it is and we have elsewhere seen, that the value of y y is _ consequently, by substitution, we get therefore, by expunging the common term \/p, and converting to an analogy, we get a^a : x^/x : : s' : s, and this, by substituting the given value of a, and the computed value of x, is 80/10~: 25.19V2539 : : 1 : 0.5. From this it appears, that the second condition of equilibrium is also satisfied ; we may therefore conclude, that the positions in which we have represented the body are the true ones ; but we may further observe, that by altering the specific gravity of the body, other posi- tions may be exhibited, provided that the expression shall never exceed fl - j* in the common or Apollonian parabola. CHAPTER XIII. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 446. IN the preceding pages, we have investigated and exemplified the method of determining the positions of equilibrium in a few of the most important cases, where the forms of the floating bodies are such as to render them of very frequent occurrence in practical construc- tions ; we shall therefore, in the next place, proceed to investigate and exemplify the conditions of Stability, or that power, by which, when the equilibrium of a floating body has been disturbed, it endeavours, in consequence of its own weight and the upward pressure of the fluid, either to regain its primitive settlement, or to recede farther from it, by revolving on an axis passing through its centre of gravity parallel to the horizon, until it arrives at some other position of equilibrium, in which the principles of quiescent floatation are again displayed. 1. DEFINITIONS AND PROPOSITIONS OF STABILITY IN FLOATING BODIES. 447. It is familiar to every person's experience, that when bodies of certain forms and dimensions, placed under particular circumstances on the surface of a fluid, have their equilibrium deranged by the action of some external force, they return to their original position after a few movements or oscillations backwards and forwards, in a direction determined by that of the disturbing impulse. It is equally obvious with regard to other bodies, that however small may be the quantity of their deviation from the original state of quiescence, they have no tendency whatever to return to it, but con- tinue to recede farther and farther from it, by revolving about a horizontal axis, until the deviating effort obtains a maximum depend- ing upon the angle of deflexion ; after which the deflecting energy 360 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. continues to decrease until it vanishes, in which case the body settles in another situation, which also satisfies the conditions of equilibrium. Again, a solid body may be so constituted with respect to shape and dimensions, that in every position which can be given to it on the surface of a fluid, it will rest in a state of equilibrium ; for in all situations and under every condition, the centre of gravity of the whole body and that of the immersed part, will occur in the same vertical line ; this being the case, it is manifest that in such a body, the equilibrium cannot be disturbed, because the external force, how- ever it may be applied, can only operate to turn the solid round an axis, passing through the centre of gravity in a direction parallel to the horizon. Homogeneous spheres are bodies of this sort, so also are homo- geneous cylinders floating with the axis horizontal; these have no tendency to solicit one situation in preference to another, and con- sequently, in whatsoever position they are placed, with reference to the axis of revolution, they are still in a state to satisfy the conditions of equilibrium, for the centre of gravity of the whole body and that of the immersed part, are always situated in the same vertical line. In the first case then, where the body has a tendency to restore itself to the original position, the equilibrium is said to be stable ; in the second case, where the body deviates farther and farther from the original state, the equilibrium is unstable ; and lastly, in the case where the body has no tendency to remain in, or to solicit one position in preference to another, the equilibrium is said to be insensible. 448. The conditions of equilibrium as here stated, are in themselves sufficiently simple and explicit, but in order that none of our readers may enter upon this important and difficult subject, with incorrect notions respecting the different species of equilibrium, and the various conditions or circumstances of floating under which a body may be placed, we have thought it expedient to subjoin the following expo- sitions and illustra- tions. Let the dotted line inj^. 1 represent a transverse section of any uniform prismatic body, placed verti- cally on the surface of a fluid, and sup- pose the specific Ffe.i. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 361 gravity of the body to be such, when compared with that of the fluid on which it floats, as to sink it to the depth mn. The body floats in equilibrio in the upright position ; suppose therefore that by the application of some extraneous agent, it is deflected into the position abed, where it is conceived to revolve about a horizontal axis, passing through G its centre of gravity, at right angles to the plane abed. If therefore, the body when thus inclined, requires the force /'to retain it in that state, or to prevent it from returning to the upright position ; then the equilibrium in which the body is originally placed, is what we understand by the equilibrium of stability. Again, let the dotted Fig. 2. line in fig. 2 represent a vertical section of any uniform homogene- ous prismatic body , float- ing upright and quies- cent on the surface of a fluid, and let the specific gravity of the solid be such as to sink it in the fluid to the depth mn\ suppose now, that by the action of some external force, the body is deflected from the vertical position into that represented, by a b cd ; it is obvious, that the revolution is made about the horizontal axis passing through G the centre of gravity, at right angles to the plane abed. Hence, if the body when thus inclined, requires the application of the force/ to retain it in that state, or to prevent it from inclining further ; then the equilibrium in which the body is originally placed, is what we comprehend by the equilibrium of instability . Finally, let abed, fig. 3, be the vertical -* 3- section, and let the specific gravity of the body be such, as to sink it to the depth md; the solid floats in equilibrio in the upright position, and in such a manner, that an evanescent force will either retain it in that state, or deflect it from it ; this is the insen- sible equilibrium, or the equilibrium of indif- ference, and the solid is said to overset. 449. Of these three species of equilibrium, bodies floating on the surface of a fluid are manifestly susceptible; but they admit of a 362 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. more perspicuous and comprehensive definition, which may be scien- tifically read in the following manner. 1 . The Equilibrium of Stability, is that property in floating bodies, by which on being slightly inclined to either side, they endeavour to redress themselves and to recover their original position. 2. The Equilibrium of Instability, is that property in floating bodies, by which on being slightly inclined from the upright position, they tumble over in the fluid and assume a new situation, in which the conditions of floating again occur. 3. The Equilibrium of Indifference, is that property in floating bodies, by which they are enabled to retain whatever position they are placed in, without exhibiting the smallest tendency, either to regain the original position, or to deviate farther from it. In addition to the different species of equilibrium described above, there are several other terms of very frequent occurrence in the doctrine of floatation, which it will be proper to explain before we proceed to develop the laws that regulate the conditions of stability. The most common and the most important of the terms here alluded to, are the following. 450. DEFINITION 1. The Centre of Effort, is the same with the centre of gravity of the entire floating body ; it is that point through which the horizontal axis passes, and about which the body is supposed to revolve. DEFINITION 2. The Centre of Floatation, or the Centre of Buoy- ancy, is the same with the centre of gravity of the immersed part of the floating body, or it is the same as the centre of gravity of the fluid displaced in consequence of the floatation. DEFINITION 3. The Line of Pressure, is the vertical line passing through the centre of effort, in the direction of which, the body is impelled downwards by means of its own weight. DEFINITION 4. The Line of Support, is the vertical line passing through the centre of buoyancy ; it is either parallel to, or coincident with the line of pressure, and is that in whose direction the body is propelled upwards by the pressure of the fluid. DEFINITION 5. The Axis of Motion, as we have already observed in treating of the positions of equilibrium, is the horizontal line passing through the centre of effort, and about which the body revolves on being deflected from its original position. DEFINITION 6. The Transverse Section of the Solid, is that indi- cated by a vertical plane at right angles to the axis of motion, and separating the body into any two parts : all the transverse sections OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 363 are parallel to one another, and the principal transverse section is that which passes through the centre of effort. DEFINITION 7. The Axis of the Section, is the straight line which passes through its centre of gravity, dividing it into two parts, which in the case of a regular body are equal and similar to one another. When this axis is vertical, it either coincides with, or is parallel to the line of pressure. DEFINITION 8. The Line of Floatation, or The Water Line, is the horizontal line in which the surface of the fluid meets a vertical trans- verse section of the floating body. DEFINITION 9. The Plane of Floatation, is the horizontal plane coincident with the surface of the fluid, and which passes through the water line, dividing the body into the immersed and extant portions. DEFINITION 10. The Equilibrating Lever, is a straight line equal to the horizontal distance between the verticals passing through the centre of effort and the centre of buoyancy ; or it is the horizontal distance between the line of pressure and the line of support. DEFINITION 11. The Stability of Floating, or the Measure of Stability, is that force by which a body floating on the surface of a fluid, endeavours to restore itself, when it has been slightly inclined from a position of equilibrium by the action of some external agent ; or it is a force precisely equal to the fluid's pressure, or to the entire weight of the floating body acting on the equilibrating lever. (See Proposition (XI.) following). DEFINITION ]2. The Metacentre, is that point in which the axis of the section and the line of support intersect each other ; it limits the elevation of the centre of effort. Upon these definitions, therefore, in combination with the following simple and obvious propositions, depends the whole doctrine of the stability of floating bodies. PROPOSITION IX. 451. It has already been admitted as a principle in the theory of hydrostatics, that every body, whatever may be its form and dimensions, if it floats upon the surface of a fluid of greater specific gravity than itself, displaces a quantity of the fluid on which it floats equal to its own weight, and consequently : 364 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. The specific gravity of the supporting fluid, is to that of the floating body, as the whole magnitude of the solid is to that of the part immersed. (See Proposition VII.) PROPOSITION X. 452. If a solid body, of whatever form or dimensions, floats upon the surface of a fluid of greater specific gravity than itself: It is impelled downwards by its own weight acting in the direction of a vertical line passing through the centre of effort ; and it is propelled upwards by the pressure of the fluid which supports it acting in the direction of a vertical line passing through the centre of buoyancy. (See Proposi- tion VI.) Therefore, if these two lines are not coincident, the floating body thus impelled must revolve upon an axis of motion, until it attains a position in which the centre of effort and the centre of buoyancy are in the same vertical line. PROPOSITION XL 453. If a solid body of any particular form and dimensions, floating on the surface of a fluid of greater specific gravity than itself, be deflected from the upright position through a given angle : The stability of the body is proportional to the length of the equilibrating lever, or to the horizontal distance between the vertical lines passing through the centre of effort and the centre of buoyancy. (See Problem LXI. following.) When the horizontal distance here alluded to is equal to nothing ; that is, when the centre of effort and the centre of buoyancy are situated in the same vertical line; then the stability, or the force which urges the body round its axis of motion vanishes, and the equi- librium is that of indifference ; for in this case, the metacentre coin- cides with the centre of effort. If the floating body be any how inclined from the upright position, and if, in consequence of the inclination, the line of support falls on the same side of the centre of effort as the depressed parts of the solid, then the length of the equilibrating lever is accounted positive, OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 365 and the pressure of the fluid operates to restore the equilibrium ; in this case, therefore, the equilibrium is that of stability. But when the line of support falls on the same side of the centre of effort as the parts of the solid which are elevated in consequence of the inclination ; then the length of the equilibrating lever is accounted negative, and the equilibrium is that of instability. Hence it appears, that the stability of a floating body is positive, nothing or negative, according as the metacentre is above, coincident with, or below the centre of effort: these consequences, however, will be more readily and more legitimately deduced from the general formula which indicates the conditions of stability, and this formula we shall shortly proceed to investigate. PROPOSITION XII. 454. The common centre of gravity of any system of bodies being given in position, if any one of these bodies be moved from one part of the system to another, it is manifest, from the principles of mechanics, that : The motion of the common centre of gravity, estimated in any given direction, is to the motion of the body moved, estimated in the same direction, as the weight of the said body, is to the weight of the entire system. Therefore, by means of these propositions and the definitions that precede them, the whole doctrine of the stability of floating bodies, with the train of consequences which immediately flow from it, may be easily and expeditiously deduced ; but in proceeding to develope the laws on which the stability of floating depends, it will be con- venient for the sake of simplicity, to consider the body as some regular homogeneous solid, of uniform shape and dimensions through- out the whole of its length ; for in that case, all the vertical transverse sections will be figures precisely equal and similar to each other ; and if the body be divided by a vertical plane passing along the axis of motion, the two parts into which it is separated will be symmetri- cally placed with respect to the dividing plane. This being premised, the principles upon which the stability of floatation depends, will be determined by the resolution of the follow- ing problem, in which all the transverse sections are trapezoids. 366 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 2. PRINCIPLES OF THE STABILITY OF FLOATING BODIES. PROBLEM LXI. 455. A solid homogeneous body of uniform shape and dimen- sions throughout the whole of its length, is placed upon a fluid of greater specific gravity than itself, in such a manner, that the centre of effort and the centre of buoyancy are in the same vertical line. It is required to determine the stability, when by the application of some external force, the body is deflected from the upright position, or from a position of equilibrium through u given angle. Let the solid to which our investigation refers be such, that the vertical transverse sections perpendicular to the axis of motion, are equal and similar trapezoids, as indicated by ABCD and abed in the annexed diagrams. The solid floats upon the surface of the fluid IK, and ABCD is its position when in a state of equilibrium ; ABFE being the extant portion of the vertical section, and EFCD the part immersed beneath the fluid's surface. The point G is the centre of effort, or the centre of gravity of the whole section, the plane of which is supposed to pass through the centre of gravity of the body, and g is the centre of buoyancy, or the centre of gravity of the part immersed below the surface of the fluid ; then since the body floats in a state of equili- brium, it follows from Proposition X., that PQ the axis of the section, which passes through the points G and g, is perpendicular to EF the line of floatation. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 367 We are now to suppose, that by the application of some external force, the solid revolves about its axis of motion until it comes unto the position represented by abed, in which state the equilibrium does not obtain. Here it is manifest that PQ, the axis of the section which was vertical in the first instance, is transferred, in consequence of the inclination, into the position pq ; and in like manner, the line EF, which before was horizontal, is transferred into the oblique position ef, and hi is now the line of floatation, or as it is otherwise called, the water line. Since the absolute weight of the body remains unaltered, whatever may be the position of floating, the area of that portion of the section which is immersed below the surface of the fluid, must also be inva- riable ; it therefore follows, that the areas hied and EFCD are equal to one another ; but the space efcd is equal to EFC D, hence the spaces hied and efcd are each of them equal to EFCD; they are therefore equal to one another, and consequently, the extant triangle hke is equal to the immersed triangle fh i. On pq the axis of the section, set off GH equal to Gg, the distance between the centre of effort and centre of buoyancy in the original position of equilibrium ; then it is manifest, that in consequence of the inclination, the point g, which is the centre of gravity of the space EFCD, will be transferred to the point n, which is the centre of gravity of the equal space efcd; and the pressure of the fluid would act upon the body in the direction of a vertical line passing through 71, if efcd were the portion of the section immersed under the fluid's surface ; but this is not the case, for in consequence of the inclination, the triangle fki, which was before above the fluid's surface, is now depressed under it, and in like manner the triangle hke, which was previously under the surface, is now elevated above it. It is therefore obvious from Proposition XII, that by transferring the triangle hke into the position fki, the point n, which is the centre of gravity of the space efcd, must partake of a corresponding motion and in the same direction ; that is, the point n must move towards those parts of the body that have become more immersed in conse- quence of the inclination, until it settles in g the centre of gravity of the immersed volume hied. Through # the centre of gravity of the immersed part hied, draw ym perpendicular to hi the line of floatation, and meeting pq the axis of the section in the point m ; then is m the metacentre, and the pressure of the fluid will act in the direction of the vertical line g'm, with a force precisely equal to the body's weight ; and according to 368 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. the principles of mechanics, it will act with the same energy at what- soever point of the line gm it may be applied. Through the point n and parallel to hi the line of floatation, draw nz cutting the vertical line gm in the point z, and through G the centre of gravity of the whole space abed, draw or perpendicular and GS parallel to nz, and let k be the point in which the lines ef and hi intersect one another; then, as we have stated above, the pressure of the fluid will have the same effect to turn the body round its axis, whether it be applied at the point g or the point s ; we shall therefore suppose it to be applied at the point s, in which case GS will represent the point of the lever, at whose extremity the pressure of the fluid acts to restore the body to its original state of equilibrium, or to urge it farther from it. Since the effect of the fluid's pressure, acting in the direction of the vertical line which passes through g the centre of buoyancy, has no dependence on the absolute position of that point, but on the horizontal distance between the vertical lines rG and gm ; it follows, that in the actual determination of the positions which bodies assume on the surface of a fluid, and their stability of floating, the situation of the centre of buoyancy in the inclined position is not required, for the horizontal distance between the vertical lines which pass through that centre and the centre of effort, is sufficient for obtaining every particular in the doctrine of floatation. Bisect the sides of the triangles h ke and/Az in the points u, v and w, a:, and draw the straight lines ku, ev and kw, ix intersecting two and two in the points / and o ; then are / and o the points thus deter- mined, respectively the centres of gravity of the triangles hke and fki. Through the points I and o draw the straight lines ly and of, respectively perpendicular to hi the line of floatation, corresponding to the inclined position of the body ; then is yt the horizontal distance through which the centre of gravity of the triangle hke has moved in consequence of the inclination ; therefore, by the principle announced in Proposition XII., we obtain area efcd : area hke : : yt : nz. It is easy to comprehend in what manner the proposition cited above applies to the case in question ; for we may assume the area efcd as a system of bodies, of which the common centre of gravity is n. One of the bodies composing this system, viz. the triangular area hke, conceived to be concentrated in the point /, is transferred, in conse- quence of the inclination from the point I to the point o, in which the OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 369 equal volume fki is similarly concentrated ; this will have the effect of moving the common centre of gravity of the system from n to z, in a direction parallel to yt, and the distance nz, through which the common centre is moved, is what the proposition determines. Let the position of the point z be supposed known ; then, if a vertical line be drawn through that point perpendicular to the line of floatation, the centre of gravity of the immersed space hied, will occur in some point of that line, as for example at g ; but we have already observed, that it is not necessary to determine the absolute position of the point in question, the horizontal distance GS or rz between the verticals Gr and mz, being all that is required. Put a zz: hied or efcd, the area of the immersed space, d z= hke or fki, the area of the triangle which has been assumed as constituting an individual body of the system ; d zzr yt, the horizontal distance through which the centre of gravity of the triangle hke has moved, in shifting to the position o in the triangle fki, S zz: Gg or Gn, the distance between the centre of effort and the centre of buoyancy, when the axis of the section is vertical ; b zz: A B or a b, the length of the greater parallel side of the trapezoidal section, /3 zz: DC or dc, the length of the lesser parallel side ; D zz: PQ or pq, the perpendicular distance between the paral- lels AB and DC, or ab and dc, c zz: EF or ef, the water line or line of floatation in the upright position, I zz: the axis of motion, or the whole length of the floating body, passing through o the centre of effort ; s zz: the specific gravity of the floating body, s' zz: the specific gravity of the supporting fluid, which in the case of water, is expressed by unity ; 5 zz: the stability of the body, or the momentum of the redress- ing force ; < zzr/Az, or nGr, the angle of deflexion, and x zz: GS or rz, the length of the equilibrating lever. Then, by substituting the literal representatives of the several quan- tities in the foregoing analogy, we shall obtain a : a' : : d: nz, f*\* > VOL. i. 2 B UN1VERS! 370 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. from which, by equating the products of the extreme and mean terms, we get aXnz~ a'd; therefore, by division, we have a'd nz . a But by the principles of Plane Trigonometry, it is rad. : 3 : : sin.0 : nr, which being reduced, gives wrzzd sin.^, and according to the construction of the figure, it is manifest that rz or G s, is equal to the difference between nz and nr, the two quantities whose values have just been determined ; consequently, by subtrac- tion, we have dd . . X ~~a asm * ' (282). 456. The equation which we have just investigated has reference only to a particular case of the general problem, viz. that in which the vertical transverse sections, throughout the whole length of the body, are equal and similar figures ; this condition, although it is a restric- tion upon the general applicability of our result, yet it allows an im- mense latitude, for the figures of bodies whose parallel transverse sections are equal and similar areas are very numerous ; and if we substitute the magnitude of the whole immersed volume, and that of the volume which becomes immersed in consequence of the inclina- tion, instead of the areas of the respective sections, the above equation becomes general, because its form and the manner of combining the terms admit of no change. The expression consists of five members on one side, one of which, that is, the angle of deflexion, must always be a given datum, or it must be directly assignable from the circumstances of the case, and the others must all be determined by means of the given dimensions, and other particulars dependent upon the figure of the section ; but the method of applying the formula, and the whole operation neces- sary for its reduction, will be sufficiently exemplified by the resolution of the following example. 457. EXAMPLE. A solid homogeneous body, of which the trans- verse parallel sections at right angles to the axis of motion, are equal and similar trapezoids, is placed upon the surface of a fluid in such a manner, that its broadest side is upwards and parallel to the OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 371 horizon ; the body floats in equilibrio in this position ; but suppose that some external force is so applied to it, as to deflect it from the upright and quiescent state through an angle of 15 degrees; it is re- quired to determine the stability or the momentum of restoration, the parallel sides of the section being respectively 40 and 30 inches, the perpendicular distance between them 20 inches, and the whole length of the body 14 feet, its specific gravity when compared with that of the fluid being as 270 to 1000, or as 0.27 to 1 ? 458. For the purpose of rendering the several steps of the operation perfectly clear and comprehensible, we shall refer to the annexed diagram, which re- presents a trans- verse section of the body in the inclined position ; ef being the line in which it is intersected by the water's surface when it is upright, and hi the corres- ponding line when it is deflected through the angle fki. PQ is the perpendicular distance between ab and cd the parallel sides of the section, QH the depth to which the body sinks in the fluid as induced by the specific gravity ; o and I are the centres of gravity of the triangles fki and hke; mt the projected distance between them on the line hi, and Gg the distance between the centre of effort and the centre of buoyancy, or the dis- tance between the centre of gravity of the whole body and that of the immersed part, when the body is upright. Now, the several parts which have to be calculated in order to resolve the question, are the areas efcd, fki, and the distances mt and og ; for which purpose, we have given a b, dc, PQ, the angle fk i and the specific gravity of the solid. Therefore, according to the principles of mensuration, the area of the whole section is expressed by 20 |(6 -f- /3) X D =(40 -f 30) X = 700 square inches, and by the nature of floatation, we have 2fi2 372 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. or by putting s' equal to unity, and substituting the respective num- bers, we obtain area efcd = a = 700 X 0.27 189 square inches. (283). 459. Consequently, by having the area of the trapezoid efcd, and one of its parallel sides dc given, the other parallel side ef and the perpendicular depth Q H can easily be found ; for by the nature of the figure and the property of the right angled triangle, we have D C / - 7 20 = b=piV * 900+ 189 3o| 6 inches. therefore, by the property of the trapezoid, we have 3(30 + c)=r 189, or by separating the terms and transposing, we get 3c = 189 90=99, and by division, it is 99 ef= c=--=33 inches. u 460. We must next endeavour to discover the point k, in which the primary and secondary water lines intersect each other, and for this purpose, \)utfk nr y, then by subtraction, we have e k = 33 y ; but by the rules of mensuration, it is and by restoring the above values of fk and e k, it becomes yXki = (33 y)Xkh. (284). Through the point c and parallel to PQ, draw en meeting ef per- pendicularly in n\ then it is manifest, from the principles of Plane Trigonometry, that *"*=="= which corresponds to the natural tangent of 75 57' 49". But by the principles of Geometry, the exterior angle cfn is equal to both the interior and opposite angles fki and fik ; consequently, by subtraction, we have angle fik = 75 57' 49" 15 = 60 57' 49", OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 373 and by Plane Trigonometry we get sin.6057'49" : sin. 75 57' 49" : : y : ki = 1.10967, and by proceeding in a similar manner with the triangle hke, we shall have sin.9057'49" : sin.75 57' 49" : : 33 y : M zz 0.9703(33 -- y) ; therefore, by substituting these values of ki and kh in equation (284), we get 1.1096/nz 0.9703(33 y)*, from which, by reciprocating the terms, we obtain and extracting the square root, it is = V 0.874434 0.9351 ; 66 y therefore, finally by reduction, we have fk zz y zz: 15.94 inches nearly. 461. Having thus determined the value of fk, the value of ki can very easily be found; for we have seen above, that ki = l.lQ96y ; consequently, by substitution, we have ^1.1096X15.94 = 17.687 inches; therefore, by the principles of Mensuration, we get area/A zzzo'zrjx 15.94 X 17.687 X 0.25882=36.47 squ. inches. (285). Let the values of a and a' as determined in equations (283) and (285), be respectively substituted in equation (282), and we shall obtain (286). but in this equation the values of d and 3 are still unknown ; in order therefore to assign their values, we must have recourse to other principles. 462. Now, since the line kw which passes through o, the centre of gravity of the triangle fki, bisects the side fi in w, we know from the principles of Geometry, that from which, by extracting the square root, we get 2 k w = and dividing by 2, it becomes kw = 374 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. But we have already found that/A zz 15. 94 inches, and k zzz 17.687 inches; consequently, their squares are 15. 94 2 rz 254.0836, and 17.687 2 zz312.83 respectively; therefore, we have kw \ - aSm '^ W; - (282-). 466. Now, in applying this expression to any particular case in practice, it is understood, that the position of the centre of gravity of the entire ship, and also the position of the centre of gravity of the immersed volume when the ship is upright and quiescent, are both known, and consequently, the distance between those centres, which is represented by the line Gg or GW, is a given or assignable quan- tity ; and moreover, the total displacement occasioned by the floating body, is supposed to have been determined by previous admeasure- ments, and hence, the weight of a quantity of water, which is equal OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 377 in magnitude to the displacement, will likewise be equal to the whole weight of the vessel. The quantity is necessarily given from the circumstances of the case, and may be of any magnitude whatever, and therefore, the only quantities required to be ascertained, for the purpose of dis- covering the momentum of the ship's stability, are v and d in the numerator of the fractional term, the one denoting the magnitude of the volume which becomes immersed in consequence of the inclina- tion, and the other, the distance through which the centre of gravity of that volume is moved in a horizontal direction, during the deflexion of the ship from the upright and quiescent position. In order, there- fore, to facilitate the determination of those quantities, the following observations are necessary. 467. If a straight line be conceived to pass through the centre of gravity of the ship, in the direction of its length and parallel to the horizon, traversing from the head to the stern of the vessel ; then, such a line is called the longer axis of the vessel ; it is the same with the axis of motion described in the fifth definition preceding, and is so called, for the purpose of distinguishing it from another line also hori- zontal, which passes through the centre of gravity at right angles to the former, and is called the shorter or transverse axis of the vessel ; it is on this axis that the vessel turns in the process of pitching, a motion which is easily understood by considering an alternate eleva- tion and depression of the head and stern. 468. A vertical plane drawn through the longer axis, when the vessel floats in an upright and quiescent position, divides it into two parts which are perfectly similar and equal to one another, and in this respect at least, the figures of vessels may be considered regular, although that their forms are not otherwise restrained to any uniform or particular proportions. From the similarity and equality of these two divisions, it necessa- rily follows, that when a vessel floats in a state of upright quiescence, the similar parts on the opposite sides of the plane of division will be equally elevated above the water's surface. A ship thus floating in a position of equilibrium, may be conceived to be divided into two parts by the horizontal plane which is coincident with the water's surface, and the section formed by this plane passing through the body of the vessel, is called the principal section of the water ; it corresponds with the plane of floatation in the particular case where the vessel is upright and quiescent, as will readily be perceived by a reference to the ninth definition preceding. 378 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. or 469. Let A B c K D represent a tranverse section of the hulk of a ship, perpendicular to the longer axis, and passing through G its centre of gravity, and suppose the vessel as it floats upon the surface of the water to be upright and quiescent; then LK the axis of the section, according to the seventh definition, is perpendicular to the horizon, and in this state the principal sec- tion of the water passes through the line D c throughout the whole length of the vessel ; which may bly be better un- derstood, if the principal water section be viewed endways, with the eye at a great distance, it will appear as if it were projected into the straight line DC. While the vessel retains its upright position and remains in a state of rest, the transverse or shorter axis, is that which is represented by the dotted line a b, and the place of the centre of gravity of the immersed portion DKC, is somewhere in the line passing through g in a direction parallel to the horizon ; for g is the place of the centre of gravity of the section DKC, which falls below the principal section of the water passing through DC. When the ship is caused to heel or to revolve about the longer axis passing through G, until it moves through an angle equal to FPC; then it is manifest, that the principal section of the water, or the plane in the ship which passes through the line DC, will be transferred into the position EF ; but the section of the water will intersect the sides of the vessel, in the direction of a plane passing through DC, which is inclined to the former plane passing through EF in an angle equal to the angle FPC. The plane which passes through the line DC in a direction parallel to the plane of the horizon, may therefore be termed the secondary section of the water, merely to distinguish it from that which formerly passed through EF, and which we denominated the principal section. The principal and secondary sections of the water must therefore intersect one another in the line denoted by the point P, or rather in the line which being viewed endways, is projected into the point P, OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 379 and stands at right angles to the plane A KB. Consequently, since the vessel is supposed to be inclined around the longer axis, it follows, that the intersection of the planes which we have supposed to be projected into the point P, will be parallel to the axis round which the vessel is supposed to revolve in passing from one position to another. But by the laws of hydrostatics, since the whole weight of the vessel is considered to be precisely the same, however much it may be deflected from the upright and quiescent position ; it follows from hence, that the volume which becomes immersed below the water's surface in consequence of the inclination, is equal in magnitude to that which is elevated above it by the same cause, and consequently, the position of the line which is represented by the point P, will depend entirely upon the form of the sides DE and CF. Now, in a ship, of which the breadth is continually altering from the head to the stern, and in no regular proportion expressible by geometrical laws, it is manifest, that the place of the point P, repre- senting the line in which the water's surface intersects the vessel in the upright and inclined positions, must be practically determined by some method of approximation, dependent upon the ordinates in the vertical and horizontal sections into which the ship is supposed to be divided. By similar modes of approximation, the other quantities necessary for the solution may also be ascertained ; but in ships of war and of burden, constructed after the forms which they generally assume at sea, the calculations necessary for the purpose are unavoidably prolix and troublesome ; and after all, they must depend for their accuracy entirely upon the skill and address of the persons by whom the requi- site ordinates are measured and registered, according to the different parts of the vessel to which they particularly belong ; for if a very nice and accurate arrangement be not preserved with regard to the magnitudes and places of the several ordinates, it is easy to be per- ceived, that the results may come out very wide of the truth, and must therefore necessarily vitiate the whole process. In our diagram, the lines DC and EF, through which the principal and secondary sections of the water pass, are supposed to bisect each other, and consequently, the point P must occur at the middle of them both ; in which case its position is known ; but the careful and atten- tive reader will easily perceive, that this can very seldom happen, unless the extreme sides of the zone which limits the angle of the ship's inclination, are equally inclined to, and similarly situated in respect of the extremities of the intersecting lines DC and EF. 380 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. When the curves D E and c F are dissimilar between themselves, and dissimilarly situated in respect of the intersecting lines DC and EF ; then it is manifest, that the point p cannot fall in the middle of either, but must occur to the right or to the left, according as it is influenced by the nature of the curves, which define the exterior contour or bound- ary of the vessel. Suppose therefore, that the intersection takes place at the point p, a little to the right of the place where the two water lines are sup- posed to bisect one another ; and through the pointy, let the straight line >?tf. L \ r y Consequently, since the several quantities w, v, S and 0, are either given d priori, or determinable from the circumstances of the case, it follows, that the momentum of stability for any angle of inclination, and for any form of body, can be found by the above formula ; but the labour and intricacy of the calculation, increases with the irregu- larity of the body to which such calculations are referred, and in particular cases, the labour required to accomplish the purpose is immensely great. PROBLEM LXII. 473. The vertical transverse sections of a ship, taken at the distance of five feet from each other along the principal longi- tudinal axis, are thirty-four in number, and are bounded by curves approaching to a parabola of a very high order ; cor- responding to these are twelve horizontal sections between the keel and the plane of floatation, taken at intervals of two feet on the vertical axis, the first section occurring at the distance of nine inches from the upper surface of the keel : It is required to determine the measure of stability, when by the action of the wind, or some other equivalent external force, the vessel is deflected from the upright position through an angle of thirty degrees; the ordinates corresponding to the several sections, being as registered in the following table. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 383 TABLE SHOWING THE ORDINATES CORRESPONDING TO THE SEVERAL SECTIONS. Horizontal sections, intervals on the vertical axis 2 feet. J & & 4 1 I 1 b o 1 h ft g I 1 1 No. 1 ft. 2 ft. 3 ft. 4 ft. 5 ft. 6 ft. 7 ft. 8 ft. 9 ft. 10 ft. 11 ft. 12 ft. 1 2 3 4 5 6 7 8 9 10 2.55 3.09 3.50 6.85 9.78 1.80 6.35 10.45 13.50 4.09 9.00 13.20 15.09 6.20 11.30 15.30 17.55 1.70 8.20 13.50 16.85 18.64 3.30 10.00 15.10 17.84 19.30 20.12 20.65 20.90 21.14 21.28 21.51 21.51 21.51 21.35 21.35 4.90 11.73 16.24 18.55 19.73 6.60 13.25 17.08 19.10 20.08 8.10 14.40 17.70 19.45 20.30 9.60 15.35 18.10 19.70 20.50 20.92 21.21 21.44 21.55 21.60 21.60 21.60 21.60 21.60 21.60 10.78 16.00 18.40 19.85 20.54 5.50 6.75 8.00 9.10 9.80 12.15 13.60 14.55 15.20 15.50 15.60 16.80 17.45 17.88 18.15 18.30 18.30 1830 18.30 18.30 17.70 18.65 19.05 19.35 19.53 19.60 19.75 19.75 19.75 19.75 18.91 19.60 20.05 20.25 20.40 20.50 20.60 20.60 20.60 20.60 19.65 20.32 20.62 20.78 20.93 20.93 21.10 21.10 21.10 21.10 21.05 21.00 20.90 20.81 20.67 20.45 20.90 21.20 21.34 21.47 20.65 21.05 21.34 21.45 21.50 20.80 21.15 21.38 21.52 21.60 20.94 21.20 21.38 21.48 21.50 21.56 21.58 21.58 21.56 21.56 21.55 21.53 21.51 21.48 21.32 11 12 13 14 15 10.50 10.50 10.50 10.50 10.50 15.90 15.90 15.90 15.90 15.90 21.51 21.51 21.51 21.51 21.51 21.59 21.59 21.59 21.59 21.59 21.60 21.60 21.60 21.60 21.60 16 17 18 19 20 10.30 9.80 9.20 8.50 8.00 15.70 15.50 15.35 15.00 14.60 18.20 18.05 17.95 17.75 17.52 19.65 19.55 19.45 19.30 19.15 20.52 20.45 20.35 20.25 20.10 21.32 21.30 21.20 21.10 21.00 21.51 21.50 21.40 21.30 21.20 21.59 21.55 21.52 21.44 21.30 21.60 21.60 21.55 21.50 21.35 21.25 21.05 20.83 20.61 20.40 21.60 21.60 21.52 21.50 21.35 21 22 23 24 25 7.20 6.40 5.90 5.10 4.20 14.20 14.62 12.90 11.90 10.60 17.25 16.90 16.40 15.70 14.80 18.90 18.62 18.28 17.75 17.10 19.90 19.70 19.30 19.00 18.46 20.50 20.32 20.05 19.75 19.30 21.85 20.68 20.40 20.15 19.85 21.05 20.89 20.65 20.40 20.10 21.18 21.00 20.75 20.55 20.30 21.24 21.08 20.82 20.61 20.43 21.22 21.05 20.82 20.61 20.44 26 27 28 29 30 3.35 2.50 1.80 1.40 1.11 9.20 7.20 5.38 3-55 2.40 13.40 11.60 9.35 6.65 4.25 16.15 14.80 12.85 10.10 7.05 17.88 16.60 15.45 13.10 10.10 6.12 2.90 1.32 0.75 18.80 18.05 17.11 15.35 12.90 19.35 18.80 18.10 16.85 15.05 19.73 19.30 18.80 17.90 16.80 20.00 19.59 19.22 18.55 17.82 20.10 19.75 19.44 19.00 18.40 20.12 19.80 19.52 19.15 18.62 20.15 19.85 19.58 19.25 18.77 31 32 33 34 0.90 0.80 0.62 0.60 1.45 0.98 0.75 0.63 2.30 1.25 0.80 0.65 3.75 1.90 1.00 0.70 9.10 4.55 1.85 0.90 12.00 7.10 2.70 1.05 14.60 10.35 4.31 1.35 16.30 13.40 7.45 1.95 17.40 15.65 14.50 3.40 17.90 16.90 14.90 7.00 18.21 17.52 1650 12.95 From these data, combined with others that are either assumed or determined by the circumstances of the case, the stability of the vessel or the momentum of the redressing force is to be found by calculation ; it will, however, be an improvement on the mode of 384 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. procedure, if in the first place, we take a brief survey of the principles of construction; for this purpose, let act represent any transverse section of the vessel, at right angles to the principal longitudinal axis passing through the centre of effort ; then is ef the breadth of this section at the water line when the .ship is loaded and the plane of the masts vertical, and hi becomes the water line, coincident with the surface of the fluid, when the vessel is deflected from the upright position through the given angle fki. It is however manifest from the ordinates in the foregoing table, that in this case, the vertical sections are all different, both in form and in magnitude, and consequently, the primary and secondary water lines do not intersect one another in the point k which bisects ef\ let p be the point of intersection, and through the point/?, draw the straight line mn parallel to hi, and making with e/the angle fpn equal to the given angle of inclination. Now, by considering the conditions of the problem, it will readily appear, that the position of the point p in any of the sections parallel to acz, cannot be determined on the same principles by which the place of that point was fixed according to the foregoing solutions, viz. by equating the areas of the triangular spaces mpe and fpn; for it is evident, that the volume which becomes immersed below the fluid's surface in consequence of the inclination, and that which emerges above by the same cause, will not now be proportional to those areas, in the same manner as they are, on the supposition of all the vertical sections being equal and similar figures. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 385 In the present instance, the vertical sections being all different, both in form and magnitude, the water's surface intersecting the vessel in the plane passing through the line mn when the vessel is inclined, will so divide the areas of the several sections, that although the space fpn may not be equal to mpe in any one of them, yet the immersed volume corresponding to all the spaces fpn, estimated from the head to the stern of the ship, shall be equal to the volume corresponding to all the emerged spaces mpe estimated in the same manner. Let ef, the breadth of the section at the water line, be bisected in the point k by the vertical line dc, and suppose a plane to pass through dc from head to stern of the vessel, such a plane will divide the vessel into two parts that are equal and symmetrical, and it will pass through the point k in all the parallel vertical sections made throughout the whole length. But it is easily shown, that at whatever distance kp from the middle point k, the plane of floatation in the inclined position, inter- sects the primary line ef in one of the vertical sections, it will intersect the corresponding line in all the other sections at the same distance from the middle point; that is, the distance kp will be the same in all the parallel sections, (the same lines and letters of reference being understood to belong to each ;) for according to the conditions of the problem, the revolution of the vessel is supposed to be made about the principal longitudinal axis, and consequently, the intersection of the two planes passing through the lines ef and mn, must be parallel to the axis of motion, and therefore parallel to the line drawn through the point k in all the sections, estimated from head to stern of the ship. We have now to determine the distance kp at which the inter- section takes place ; and for this purpose we must consider, that according to the given conditions, whatever may be the position of the point p in all the sections, if lines mn are drawn through those points, making with ef, an angle equal to the given angle of inclina- tion ; then it is manifest, that the same plane will pass through all the lines mn that occur betwixt the head and stern of the vessel. It is therefore required to determine, at what distance kp from the middle points k, the plane of floatation corresponding to the inclined position of the vessel must pass, so as to cut off a volume on the depressed side fpn equal to that which rises above the water on the side mpe. In each of the parallel vertical sections, let the common line hi VOL. i. 2 c 386 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. be drawn through k, the middle point of ef, and inclined to ef at an angle equal to that of the vessel's deflexion ; then, from what we have stated above, all the lines hi will lie in the same plane; that is, the same plane will pass through the line hi in all the sections. If there- fore, the areas of the spaces fk i and h k e in each of the vertical sections, be determined by some mode of mensuration adapted to the particular case, it is easy from these equidistant areas, to ascertain the solidity of the volumes contained between the planes passing through the lines kf, ki and he, kh. Put m the magnitude or solid contents of the volume, bounded by the side of the vessel and the planes passing through A/and ki, m' the magnitude or solid contents of the volume, bounded by the side of the vessel and the planes passing through ke and k A, A the area or superficial contents of the plane passing through the line hki in all the sections, estimated from head to stern of the vessel, which area is determined by having given all the lines hi; (j) ~fki, the angle through which the vessel is inclined from the upright and quiescent position, and e nr m m', the difference of the volumes or solidities, denoted by the symbols m and m'. Then, if upon the line kf, which coincides with the line of floata- tion when the vessel is upright and quiescent, there be set off in each of the parallel sections, the line kp f_ AXsin.<^' and through all the points p thus found, let lines mpn be drawn parallel to hi, and consequently, cutting ef in the points p, at an angle equal to that of the vessel's inclination; then, if a plane be drawn through all the lines mpn, it will so divide the vessel, that the solidity of the volume contained between the planes passing through the lines fp and np, will approximate to an equality with the volume contained between the planes passing through the lines ep and mp. Therefore, since the surface of the water coincides with the plane passing through ef when the vessel is upright, it will also coincide with the plane passing through all the lines mpn, when the vessel is deflected through the angle fpn, whose magnitude is given. It is very easy to show, that by setting off the distance kp in all the OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 387 sections, as determined by the preceding equation, and thereby draw- ing a plane through all the lines mpn, the plane thus drawn is coincident with the water's surface, and is situated very nearly in its true position. For through the point /<:, draw the line kr meet- ing mn perpendicularly in the point r ; then is krp a right angled triangle in which the angle kpr is given, and by the principles of mensuration, it is manifest, that the solid contained between the planes passing through the lines hki and mpn from head to stern of the vessel, is very nearly equal to the area of the plane passing through hkij drawn into kr the perpendicular thickness of the solid. Now, the solid of which hinm is a section, is obviously equal to the difference of the solids of which fki and hke are sections; hence we have m m e AXkr nearly ; consequently, by division, we obtain and by the principles of Plane Trigonometry, we get kr : kp : : sin. kpr : rad., or by restoring the analytical values, it is e : kp : : sm.^> : rad. ; and from this, by reducing the proportion and putting radius equal to unity, we obtain * _ e _ -AXsin.0' (289). An equation which is very nearly true for small inclinations, and this being the case, it fully establishes the propriety of the above construction ; if the areas of the planes passing through the lines hki and mpn are equal to one another, the construction as thus effected would be rigorously correct. 474. In pursuing the construction, it will be necessary, in order to avoid confusion in the lines and letters of reference, to redraw that part of the section which includes the angle of the vessel's inclination, viz. the space contained between the sides of the vessel me, nf and the dotted lines en, inf. We shall not, however, attempt to preserve the due proportion between the several parts of the figure ; this indeed would be troublesome and altogether unnecessary, since it is the prin- ciples of construction only that we mean to illustrate, and not the actual solution of any particular example. 2c2 388 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. .7 Let enfm be the space in question, including the angle of the vessel's inclina- tion ; draw the lines me and nf cutting off the curvilinear areas mre and nsf; bisect the sides me, pe in the points u and TT, and draw the lines pu and mir intersecting each other in l\ /is the centre of gravity of the triangular space mpe. Suppose z to be the centre of gravity of the curvilinear segment mre, and through the points I and z, draw lq and zy respec- tively perpendicular to mn, the line of floatation in the inclined posi- tion of the vessel. Again, bisect the sides w/and/p in the points w, $, and draw pw and n(f> intersecting each other in the point o\ o is the centre of gravity of the triangular space npf. Let v be the centre of gravity of the curvilinear area n sf, and through the points o and v, draw the straight lines ot and vx respectively perpendicular to the water line mpn; then, in the line tx intercepted by the perpendiculars ot and vx, take tc such, that it shall be to 0: in the same proportion, as the curvilinear space nsf, is to the compound space pnsf, and by the property of the centre of gravity, c will be the point in mn, where it is intersected by the perpendicular through the common centre of the triangular and curvilinear spaces npf and n sf. Through the point p in all the sections, let a line PQ be drawn at right angles to mn ; then, the same plane will pass through all these lines, and cp will be the perpendicular distance of this plane, from the centre of gravity of the mixed space pnsf. Therefore, if the products arising from multiplying each area, into the distance pc of its centre of gravity from the plane passing through PQ, be truly cal- culated in all the sections contained between the head and stern of the vessel ; then, by the principle announced and demonstrated in Proposition (A), Chapter I, the distance of the centre of gravity of the volume, whose sections are represented by all the areas pnsf, from the vertical plane passing through PQ can easily be ascertained. Let pR be that distance, and by a similar mode of computation, suppose that pE is found to be the corresponding distance of the OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 389 centre of gravity of the volume whose sections are the areas pmre ; then is B R the horizontal distance between the centres of gravity of the volumes that are respectively immersed and emerged, below and above the water's surface, in consequence of the vessel being deflected from the upright position, through an angle of which the magnitude is known. 475. The solid content of the entire volume immersed, or the quan- tity of water displaced by the immersed part of the vessel, is to be obtained from the areas of the several horizontal sections ; for the ordinates drawn in the several sections being arranged in regular order, after the manner which we have adopted in the preceding table, the area of any section can readily be assigned, by methods of approximation adapted to the particular case, and from these areas the solidity of the immersed volume is to be inferred ; making allowance for the irregularities of the vessel towards the head and stern, if it be at all necessary to take those parts into the account ; in all practical cases, however, they may safely be omitted. That part of the immersed volume, comprehended between the keel and the lowest horizontal section, is obtained, by first finding the areas of the several vertical planes, between the keel and the nearest ordi- nates, and from these areas, by means of some appropriate mode of approximation, the magnitude of the part cut off by the lowermost horizontal plane will be determined ; which being added to the solidity of the part contained between the extreme planes, will give the mag- nitude of the immersed volume, or the quantity of fluid displaced. 476. Referring to the original diagram, it will be observed, that from the areas of the several horizontal sections, made between the keel of the vessel and the plane of floatation, the distance kg, that is, the distance between the water line ef and the centre of buoyancy, or the centre of gravity of the immersed volume, can also be determined by the application of particular approximating rules, and the best with which we are acquainted for this purpose, are those given by Stirling in his " Methodus Differ enlialis" and by Simpson in his " Essays ;" these rules may be expressed in general terms as follow. RULE 1. Jyx=i(?\S}Xr, where x is the fluxion of the abscissa, y the perpendicular ordinate, expressing a general term or function of x ; r the common distance between the ordinates ; S the sum of the first and last ordinates, and p the sum of the whole series. 390 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. RULE 2. y#* = (S-f4p-|-2Q)Xir, where x, y, r and 5 denote as in rule 1st; p the sum of the 2nd, 4th, 6th, 8th, &c. ordinates, and Q the sum of the 3rd, 5th, 7th, 9th, &c. (the last of the series excepted). RULES. / X =(S + 2p + 3g)X|r, here again, x, y, r and S denote as in the preceding cases ; P the sum of the 4th, 7th, 10th, 13th, &c. ordinates (the last excepted), and Q the sum of the 2nd, 3rd, 5th, 6th, 8th, 9th, &c. With respect to the applicability of the above rules, it may be observed, that the first approximates to the fluent, whatever may be the number of the given ordinates, and the second only requires that the number of ordinates shall be odd. But in order to apply the third rule, it is a necessary condition, that the number of given ordinates shall be some number in the series 4, 7, 10, 13, 16, &c. ; that is, the number of ordinates must be some multiple of 3 increased by unity. In every case, however, the approximate fluent can be obtained, either from the second or third rule considered separately, or from both taken conjointly. 477. But to return from this short digression, we may remark, that the position of the point G, which marks the centre of effort, or the centre of gravity of the whole vessel, depends partly on the equipment and construction, and partly upon the distribution of the loading and ballast; which circumstances, therefore, determine G#, the distance between the centre of effort and the centre of buoyancy when the vessel is upright. These several conditions having been determined, the remaining part of the construction, limiting the measure of the vessel's stability, may be effected as follows. Through g the centre of buoyancy, or the centre of gravity of the immersed volume, draw gt parallel to mn, and make gt to BR (see the subsidiary figure), as the volume immersed in consequence of the inclination, is to the whole immersed volume induced by the weight of the vessel ; through G the centre of effort, draw GZ parallel and 05 perpendicular to gt, and through t draw tm parallel to GS, and meet- ing the axis cd in M ; then is M the metacentre, and GZ the measure of the vessel's stability when inclined from the upright position through the angle fpn or gGs. The principles of the preceding construction are general, and can be applied in all cases, whatever may be the figure of the vessel, or OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 391 the nature of its bounding curves ; but the arithmetical operations, as applied to any particular case, are unavoidably tedious, and necessa- rily extend to considerable length ; they are, however, very far from being difficult, as the ensuing process will fully testify. 478. By referring to the table of ordinates, it will appear, that the greatest, or principal transverse section, intersects the longer axis, at about the distance of 60 feet, or 12 intervals from the section nearest to the head of the ship ; we shall therefore delineate that section, and in order that nothing may be wanting to the proper understanding of the subject, we shall also delineate the plane of floatation, which cor- responds to the twelfth horizontal section in the preceding table of ordinates. The ordinate in the table opposite the twelfth vertical and under the twelfth horizontal section, is 21.58 feet, and the whole vertical distance between the keel and the plane of floatation, is 22.75 feet; therefore, draw the horizontal line a a which make equal to 43.16 feet, and bisect a a per- pendicularly by AK equal to 22.75 feet. Divide the vertical axis A K into twelve parts, eleven of which are 2 feet each, and the first or lowermost only three fourths of a foot, or nine inches ; then, through the several points of division 1, 2, 3, 4, 5, 6, &c. and parallel to aa, draw the several ordinates, taken from the twelfth horizontal row in the preceding table, which set off both ways, and through the extre- mities of the several ordinates, let the curve line ana be drawn, which will represent the boundary of the principal lateral section, so far as it is immersed below the fluid's surface. And exactly in the same manner may the whole of the 34 vertical sections, into which the longer axis is divided, be delineated ; but the above being sufficient for illustration, we shall next proceed to describe the twelfth horizontal section which is coincident with the water's surface, and of which the greatest ordinate is 21.58 feet, correspond- ing to a A in the above vertical section. 479. Since the body of the vessel is divided at intervals of 5 feet into 34 vertical sections, it follows, that between the first section adjacent 392 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. to the head, and the 34th section adjacent to the stern, there must be 33 intervals, or 33x5 = 165 feet; therefore, draw the horizontal line mn to represent the longer axis of the plane of floata- tion, and make be equal to 165 feet, which divide into 33 equal intervals of 5 feet each; then at right angles to mn and through the several points of divi- sion 1,2,3,4, 5, &c. to 34, draw the ordinates dd, ee, //, a a, gg, kh, &c. to ii t and from a scale of equal parts, of the same dimen- sions as that from which be is taken, set off both ways, (beginning at the 1st division adjacent to the head of the vessel), the numbers contained in the twelfth column of the pre- ceding table; then, through the extremities of the seve- ral ordinates, let the curve man a be drawn, and it will represent the plane of floatation when the vessel is upright, according to the foregoing tabulated mea- surements. It is manifest, that by a similar mode of procedure, the eleven remaining horizontal sections, situated between the keel and the plane of floatation, might also be delineated; it is, however, unnecessary to pursue the subject of construction farther, since what has already been done, is quite sufficient to show the reader, the method and nature of the delineation when pursued throughout the entire vessel.* * It may be proper to remark, that the scales from which the vertical and horizontal sections have been constructed, are to one another as 2 to 1 ; the one for the vertical section being l-20th of an inch to a foot, and the other l-10th. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 393 480. It now only remains to calculate the measure of stability, when the vessel is deflected from the upright position through an angle of 30 degrees, and for this purpose we must again refer to the original diagram, where, on the supposition that it is correctly constructed, the lines ki and kh are to be carefully measured in each of the sec- tions, on the same scale with the original dimensions; then, if the lines fi and he be drawn, the areas of the triangular spaces fki and like, can easily be determined from the two sides and the included angle ; and if a series of perpendicular ordinates be measured on the lines fn and he, the areas of the curvilinear spaces/#i and hye may from thence be found, which being added to the triangles fki and hke, the sums will be the whole areas of the compound spaces fki x and hkey. Pursuing a similar process throughout the 34 vertical sections, we shall at last arrive at the magnitudes of the volumes which are con- tained between the planes passing through kf, ki and kh, he, a knowledge of these volumes being necessary to determine the posi- tion of the point p. It is presumed that it will be sufficient for the exemplification of the rules, to exhibit the calculation of one of the spaces fk ix ; to perform the operation for the whole series, would be a very tedious and at the same time a superfluous proceeding; and for this reason, that the constructions and calculations founded on them, for inferring the results in any one of the sections, are similar to those required for obtaining the corresponding results in any other section ; and this being the case, the representation of one process will suffice for all the rest. 481. But to proceed, the line ki being taken in the compasses, and applied to an accurate scale of the proper dimensions, it is found to indicate 22.6 feet, and according to the table of ordinates, the line fk is 21.58 feet; and moreover, according to the conditions of the problem, the angle of inclination, or that contained between the lines kf and ki, is equal to 30 degrees, of which the natural sine is j, radius being unity; consequently, if a denote the area of the triangle fki, we have by the principles of mensuration, a=|(22.58x21.6)=: 121.927 square feet. Now, according to the principles of Plane Trigonometry, the line/i is expressed by the equation ; V/22.6H-21.58 2 2x22.6X21.58xcos.30zz: 11.55 feet. 394 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. consequently, if this line be divided into six equal parts of 1.925 feet each, and perpendicular ordinates be erected thereon, they will be found to measure as follows, viz. No. of ordinates Feet - - - 0.15 3 0.30 4 0.43 5 0.38 6 0.23 Therefore, if a' denote the area of the curvilinear space fxi ; then by the second of the preceding approximating rules, we have S 4- 0, the sum of the extreme ordinates, 4p = 4(.15 4- .43 4- .23) = 3.24, the second term of the series, 2Q = 2(.30 4- .38) 1.36, the third and last term ; hence, by addition, we get S _|_ 4? 4- 2 Q = 4- 3.24 + 1 .36 = 4.6 ; but one third of the common interval is 0.642 of a foot nearly ; con- quently, by multiplication, the area of the curvilinear space fxi, becomes a 1 = .642 X 4.6 = 2.9532 square feet, which being added to the area of the triangle above determined, the area of the compound space fk i x becomes a 4- a' = 121.927 4- 2.953= 124.88 square feet. Again, if we put b to denote the area of the triangle hke, and b' the area of the curvilinear space hye; then, by proceeding in a manner similar to the above, the area of the mixed space hkey becomes b 4- b' = 1 33.68 square feet. Now, if in this way, the values of a 4- a' and b 4- b' be calculated for each of the 34 vertical sections, the several results will be as exhi- bited in the following table. OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 395 Table of Areas for the thirty -four Vertical Sections. No. of Values of Values of No. of Values of Values of sections. a -\- a' & + &' sections. a + a' 6 + 6' feet. feet. feet. feet. 1 42.86 23.61 18 124.62 132.53 2 81.53 58.92 19 123.91 131.05 3 100.80 86.80 20 123.21 129.57 4 114.16 105.27 21 121.06 127.48 5 121.56 115.70 22 118.91 125.40 6 121.75 120.90 23 117.50 122.66 7 123.47 125.36 24 116.10 119.93 8 125.20 129.82 25 114.01 116.88 9 124.87 131.04 26 111.91 113.83 10 124.54 132.27 27 108.96 109.81 11 124.69 132.97 28 106.01 105.80 12* 124.88 133.68 29 101.82 98.92 13 124.87 133.68 30 97.24 91.71 14 124.87 133.68 31 92.41 79.95 15 124.82 133.42 32 86.31 66.06 16 124.78 133.17 33 81.60 48.20 17 124.20 132.85 34 68.35 17.92 Sums 1953.85 1963.14 Sums 1813.93 1737.70 Therefore, the sum of all the (a -f- a')"zz 3767.78 ; and the sum of all the (b -f i'yzz 3700.84, and by the conditions of the problem, the vertical sections intersect the principal longitudinal axis at intervals of 5 feet ; therefore, by applying the third of the preceding approxi- mative rules, the solid contents of the volume contained between the planes passing through fk and ki, will be found as follows. S= 42.86 + 68.35 zz 1 1 1.21, the sum of the extreme ordinates, 2pzz(611.82 -f 555.25) X 2= 2334. 14, the 2nd term of the series, or twice the sum of the 4th, 7th, 10th, 13th, &c. ordinates, 3Qzz(1299.17 -\- 1 190. 33) X 3 zz 7468. 5, the third term of the series, or the sum of the 2nd, 3rd, 5th, 6th, 8th, 9th, &c. ordinates ; consequently, by addition, we have S -f 2r -f SQ zz 1 1 1.21 -f 2334. 14 -f 7468.5 zz 9913.85 ; * This is the vertical section for which we have exhibited the process of com- putation. 396 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. and finally, by multiplying by three eighths of the common interval, the magnitude of the volume becomes m = (S4-2p-f3Q)X5Xf z= 9913.85 x V = 18588.47 cubic feet very nearly. Proceeding exactly in the same manner with the areas (b -\- >'), the solidity of the space comprehended between the planes passing through the lines kh and ke, and the intercepted side of the vessel, becomes m'=. (5-f 2p 4- 3Q)X y = 18433.47 cubic feet; therefore, by subtraction, we obtain m m'= e = 18588.47 1 8433.47 = 155. 482. In the next place, we have to determine the area of the plane passing through all the lines hki in the several vertical sections ; this is effected by measuring all the ordinates in that plane, taken at the common interval of 5 feet along the axis passing through k from head to stern of the vessel. When this operation is performed in a dexterous manner, the area of the plane will be found to be 7106 square feet very nearly ; that is, A = 7106 square feet; consequently, by equation (412), we have Hence it appears, that the distance of the pointy from the middle point k, is too small to cause any material error in the result, we shall therefore suppose that the plane of floatation corresponding to both positions of the vessel, intersect each other in the axis passing through k from head to stern of the vessel. Taking, therefore, the mean between the two foregoing solidities, we shall have $(ro 4- m') |(18588.47 -f 18433.47) = 18510.97 cubic feet. This, therefore, is the solidity of the volume which becomes immersed in consequence of the inclination ; and by pursuing a similar mode of procedure with respect to the areas of the twelve horizontal sections, the solidity of the whole volume immersed, will be found to be 119384 cubic feet very nearly ; and moreover, by referring to the subsidiary figure employed in the construction, and introducing the principles by which the distance BR is ascertained, we shall have BRzr27.32 feet; consequently, by Proposition XII., Chapter XIII., the distance gt in the original figure is thus found, OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 397 119384 : 27.32 : : 18510.97 : 4.25 feet, 483. But in order to infer the stability of the vessel from the value of gt just discovered, it is necessary to have given the distance eg, or the distance between the centre of effort and the centre of buoyancy ; now it is obvious, that the position of this latter point is regulated entirely by the form and dimensions of the immersed portion of the vessel, and consequently, it may be considered as absolutely fixed with respect to the plane of floatation ; but since the position of the centre of effort is regulated partly by the construction and equipment, and partly by the distribution of the loading and ballast, it can only be assumed on the ground of supposition, unless in cases where the position of that point has been actually ascertained by accurate mensuration. In several instances, the distance Gg has been measured, and found to be equal to about one eighth of the greatest breadth at the plane of floatation ; therefore, by assuming this to be the case generally, we have eg == J- of 43.16 = 5.396 feet, therefore, by Plane Trigonometry, it is rad. : 5.396 : : sin. 30 : gs, from which, by reducing the proportion, we obtain #5 = 2.698 feet, which being subtracted from g t, the remainder is S=:GZ = 4.25 2.698=:1.552 feet, the measure of the vessel's stability, or the length of the equilibrating lever. But the whole weight of the vessel, as found from the solidity of the immersed part, is 119384 w nr 341 1 tons very nearly ; 35 cubic feet 35 of sea water being equal to one ton weight ; consequently, the momentum of the redressing force, or the power which the pressure of the water exerts to restore the vessel to the upright position, is equal to 3411 tons acting on a lever whose length is 1.552 feet; or it is equivalent to a force or pressure of 245 tons acting at the dis- tance of half the greatest breadth of the vessel from the axis ; for by the principles of the lever, it is 21.58 : 3411 : : 1.552 : 245 tons nearly. The preceding is the method of determining the stability of a ship, on the supposition that the data are all assignable ; the process con- 398 OF THE STABILITY OF STEAM SHIPS. sidered in its full extent is unavoidably tedious and prolix, we have merely pointed out the method of conducting the calculations ; but when it is necessary to determine the stability of a ship in actual practice, every individual process must be separately performed, and the result obtained as above, will approximate very nearly to the truth. 484. Those who have ever witnessed the spectacle of a ship tossed in a tempest, or have read any of the brilliant accounts which maritime tales afford, will appreciate the subject we have just investigated. They may have seen, moreover, the vast bulwark slide from her cradle into the calm water, on which she first swung round and heeled till she regained her stability of equilibrium ; giving the imagination a contrast of the stormy element on which she was soon to ride in awful grandeur. But seamen will best appreciate our labours, especially those who in the days of battle and the nights of danger have had to manage the noblest work of art and skill ; and who in their country's cause have encountered all weathers and every clime, traversing the wide expanse of ocean's bosom, visiting all the ends of the earth, and identifying themselves as part of the stupendous ship which figuratively has to do and to suffer for her country, and which in peace or in war, in sunshine or in storm, carries with her the benediction of mankind pronounced as on a living being, when she was first launched in pre- sence often thousand enthusiastic spectators, one and all sympathizing in the national solemnity. 4. PRINCIPLES OF THE STABILITY OF STEAM SHIPS. 485. When a ship is set afloat upon the surface of the waters, and impelled by some power acting in the direction of its length, as is the case with steam vessels, now so extensively employed, the subject of stability becomes of very great importance. This remark does not strictly apply to vesels navigating still waters, or rivers where the tides produce but small effects ; but it is well known that the natural motion of the sea, even in its calmest state, causes a considerable lateral motion in a vessel placed upon its surface, and in consequence of this motion, the paddles are made to dip unequally in the water, by which means some part of the impelling power is lost. It is with the view of avoiding this waste of power, that the subject of stability acquires such vast importance when referred to steam vessels ; and it is easy to perceive, that the best method of attaining this object, is to adapt the form and capacity of the vessel to the OF THE STABILITY OF STEAM SHIPS. 399 several circumstances by which the floatation is regulated, and on which the mode of action depends. The late Thomas Tredgold has considered this subject in his work on the STEAM ENGINE, and his views in this case, as in all others where the powers of his comprehensive and refined mind have been called into action, are concise, elegant, and original ; and we cannot close this chapter to greater advantage than by adopting his theory, which however we shall modify to suit the plan and arrangement of the present work. 486. In order to simplify the investigation of stability, Tredgold considers the vessel to be a solid homogeneous body of the same density as water, with vertical or circular surfaces at the water lines when the vessel is in a state of quiescence. Now, it is obvious, that with regard to a ship which is designed to carry burdens at sea, the first of these conditions cannot obtain ; this however is of no conse- quence as respects the result of the inquiry, for in reality it refers to a mass of water equal in bulk to the immersed portion of the floating body. As another means of simplification, he supposes the transverse sections of the ship at right angles to the axis of motion to be in the form of a parabola, of which the equation is px=iy n , and for the purpose of contrasting the extremes of form, he branches the subject into the two following varieties, viz. 1 . When the ordinates are parallel to the depth, and 2. When the ordinates are parallel to the breadth. For each of these cases a general equation is deduced, involving the sine of the angle of inclination, the breadth and depth of the vessel, and the index or exponent by which the order of the parabola is expressed. 487. Having already investigated an expression by which the stability of a floating body is indicated, we do not consider it neces- sary to trace the steps of inquiry in the present instance, for the intelligent reader will at once perceive, that although the form of the equation is somewhat different, by reason of its involving different parts and different data, yet the principles upon which the investiga- tion proceeds, are, and necessarily must be, the same, or nearly the same as before. 488. When the ordinates of the parabola are parallel to the depth, the general equation by which the stability is indicated, becomes (290). 400 OF THE STABILITY OF STEAM SHIPS. where > DC is the breadth of the vessel at the water line when upright and quiescent, C?=LK the corresponding depth, FPC the angle of inclination from the upright position, n the exponent denoting the order of the parabolic section, and S the stability. 489. If we examine the structure of the above equation, it will readily appear, that while b* is greater than - ' , the stability is n | A> positive, and the vessel endeavours to regain the upright position ; if these two quantities are equal to one another there is no stability ; and if the latter exceeds the former, the stability is negative, and the vessel oversets. Hence it appears, that between the breadth and depth of the vessel, a certain relation must obtain to render it fit and sufficiently stable for the purposes of navigation; and it is further manifest, that the stability increases directly as the exponent of the ordinate, so does the area of the transverse section ; but in order to give the proper degree of stability, the breadth must increase more rapidly than the depth. By giving different values to the symbol n in the preceding general equation, we shall obtain expressions to indicate the stability for sections of different forms ; thus for instance, if n zz 1 the section is a triangle, and the expression for the stability becomes (291). 490. This equation is very simple, and can easily be illustrated by an example ; the practical rule for its reduction may be expressed in the following terms. RULE. From the square of the breadth of the water line when the vessel is upright, subtract twice the square of the corresponding depth ; multiply the remainder by the breadth drawn into the natural sine of the angle of inclination, and one twelfth of the product will express the stability. 491. EXAMPLE. A floating body in the form of a triangular prism, has its breadth at the water line equal to 28 feet, the corresponding depth under the water equal to 19| feet, and its density equal to the density of water; now, suppose the body to be in a state of equili- brium when the axis is vertical ; what will be its stability, or what is the relative value of the force by which it would endeavour to regain the upright position, on the supposition that it has been deflected from it through an angle of 15 degrees ? This is obviously a case that is not likely to occur in the practice OF THE STABILITY OF STEAM SHIPS. 401 of steam navigation, because the form is altogether unsuitable for vessels of that description, and our only object forgiving it here is to show the method of reducing the equation ; this being the more necessary for the sake of system, as it forms a particular case of the general problem, and is deducible from it by merely assuming a particular value for the exponent of the parabolic ordinate. By the rule, we have (b* 2e? 2 ) =r 784 - 380.25X2 = 23.5 ; therefore, by multiplication and division, we obtain b sin. rf> ^(^2rf a )zz28x0.25882X23.5-rl2z=5 14.19 very nearly. 492. Returning to the general equation, if we suppose wzr2, then the section is in the form of the common or Apollonian para- bola, as represented in the an- nexed diagram, wherein AB is the base or double ordinate of the parabolic section, DC its axis, FII the water-line, or double ordinate of the immersed portion FDH, DE the corresponding abscissa, and IK the horizontal surface of the fluid. Then, with ram 2, the expression for the stability becomes (292) . The form of the vessel of which the stability is expressed by the above equation, is much better adapted for the purposes of steam navigation, than the triangular form already discussed; but it is obvious from the relation of the parenthetical terms, that it requires a much greater breadth at the water line under the same depth and inclina- tion, to give an equal degree of stability ; and the breadth necessary for this purpose maybe determined by reversing the expression, which will then assume the form of a cubic equation, wanting the second term, and whose reduction will give the necessary breadth. 493. Now, by the preceding calculation we have found the stability to be 14.19 very nearly, while the depth is 19| feet, and the inclina- tion from the upright position, 15 degrees, of which the natural sine is 0.25882; consequently, by substitution we obtain 0.021576 3 24.66=14.19, and if this equation be reduced, we shall find the value of b or the breadth of the vessel at the water line, to be a very small quantity in VOL. i. 2 D 402 OF THE STABILITY OF STEAM SHIPS. excess of 34 feet ; but taking it at 34, the value of the stability for a vessel in the form of a common parabola becomes s _ 34X0.25882 (1156 _ 1140t75)= 1M84; hence it appears, that the breadth at the water line, in the case of the parabola, requires an increase of more than 6 feet, to give the same stability as the triangle under the same depth and deflexion. 494. If the equation for the stability in the case of the parabola, be compared with that for the triangle, it will be seen that 3d 2 occurs in the one case, instead of 2d 2 in the other ; consequently, the practical rule as given for the triangle, will also apply to the parabola, if the phrase " thrice the square of the corresponding depth" be substituted for " twice the square," as it is now expressed ; the repetition of the rule is therefore unnecessary. 495. Again, if we put wnr3, then the transverse section of the vessel is in the form of a cubic parabola, and the general equation for the value of the stability becomes 496. This form is greatly superior to the preceding one for a steam vessel, as it gives the surfaces in contact with the water a less degree of curvature ; but it requires a greater increase of the breadth at the water line in proportion to the depth to obtain the same degree of stability, which is manifest from the increase of the negative co- efficient, the form of the equation being in every other respect the same as before. The practical rule for this form, may be expressed in words at length as follows. RULE. From the square of the breadth at the water line when the vessel is upright, subtract 3.6 times the square of the corresponding depth ; multiply the remainder by the breadth drawn into the natural sine of the angle of inclina- tion r and one twelfth of the product will express the stability. 497. EXAMPLE. Let the breadth of the water line be 38 feet, and let the depth and the deflexion, as well as the density of the vessel, be the same as before ; what then will be the value of the stability ? Here by the rule we have (6 2 3.6d 9 )i=38 2 3.6 X19.5 2 =: 75.1 ; consequently, by multiplication and division, we have OF THE STABILITY OF STEAM SHIPS. 403 12 v - ;= 38x0.02157x75.1 n: 6 1.6 nearly. 498. In order to pursue the inquiry a step further, let us suppose that 7zz=5; then, by substituting this value of n in the general equation for the value of the stability, we shall get (294). an equation which differs in nothing from those that precede it, but in the value of the constant co-efficient of the negative term within the parenthesis, a quantity which indicates the increase of breadth at the water line, necessary to give the vessel the same degree of stability, under the same depth and deflexion, which it possesses when bounded by curves of the lower orders. 499. If the curves which we have just considered were delineated from a fixed scale, according to the relation that subsists between the ordinates and the corresponding abscissas, it would be seen, that the breadths towards the vertex become greater and greater as the exponent of the ordinate increases; the figure therefore approaches continually to the form of a rectangular parallelogram, and essentially coincides fc' with it, when the value of n becomes infinite, as in the parabola A KB, wherein DC is the breadth, and LK the depth of the vessel ; E F the water line, and k'k the line of sup- port in the inclined position ; y the ordinate parallel to the depth, and x the abscissa; DPE the immersed triangle, I and F PC the extant triangle. This extreme case has a manifest rela- tion to the subject of stability ; for whatever may be the effect of giving to the sides of vessels the forms of the higher orders of parabolas, it is evident, that as the exponent of the ordinate is increased, the stability will approach to that which would obtain if the sides were made parallel to the plane of the masts. Now, it may easily be shown, that when the sides of the vessel are made to coincide with the form of a conic parabola, (fig. art. 492,) the stability is the same as when the sides are parallel planes ; hence it is inferred, that if the sides of a vessel be formed to coincide with a parabola of the lowest order, and another to coincide with one of the highest, all other circumstances being the same, the stabilities will be equal in these two cases. 2 D 2 404 OF THE STABILITY OF STEAM SHIPS. 500. But we must now proceed to consider the second variety, in which the ordinates are parallel to the breadth of the vessel at the water line when the vessel is placed in an upright and quiescent position ; and in this case, the general equation expressing the value of the stability, is ^sin^x .12nd 2 \ "ITA 6 ~n*+3rc + 2/' (295). where the several letters which enter the equation indicate precisely the same quantities, and refer to the same parts of the vessel as before ; and by giving particular values to the quantity n, we shall obtain another series of equations, indicating the stability according to the order of the parabolic curve by which the vessel is bounded. 501. If we put w:nl, then the transverse section of the vessel becomes a triangle, and the equation expressing the value of the stability in that case, is (296) . which is manifestly the same expression as that which we obtained for the triangle in the first variety, where the ordinates were supposed to be parallel to the depth; hence, the value of the stability when estimated in numbers will also be the same. 502. Again, if we suppose the bounding curve of a cross section to be the same as the common parabola, then 11=12, and this being sub- stituted in the general equation, the expression for the stability in this case, is (297) . the very same as for the triangle ; hence it appears, that when the ordinates are parallel to the breadth, the stability for a triangular section is the same as it is for a section in the form of the common or Apollonian parabola. 503. But when the boundary of the section is in the form of a cubic parabola, then n =: 3, which being substituted in the general equation, the expression for the value of the stability in this case, is s= * (*.!.>. .'.': r. (298) If this equation be compared with the corresponding one for the cubic parabola, in the case when the ordinates are parallel to the depth, it will be seen that the present form is superior in point of stability, since it requires a less breadth in proportion to the depth to offer an equal resistance. This inference is drawn from a comparison OF THE STABILITY OF STEAM SHIPS. 405 of the constants belonging to the negative term, for in the one case it is double of what it is in the other, and consequently, in the latter case, a less breadth is necessary to give a positive result. 504. Lastly, if n zr 5, then the parabola which bounds the trans- verse section A KB of the vessel, is defined by the equation px=iy s , as in the annexed figure, in which the several letters indi- cate the parts already men- tioned, viz. DC the breadth, LK the depth of the vessel, E F the water line, k' k the line of support, y the ordinate parallel to the breadth, x the abscissa ; then, the expression for the stability in this case, becomes .,/ (299 , from which it appears, that the higher the order of the parabola, the less increase of breadth is necessary with the same depth to obtain an equal degree of resistance ; but in the case when the ordinates are parallel to the depth, as in the first variety, the contrary takes place, a greater increase of breadth being necessary for the same purpose. Hence we conclude, that the higher the order of the parabola, the greater is the degree of stability ; but the form in which the ordinates are parallel to the breadth, is preferable to that in which they are parallel to the depth; and, as Mr. Tredgold justly remarks, " this species of figure may be easily traced through all the varieties of form, and it has obviously a decided advantage in point of stability, and it is so easy to compute its capacity and describe it by ordinates, that it is much to be preferred to the elliptical figures which foreign writers have chosen for calculation." 505. In order that the stability may be the same at every section throughout the whole length of the vessel, this being a necessary condition in the most advantageous cases, the breadth should be every where in the same ratio to the depth ; for when this is the case, the vessel will suffer no lateral strain from a change of position. The preceding determinations relate to the vessel's stability when the inclination is made about the longer axis ; but the position of the shorter axis, round which the ship revolves in pitching, in all cases of practical inquiry, must also be considered ; but since the investigation sf the several conditions would be similar to that which refers to the longer axis, we deem it unnecessary to extend the inquiry any further. CHAPTER XIV. OF THE CENTRE OF PRESSURE. 506. THE subject of the present chapter might have been placed in juxtaposition with the doctrine of pressure on plane surfaces ; but we chose to reserve it for the conclusion of fluid equilibrium, and in as brief a manner as possible we shall now view the centre of pressure, by illustrating a few select examples dependent upon its principles. The Centre of Pressure of a plane surface immersed in a fluid, or sustaining a fluid pressing against it, is that point, to which, if a force be applied equal and contrary to the whole pressure exerted by the fluid, the plane will remain at rest, having no tendency to incline to either side. It is manifest from this definition, that if a plane surface immersed in a fluid, or otherwise exposed to its influence, be parallel to the horizon ; then, the centre of pressure and the centre of gravity occur in the same point, and the same is true with respect to every plane on which the pressure is uniform; but when the plane on which the pressure is exerted, is any how inclined to the horizon, or to the surface of the fluid whose pressure it sustains ; then, in order to determine the centre of pressure, we must have recourse to the resolution of the following problem. PROBLEM LXIII. 507. Having given the dimensions and position of a plane sur- face immersed in a fluid, or otherwise exposed to its influence : It is required to determine the position of the centre of pressure, or that point, to which, if a force be applied equal and opposite to the pressure of the fluid, the plane shall remain in a state of quiescence, having no tendency to incline to either side. 01' THE- CENTRE OF PRESSURE. 407 Let ABC be a cistern filled with an incompressible and non-elastic fluid, and let abed be a rectangular plane immersed in it at a given angle of inclination to its surface ; produce the sides da and cb di- rectly forward to meet the surface of the fluid in the points e and /; join ef, and through the points e and /, draw es and/r respectively perpendicular to the plane produced, and coinciding with the surface of the fluid in ef; draw also ds and cr, meeting es andyY at right angles in the points s and r\ then is des or c/r, the angle of the plane's inclination, and ds, cr are the per- pendicular depths of the points d and c. Let P be the position of the centre of pressure, and through p draw pm and PW, respectively perpendicular to cb and cd the sides of the rectangular plane ; then are cb and cd the axes of rectangular co-ordi- nates originating at c, and pm, PW are the corresponding co-ordinates, passing through p the centre of pressure, supposed to be situated in that point. Now, it is manifest from the nature of fluid pressure demonstrated in the first chapter, that the force of the fluid against d : Is equal to the weight of a column of the fluid, whose base is the point d, and altitude the perpendicular depth of that point below the upper surface of the fluid. ^ Consequently, the force against the point d, varies as dx ds ; but by the principles of Plane Trigonometry, we have rad. : ed : : sin. des : ds; hence by reduction, we get ds=:ed$m.des; therefore the pressure on the point d varies as d XedXsin.des, and the effort or momentum of this force, to turn the plane about the ordinate pm, manifestly varies as d Xee?Xsin.c?esXPw, where p n is the length of the lever on Avhich the force acts. But by subtraction, p n -=.ed /m, for edfc; therefore by sub- stitution, the force to turn the plane about the ordinate pm, varies as 408 OF THE CENTRE OF PRESSURE. dXed* Xsm.de s e?Xec?X/wXsi therefore, the accumulated effect of all the forces, to turn the plane round rw, must be proportional to the sum of {dXed*}Xs\n.des /mXsum of {dXed}Xsin-des, and this by the definition is equal to nothing; hence we get yVwXsum of {c?Xec?}sin.rfeszzsum of {dXed*} sin.e?es ; therefore, by division, we obtain _ sum o ~ su But the sum of {dXed}, is obviously the same as the body, or sum of all the constituent particles, multiplied into the distance of the common centre of gravity ; and therefore, by the principles of mechanics, fm is also the distance of the centre of percussion, if ef, the common intersection of the plane with the fluid, be considered as the axis of suspension, the plane being supposed to vibrate flat-ways. By reasoning in the same manner as above, it will readily appear, that the effect or momentum of the pressure on c?, to turn the plane about the ordinate PW, varies as dXe dXdnX$in.de s; but by subtraction, it is dn cd en ; therefore by substitution, the force on d to turn the plane round PW, varies as and consequently, the effect of all the forces to turn the plane around pn, must be proportional to the sum of {dxdeycd} siu.des crcXsum of {dxed} s'm.des ; but by the definition, the sum of these forces is equal to nothing; for the plane has no tendency to incline to either side, being sustained in a state of quiescence by means of the fluid, and the equivalent oppos- ing force applied at the centre of pressure ; hence we get cwXsum of [dXed}~ sum of {dXdeXcd} ; therefore, by division, we shall have __sum of {dXdeXcd} sumof{dXde} ' (301). which expression also indicates the distance of the centre of percus- sion ; from which it is manifest, that the centres of pressure and percussion coincide, when the line of common section between the plane or the plane produced, and the surface of the fluid is made the axis of suspension. This being the case, it is evident, that the formulse OF THE CENTRE OF PRESSURE. 409 which are employed to determine the centre of percussion, may also, and with equal propriety, be employed to determine the centre of pressure. Now, the writers on the general principles of mechanical science have demonstrated, that if . xmed, the side of the plane extending downwards, and ?/:zicc?, the horizontal side parallel tofe; t\\Gnfm and pm, the respective distances of the point P, fromfe and fc the sides of the plane, are generally represented by the following fluxional equations, viz. / y* fm - - , andpwzn Jxyx ZJyxx (302). From these two equations, therefore, the centre of pressure cor- responding to any particular case, can easily be found, as will become manifest, by carefully tracing the several steps in the resolution of the following problems. PROBLEM LXIV. 508. A physical line of a given length, is vertically immersed in a fluid : It is required to ascertain at what distance below the surface of the fluid the centre of pressure occurs. Let be be a physical line, perpendicularly immersed in a fluid of which the surface is AB, and produce cb to f, so that the point f may be considered as the centre of suspen- sion, and let m be the centre of pressure, or the centre of percussion ; then by equation (302), we shall obtain * -^ f' ' 9 in which equation y is constant ; therefore, Put d fc, the distance of the lower extremity below AB, 8 /, the distance of the higher extremity, and I ~bc, the whole length of the line. 410 OF THE CENTRE OF PRESSURE. Therefore, by addition we have d= I -f- , and by taking the fluent of the above equation, we get fm ._ J - 3 (a*- ar and when a; me?, we shall obtain (303). The equation as it now stands, is general in reference to a line of which the extremities are both situated below the surface ; but when the upper extremity is coincident with it, then 8 vanishes, in which case c? Z, and our equation becomes fm = \L* ( 304 >- 509. This last form of the expression is too simple to require any illustration ; but the form which it assumes in equation (303), may be expressed in words at length in the following manner. RULE. Divide the difference of the cubes of the depths of the extremities of the given line below the surface of the fluid, by the difference of their squares, and two thirds of the quotient will give the distance of the centre of pressure below the surface ; from which, subtract the depth of the upper extremity, and the remainder will show the point in the line where the centre of pressure is situated. 510. EXAMPLE. Required the position of the centre of pressure in a line of 4 feet in length, when immersed vertically in a fluid, the * Now what is here true of a physical line is true also of a plane, which, if it reach the surface of the fluid whose pressure it sustains, will have its centre of pressure at a distance equal to two thirds of its breadth or depth from the upper extremity ; and this holds true also, whatever may be its inclination, its centre of pressure will he distant from the upper edge by two thirds of its surface or breadth. A single force, therefore, applied at that distance, and exactly in the middle of the length of the plane, would hold it at rest. And the same would manifestly be the case, if the rod, in place of being applied longitudinally at a single point, were placed across the plane over the point which indicates the position of the centre of pressure. All that is required in order to procure the equilibrium is, that a sufficient balancing force be applied to that centre ; thus, a sluice or floodgate may be held in its place by the pressure of a single force, applied at one third of its length from its base, and at two thirds of its length below the surface of the fluid. And this suggests the practical importance of placing the beams and hinges of flood and lock-gates at equal distances above and below the centre of pressure, which is at two thirds the depth of the gate. See Problem LXVI. p. 416. OF THE CENTRE OF PRESSURE. 411 upper extremity being 2 feet below the surface, and the lower extremity 6 feet ? Here we have d 3 3 :z: 6 s 2 3 208, and (F c~ 6 2 2'zz: 32 ; consequently, by division, we obtain 208 -55= 6.5 feet, and by taking two thirds of this, we get fm i= -| of 6.5 rz 44 ft. and finally, by subtracting the depth of the upper end, we obtain 511. If the upper extremity of the line had been in contact with the surface of the fluid, then would imzzfof 4z=2tft. This is manifest from equation (304), and if a rectangle be described upon the vertical line, the distance of its centre of pressure below the surface of the fluid will be expressed by the equation (303 or 304), according as the upper extremity is situated below, or in contact with the surface, and this distance will obviously be measured in the line by which the rectangle is bisected. 512. If the upper side of the rectangle coincides with the surface of the fluid, as in the annexed diagram, where adfe is the rectangular parallelogram, having T _ _ the upper side ad in contact with IK ; then, according to equation (304), bm the distance of the centre of pressure, is equal to two thirds of be, the whole length of the paral- lelogram, and consequently, by subtraction, we have mc^bc; therefore, the tendency of the plane to turn about the base ef, is equal to the pressure which it sustains, drawn into the length of the lever mc \l, where I denotes the whole length of the plane. Put b nr ad or ef, the horizontal breadth of the plane, p =: the entire pressure which it sustains, and s iz: the specific gravity of the fluid in which it is immersed. Then, by equation (8), Problem III. Chapter II. the whole pressure sustained by the immersed plane, is expressed by 412 OF THE CENTRE OF PRESSURE. and this pressure being applied at m, operates on the lever me to turn the plane about ef, with a force which is equal to Through the point m, draw mn parallel to ef, the base of the immersed plane ; then, the tendency to turn round the vertical side a e, is equal to the whole pressure upon the plane, drawn into the length of the lever mn ; but mn=i \b, and we have seen above, that the pressure is expressed by %bl*s; consequently, the tendency to turn round ae, is let these two momenta be compared, and we shall have 513. This is obvious, for by casting out the common factors, and assimilating the fractions, the ratio becomes 21 : 3b ; and when I and b are equal to one another, or when the immersed plane is a square ; then the ratio is simply as 2:3; that is, the tendency of the plane to turn round the lower horizontal side, is to its tendency to turn round a vertical side, as 1 : 1 J, or as 2 : 3. PROBLEM LXV. 514. A semi-parabolic plane is immersed vertically in a fluid, in such a manner, that the extreme ordinate is just in contact with the surface : It is required to determine the position of the centre of pressure, both with respect to the axis and the extreme ordinate , which is coincident with the surface of the fluid. Let IK be the surface of the fluid, and acd the semi-parabola vertically immersed in it, in such a manner, that ad the extreme ordinate coincides with i K, while the axis ac is perpendicular to it. Let m be the point at which the centre of pressure is supposed to be situated, and through m draw mb and mn, respectively perpen- dicular toad and ac, and suppose the axes of co-ordinates to originate at a ; then, if we OF THE CENTRE OF PRESSURE. 413 Put I ac, the axis of the semi-parabola, b zr ad, the extreme ordinate, which is in contact with IK, x =z any abscissa estimated from the vertex at c, and y zz the corresponding ordinate, then is I x the distance between the ordinate and the origin of the axes, corresponding to x in the general investigation. Problem LXIII. ; but by the property of the parabola, we have l:bl::x:y*; and from this, by reduction, we get Therefore, if l x and by |, be respectively substituted for and y in the equations of condition numbered (302), we shall have and for the corresponding co-ordinate, it is / mn~ But by the writers on the fluxional analysis, the complete fluents of these expressions are respectively as follows, viz. - , bm=: - ^ - ST- - ,and?wn 351 the correction in both cases being equal to nothing; but when arrr/, we get imzz|/, and mn T s jb, (305). and from these values of the co-ordinates, is the centre of pressure to be found. 515. EXAMPLE. A plane in the form of a semi-parabola, is immersed perpendicularly in a fluid, in such a manner, that the extreme ordinate coincides with the surface; whereabouts is the centre of pressure situated, the axis being 9 and the ordinate 6 inches ? 414 OF THE CENTRE OF PRESSURE. Here then we have given I 9 inches, and b zz: 6 inches ; conse- quently, by the equations numbered (305), we have bm $ of 9 zn 5j- inches, and mn^n T \ of 6 zz l inches. Therefore, with the abscissa ac:=9 inches, and the ordinate ad 6 inches, construct the semi-parabola adc, by means of points or otherwise, as directed by the writers on conic sections ; then, on the axis ac and the ordinate ad, set off an and a b, respectively equal to 5-f and If inches, as obtained by the pre- ceding calculation ; and through the points n and b as thus determined, draw nm and bm, respec- tively parallel to ad and ac, intersecting each other in m; then is m the place where the centre of pressure occurs, as was required by the question. 516. It would be easy to multiply cases and examples, respecting the parabola and other curves of a kindred nature, considering them either entire or in part, and situated in different positions, as referred to the surface of the fluid ; but since the resolution in every instance, depends upon the integration of the general fluxional equations num- bered (302), when accommodated to the particular figure, we think it quite unnecessary to dwell longer on this part of the inquiry ; we therefore proceed to resolve a problem or two that depend upon similar principles, and consequently, are well adapted for illustrating the manner in which the inquiry is to be extended. PROBLEM LXVI. 517. A vessel in the form of a parallelopipedon with the sides vertical, has one side loose revolving on a hinge at the bottom, and is kept in its position by a certain power applied at a given point: It is required to determine how high the vessel must be filled withjtuid, before the revolving side is forced open. Let ABC represent the vessel in question, and let avdc be the loose side moveable about the hinges at e andy*; bisect c d in c, and draw cb perpendicular to cd, and let n be the point at which the given power is applied ; then, because the side axdc is just sustained by means of the power acting at n, it follows, that the whole force of the fluid acting at the centre of pressure must produce the equipoise. OF THE CENTRE OF PRESSURE. 415 Suppose m to be the centre of pres- sure, and make mb equal to twice me; then by equation (304), the point b must coincide with the sur- face of the fluid. Through the point b and parallel to B or cc?, draw the straight line rs, which marks the height to which the vessel must be filled with fluid, before the side is forced open. C Put b = au, the breadth of the loose side of the vessel, S zz en, the distance from the bottom at which the force is applied, yzz: the magnitude of the force applied at the point n, s zr the specific gravity of the fluid contained in the vessel, p HZ the pressure of the fluid against its side, and z = cb, the height to which the fluid rises. Then, by the principle indicated in equation (8), Problem III. Chapter II. we have and this takes place at the centre of pressure, which, according to equation (304), is situated at two thirds of the depth below the surface, and consequently, its effect to turn the side andc on the hinges e and /*, is, according to the principle of the lever, expressed by Now, the effect of the force applied at w, to prevent the side from being thrust open by the pressure of the fluid, is expressed by the magnitude of the given force, drawn into en the length of the lever on which it acts, and is precisely equal to the effect of the fluid acting at the centre of pressure ; hence we get /8 =#*,; and by division this becomes Z - Z - -7 * bs from which, by extracting the cube root, we obtain bs 416 OF THE CENTRE OF PRESSURE. This is the general form of the equation, corresponding to a fluid oi any density whatever denoted by s; but when the fluid is water, of which the specific gravity is unity, the above equation becomes (306). The method of reducing this equation, may be very simply expressed in words as follows. RULE. Divide six times the momentum of the given force,* by the horizontal breadth of the side to which it is applied, and the cube root of the quotient will be the height to which the vessel must be filled. EXAMPLE. The horizontal breadth of one side of an oblong pris- matic vessel, is 30 inches ; now, supposing this side to be loose and moveable about a Hinge at the bottom ; how high must the vessel be filled with water, in order that the pressure of the water, and a force of 400 Ibs. applied at the distance of 12 inches from the bottom, may exactly balance each other ? Here we have given b = 30 inches ; 3 zr 12 inches ; and /zz 400 Ibs ; consequently, we. obtain z=f/ 6x4QQ Xl2 30 9.865 inches. But the centre of pressure, at which the weight of the fluid is sup- posed to be applied in opposition to the given force, is situated at one third of the above distance from the bottom of the vessel ; hence we have cm zzi of 9.865 3.288 inches. Therefore, the pressure of the fluid acting on a lever of 3.288 inches, must be equal to a force of 400 Ibs. acting on a lever of 12 inches ; that is 400 X 12 = 30 X 9.865 s -:- 6. * The momentum of the given force, is equivalent to its magnitude, drawn into the distance above the bottom of the point at which it is applied. OF THE CENTRE OF PRESSURE. 417 PROBLEM LXVII. 518. A vessel in the form of a tetrahedron is entirely filled with water, and has one of its planes bisected by a line drawn from the vertex to the middle of the opposite side ; now supposing one half of the bisected plane to be loose, and moveable about a hinge at its lower extremity : It is required to determine the magnitude of the force, the point of application, and the direction in which it acts with respect to the horizon, when the moveable half of the contain- ing plane is just retained in a state of quiescence. Let ACB be one side of the vessel, divided by the line CD into two parts, which are equal and similar to one another; and let the part BCD be moveable about the hinges at e and/. Suppose the centre of pressure of the loose part B c D to be at the point m, and through m draw the straight lines mb and mn, respectively paral- lel to CD and AB; in CD take any other point E, and through E draw EF perpendicular to CD, or parallel to DB, making the triangles CDB and CEF similar to one another. Put I =z AB, BC or AC, the side of the containing plane, or the edge of the tetrahedron, d r= CD, the length of its perpendicular, 5 =z the perpendicular depth of its centre of gravity below the vertex at c, p = the pressure of the water on the loose triangle CDB, ar^: CE, any variable distance, y zz EF, the corresponding co-ordinate, and (f> z= the angle which the direction of the retaining force makes with the horizon. Then it is manifest from the nature of the .problem, that in the case of an equilibrium, the magnitude of the retaining force must be equal to the whole pressure of the water upon the moveable triangle CDB ; VOL. i. 2 E 418 OF THE CENTRE OF PRESSURE. but by Problem XVII. Chapter V. equation (56), the whole pressure upon the side ACB is expressed by / 8 \/2 ' consequently, the pressure upon the triangle CDB, becomes This is manifest, for by Proposition I. Chapter I. the pressure upon the triangle ACB, is equal to its area drawn into the perpendicular depth of the centre of gravity, the specific gravity or density of the fluid being expressed by unity ; but by Problem XVII. Chapter V. the perpendicular depth of the centre of gravity of the side of a tetrahedron below the vertex, is and according to the writers on mensuration, the area of an equilateral triangle, is expressed by one fourth of the square of the side, drawn into the square root of 3 ; therefore, we have where a denotes the area of the triangle ACB; consequently, by multiplication we obtain from which, by division, we get p-=L -rW's. (307). 519. This equation satisfies the first demand of the problem, and the second manifestly requires the determination of the centre of pressure ; for by the definition, the point of application of a force, equal in intensity to the pressure of the water, must occur at the centre of pressure of the plane CDB, on which that pressure is exerted. Now, by the principles of Plane Trigonometry, the length of the perpendicular c D is thus determined, rad. : I : : sin. 60 : CD, from which, by reduction, we get but sin.60=r %\/3, and consequently, by substitution, we get Therefore, since the triangles CDB and CEF are similar, by the property of similar triangles, we have JJ/3 : ' x : y, and by reduction, we get OF THE CENTRE OF PRESSURE. 419 Let this value of y be substituted in the first of the equations of condition (302), and we shall have, for the value of the distance en, as follows, viz. > ,. Jxyx fx*x the correction being equal to nothing ; and when a: zr c D or d, we have en = 1^*3. (308). 520. Again, for the horizontal co-ordinate nm, by substituting the above value of y in the second of the equations of condition, we obtain here again the correction is nothing, and in the limit when a? = CD or dj we have T 3 G L ( 309 )- 8\/3 521. It is shown by the writers on mensuration, that the planes com- posing a tetrahedron, are inclined to each other in an angle whose sine is equal to V 2 > an d by the principles of mechanics, the direction of the force applied at m, must be perpendicular to the plane ; itjs therefore inclined to the horizon at an angle whose cosine is $^/2 ; but by the principles of Trigonometry, we have sin.^> -v/ 1 cos 2 .0, or by substituting as above, we get sin.^ i= v/l f = $ = .33333, corresponding to the natural sine of 19 28' 15". Hence it appears, that at whatever point in the plane the retaining force may be applied, its direction will be inclined to the horizon, at an angle of 19 28' 15" ; the third demand of the problem is therefore satisfied, and we have seen that equation (307) fulfils the first, while the second requires the application of equations (308) and (309), and the method of reduction will become manifest from the resolution of the following example. 2E2 I 420 OF THE CENTRE OF PRESSURE. 522. EXAMPLE. A vessel, in the form of a tetrahedron has the length of its edges equal to 15 inches; now, supposing it to be filled with water, and placed upon its bottom with the axis vertical ; conceive one of its sides or containing planes to be bisected by a line drawn from the vertex to the middle of one of the bottom edges, and let one half of this plane be considered as loose, and moveable about a hinge at the bottom ; what must be the magnitude of a force that will just retain it in its place, and at what point must it be applied ? By equation (307), the magnitude of the required force is precisely equal to the pressure of the fluid upon the moveable plane ; therefore, by substituting the datum of the above example, we have p z= T VX 15 s X -/2 397.74 cubic inches of water ; which being reduced to Ibs. avoirdupois, becomes p = 397.74 X62.5 -4-1728 14.38 Ibs. 523. Again, for the point at which this force must be applied, in order to counteract the pressure of the water, we have by equation (308), cw~|7^/3" of 15 X -v/3 9| inches nearly. And in like manner, for the corresponding co-ordinate nm, we have by equation (309), nm ^l -?^ of 15 2.8125 inches, from which the position of the point m can easily be ascertained. 524. If the vessel should be placed with the bottom upwards and parallel to the horizon, then we shall have j9 1=7.19 Ibs. ; DW 6.495 inches, and nm~ 1.875 inches. A great variety of useful and interesting problems similar to the preceding, might be proposed in this place ; but as we have already overleaped the limits of this subject in the present volume, we must defer their consideration till another opportunity. CHAPTER XV. OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 525. THE subject of Capillary Attraction, and the Cohesion of Fluids, considered merely as a branch of philosophical inquiry, is exceedingly seductive and interesting ; but when viewed in the light of a demonstrative and practical department of physical science, its application is necessarily very circumscribed, and its character is unimportant as an analytical theory. It has, however, been very extensively studied, both in this and in foreign countries, and numerous philosophers of the greatest eminence, possessed of the loftiest conceptions and the most profound mathema- tical attainments, have deemed it a topic worthy of their most atten- tive consideration, and have ascribed to its influence, a numerous class of phenomena, in reference to the operations of nature on the various objects of our sublunary world. To this it is owing, that the rains which fall on the higher elevations, do not immediately descend and run to the sea with an increasing velocity, but are retained by the soil, and being slowly filtered down through it, are cleansed from their impurities, and delivered in springs and fountains at the foot of the hills, so as to afford a constant and nearly uniform supply of moisture to the lower levels. By capillary attraction, does the oil or melted tallow rise slowly through the wick of a lamp or candle, where it is converted into vapour by the heat of the surrounding flame, and rushing out in every direction, is ignited when it comes in contact with the circumambient air. By capillary attraction, the juices of the earth are absorbed by plants, and carried through their numerous ramifications to the remotest leaf, where they are again partly discharged by evaporation, after a similar manner to that in which the oil is dissipated from the wick of a lamp, or the melted tallow from the wick of a candle. 422 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. It is also by the principles of capillary attraction, that the lymph and other fluids are taken up, and transferred through the ramifying vessels to every part of the animal frame ; other causes dependent on the organical structure both of plants and animals, may assist in producing this effect ; but it is abundantly proved by observations, that by far the greatest part of it is produced by capillary attraction alone. It is solely owing to it, that a piece of dry wood absorbs a considerable quantity of moisture, and in consequence of this absorp- tion it swells with a force almost irresistible, thereby splitting rocks and other bodies of inconceivable hardness and tenacity. Consequently, since the principles of capillary attraction are found to exercise such extensive influence in the operations of nature, philo- sophers are justified in attempting to acquire a more precise and comprehensive knowledge of the manner in which it acts, and of the laws by which that action is regulated during the period of its opera- tion on natural bodies. 526. DEFINITION. Capillary Attraction is that principle in nature, by which water and other liquids are made to ascend in slender tubes, to heights considerably above the level of the fluid in the containing vessel ; it is so called, because its influence is only sensible in tubes whose bore is extremely small, in general very little exceeding the diameter of a hair, but never greater than one tenth of an inch. The tube thus limited, and in which the fluid is found to ascend, is called a capillary tube, from the Latin word capillus, a hair. The principles of capillary attraction, and the theory which it unfolds, together with its application to tubes of various forms and diameters, we shall very briefly consider in the present chapter ; the chief and most im- portant properties peculiar to this subject are detailed in the following experiments. 527. EXPERIMENT 1. There is an attraction of cohesion between the constituent particles of glass and water. This is manifest, for if a very smooth plate of glass be brought into contact with water, and then gently removed from it, it will be found that a small portion of the fluid adheres to the glass, and remains suspended from the lower surface when placed in a horizontal posi- tion ; hence the existence of an attraction is inferred, and its intensity must be such, as to balance and sustain the gravitating power of the water. And again, if a smooth plate of glass be suspended horizontally from one arm of a lever, and kept in equilibrio by a weight applied at the other arm ; then, if the glass be brought into contact with the OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 423 surface of the water, it will be found that an additional weight must be applied to the opposite arm to effect a separation ; and the magni- tude of this additional weight, is a precise measure of the force of cohesion. If the water and the glass be placed in a vacuum, and then brought into contact, the same effect will be found to obtain, and conse- quently, the. cohesion is not produced by the pressure of the atmos- phere ; hence an attraction must exist between the particles of the water and the glass. 528. EXPERIMENT 2. The constituent particles of a mass of fluid are mutually attracted ; that is, they have an attraction towards each other. According to the preceding principle, when a smooth plate of glass is brought into contact with the water and gently withdrawn from it, a thin stratum of fluid adheres to its lower surface ; now, if this stratum of fluid be carefully weighed, it will be found that its weight is much less than that which is required to detach it ; consequently, an attraction necessarily exists, which would keep the stratum united to the fluid in the vessel independently of its weight, and hence it is inferred, that the particles are mutually attracted ; that is, they have an attraction towards each other. 529. EXPERIMENT 3. The constituent particles of a mass of mer- cury have an intense mutual attraction ; that is, they are strongly attracted towards each other. This becomes manifest from the circumstance of the smallest quan- tity constantly assuming a globular form, and from the resistance which it opposes to the separation of its parts. Another circumstance which proves the attractive principle in the particles of mercury, is, that if a quantity of it be separated into a great number of parts, they will all of them be spherical ; and if any two of them be brought into contact, they will instantly unite, and constitute a single drop of the same form which they separately assumed. 530. EXPERIMENT 4. The attractive power which is evolved between the particles of glass and water, is sensible only at insensible dis- tances ; that is, 'the attraction between the particles is imperceptible, unless the distance between them be very small. This is inferred from the following circumstance, viz. whatever may be the thickness of the plate of glass which is brought in contact with the water, the force required to detach it is always the same. This indicates that any new laminte of matter that may be added to the 424 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. plate, have no influence whatever upon the fluid with which it is brought in contact ; whence it follows, that the indefinitely thin lamina of fluid which attaches to the surface of the plate, interposes between it and the rest of the fluid in the vessel, a sufficient distance to prevent any sensible effect from their mutual attraction ; and furthermore, it appears that the force which is requisite to detach all the equal laminae of the fluid is the same, being that which is required to separate an individual film of the fluid from the rest. Again, it is manifest from observation, that water of the same tem- perature, rises to the same height in capillary tubes of the same bore, whatever may be the thickness of the glass of which they are con- stituted; from this we infer, that the laminse of the glass tube, however small their distance from the interior surface, have no influence in promoting the ascent of the fluid. If the inner surface of a capillary tube be covered with a very thin coating of tallow, or some other unctuous substance, the water will not ascend, for in that case the capillary attraction is destroyed ; hence we conclude, that the action of gravity and capillary action are different in their nature, for if they were similar, the capillary force, like the force of gravity, would act through media of all kinds, and consequently, would cause the fluid to rise in the tube, notwithstand- ing its inner surface being coated with grease. 531 . From the preceding experiments, and others of a kindred charac- ter, it is inferred, that the force of attraction in a capillary tube, when it exceeds the mutual attraction of the fluid particles, extends its influence no farther than to the fluid immediately in contact with it, which it raises; and the water thus raised, by forming an interior tube, in virtue of its own attraction, raises that which is immediately in contact with itself, and this again, by extending its influence to the lower particles, continues the operation to the axis of the tube. The direction of the first elements of the fluid, depends entirely upon the respective natures of the fluid, and the solid with which it comes in contact ; if these are the same in all cases, the direction is invariable, whatever may be the form of the attracting surface, whether it be fashioned into a tube, or retains the simple form of a plane ; but the direction of the other elements, or those which are situated out of the sphere of sensible activity of the attracting surface, depends solely on the mutual effect of the fluid particles, and the form which the surface of the fluid assumes, is also regulated by the same cause. 532. From numerous experiments and careful micrometrical observa- tions^ has been ascertained, that when water moves freely in a capillary OF CAPILLARY ATTRACTION AND THE COHESION OP FLUIDS. 425 tube, the surface forms itself into a hemisphere, with its vertex down- wards and its base horizontal, in which position it nearly touches the interior surface of the tube ; but when the fluid rises between two planes, the surface assumes a circular form, having for tangents the planes by which it is attracted. These experiments and particulars being premised, we shall now proceed to develope the theory of calculation ; and in order that it may be invested with all the interest of which it is susceptible, we deem it advisable to adopt the method of M. le Comte La Place, one of the most profound and sagacious philosophers that have existed in this, or in any preceding age. PROBLEM LXVIII. 533. A cylindrical tube of glass, whose diameter is exceedingly small, has its lower extremity immersed in a vessel of water, and its axis vertical : It is required to determine with what force the water rises in the tube, by means of the attractive influence of its surface. Let abed represent a vertical section of a vessel, filled with water to the height BC, and let AB be the corresponding section of a small cylin- drical tube immersed in it at the lower extremity, and having its axis perpen- dicular to the surface. The fluid rises in the tube above its natural le^vel, a thin film being first raised by the attraction of the inner surface of the glass ; this first film of fluid raises a second, and the second a third, and so on, until the weight of the elevated fluid exactly balances all the forces by which it is actuated, viz. the attractive influence of the glass, and the mutual adherence of its own particles. Let us now suppose that the inner surface of the tube is produced to E, then carried horizontally to D and vertically to c, and let the sides of this extended tube be conceived to be so extremely thin, as to have no action whatever on the contained fluid, and not to prevent the reciprocal attraction which obtains between the real tube AB and the particles of the fluid ; that is, let the portion BEDC of the tube be so 426 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. circumstanced, as neither to attract nor repel the fluid particles, and consequently, the circumstances of the problem will not be at all affected by supposing the tube to assume the form represented in the diagram. Now, since the fluid in the tube A E is in equilibrio with that in the tube CD, it is manifest, that the excess of pressure in AE, arising from the superior height of the column, is destroyed by the vertical attrac- tion of the tube, together with the mutual attraction of the fluid particles in the tube AB ; in order therefore to analyze these different attractions, we shall first consider those that take place under the tube AB, in which the fluid rises above its natural level. 534. In the first place then, it is evident, that the fluid in the imaginary tube BE, is attracted, 1. By the reciprocal action of its own particles, 2. By the exterior fluid surrounding the tube BE, 3. By the vertical attraction of the fluid in AB, and 4. By the attraction of the glass in the tube AB. Now, the first and second of these attractions, are obviously destroyed by the equal and similar attractions experienced by the fluid in the opposite branch DC; consequently, their effects may be entirely disregarded. But the vertical attraction of the fluid in the tube AB, is also destroyed by an equal and opposite attraction exerted by the fluid in BE, so that these balanced effects may likewise be neglected, and there remains the attraction of the glass in AB, which operates to destroy the excess of pressure exerted by the elevated column BF. 535. Again, the fluid in the lower portion of the cylindrical tube AB, is attracted, 1. By the reciprocal action of its own particles, 2. By the fluid in the imaginary tube BE, 3. By the attraction of the glass in the tube by which it is contained. But since the reciprocal attractions of the particles of a body, do not communicate to it any motion if it is solid, we may, without altering the circumstances of the problem, imagine the fluid in AB to be frozen ; then, since the fluid in the lower part of AB, and that in the imaginary tube BE, are acted on by equal and opposite attractions, these attractions destroy each other, and consequently, their effects may be neglected ; hence, the only effective force which remains to actuate the fluid in AB, is the attraction of the glass containing it. Let this OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 427 force be denoted by /, which obtains equally in both the cases above stated ; therefore, if F denote the intensity of the vertical attractive force, we shall have * = 2f. 536. But there is a negative force acting- in the opposite direction, by which this value of F is influenced, and which arises from the attrac- tion exerted by the fluid surrounding the imaginary tube, on the lower particles in the column BE, and the result of this attraction is a vertical force acting downwards, in opposition to the force 2/"; let this antagonist force be denoted by/', and we shall obtain FZZ2/-/'. Put m zz the magnitude, or solid contents of the column B F, I zz the density or specific gravity of the fluid, and g zz the power of gravity. Then by multiplying these quantities together, the weight of the elevated column is expressed by m%g ; but in the case of an equili- brium between this weight and the attractive forces by which it is elevated, it is manifest that they are equal ; hence we have mlglff. (310). If the force 2/ be less than /', the value of m or the magnitude of the attracted column will be negative, and the fluid will sink in the tube ; but whenever the force 2/ exceeds /', the value of m will be positive, and the fluid will rise above its natural level. 537. Since the attractive forces, both of the glass and the fluid, are insensible at sensible distances, the surface of the tube A B will have a sensible effect on the column of fluid immediately in contact with it; this being the case, we may neglect the consideration of curvature, and conceive the inner surface to be developed upon a plane ; the force f will therefore be proportional to the width of this plane, or which is the same thing, to the inner circumference of the tube. Put d zz the inner diameter of the tube, TT zz the ratio of the circumference to the diameter, zz a constant quantity, representing the intensity of the attraction of the tube upon the fluid, and 9' zz another constant, representing the intensity of attraction which the fluid exerts upon itself. Then, by the principles of mensuration, we have tin equal to the inner circumference of the tube, and also to the exterior circum- ference of a column of fluid of the same diameter ; therefore, it is 428 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. f=i dirty, which being substituted for /and/' in equation (310), gives m$g = dir('2 f). (311). This is the general equation that expresses the force by which the water is raised in a cylindrical tube, and its application to particular cases will be exemplified by the resolution of the following problems. PROBLEM LXIX. 538. In a cylindrical capillary tube of a given diameter, the top of the elevated column is terminated by a hemisphere : It is therefore required to determine the height to which the 1 fluid ascends above its natural level. Let abed be a section passing along the axis of a very small cylindrical tube, of whioh the diameter is ab; let the tube be vertically immersed in the fluid whose surface is IK, and suppose that in consequence of the immersion, the fluid rises in the tube to e on a level with the surface IK, and from thence it is at- tracted by the glass in the tube, together with the mutual action of its own particles, until it arrives at ab, where it forms the spherical meniscus abfg, and in which position, the weight of the elevated column is in equilibrio with the attractive forces. Now, the problem demands the height to which the fluid rises in the tube in consequence of the attraction, and on the supposition that its diameter is very small. Put r = am, the radius of the interior surface of the tube, h =n en, the height of the uniform column, or the distance between the surface of the fluid and the lowest point of the spherical meniscus, A'n: ev, the mean altitude, or the height at which the fluid would stand, if the meniscus were to fall down and form a level surface, TT the ratio of the circumference to the diameter, and m i= the magnitude of the whole elevated column. OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 429 Then, by the principles of mensuration, it is manifest that the inner circumference of the tube is 2r7r, and the solidity of the uniform column whose height is en, becomes r*/nr; now, the solidity of the meniscus ganbf, is obviously equal to the difference between the cylinder abfg and its inscribed hemisphere an b. But by the rules for the mensuration of solids, we know that tl>e solidity of the cylinder abfg is r 8 7r, and that of the inscribed hemi- sphere is fr 3 7r ; consequently, the solidity of the meniscus is r s 7r fr 3 7r = r 8 7r, which being added to the solidity of the uniform column, gives wzrr 2 /i7r -f- T^TJ from which, by collecting the terms, we get Now this is equivalent to the solidity of a cylinder, whose radius is r and altitude evrz h' ; consequently, we have m = r*K (h + r) zz rV h' ; whence it appears, that h' = h + ir. (312). 539. Instead of din the equation (3 11), let its equal 2r be substituted, and instead of m in the same equation, let its equivalent h'r^ir be introduced, and we shall obtain h' r 9 TT $g = 2r TT (20 0'), and from this, by casting out the common factors, we get A' r 30 = 2(20 A and dividing by 5#, it becomes Now, since the symbols , 0', 3 and g are constant for the same fluid and material, it follows that the whole expression is constant ; hence, the height to which the fluid rises, varies inversely as the radius of the tube. 540. Instead of h' in the equation (313), let its equivalent (h -f- r) in equation (312) be substituted, and we shall obtain " ' 2(2<6 0') Hence it is manifest, that the constant quantity -'- , is equal to the mean altitude of the fluid multiplied by the radius of the tube ; and it has been shown in equation (312), that the mean altitude 430 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. is equal to the observed altitude of the lowest point of the meniscus, increased by one third of the radius of the tube, or which is the same thing, by one sixth of the diameter ; the value of the constant quantity, can therefore only be determined by experiment, and accordingly we find, that various accurate observations have been made for the purpose of assigning the value of this element; the mean of which, according to M. Weitbrecht, gives hence, finally, we obtain #r=.0214. (315). 541 . The equation (315), it may be remarked, is general for cylindrical tubes, if the elevated column of fluid is terminated by a hemispherical meniscus, and the practical rule which it supplies, is simply as follows. RULE. Divide the constant fraction .0214 by the radius of the capillary tube, and the quotient will express the mean altitude to which the Jluid rises above its natural level. If it be required to determine the highest point to which the fluid particles ascend, it will be discovered, by adding to the mean altitude two thirds of the radius of the tube, or one sixth of the diameter. 542. EXAMPLE. The diameter of a cylindrical tube of glass, is .06 of an English inch ; now, supposing it to be placed in a vertical position, with its lower extremity immersed in a vessel of water ; what is the mean altitude to which the fluid will ascend, and what is the altitude of the highest particles ? Since, according to the question, the diameter of the tube is .06 of an inch, the radius is .03 or half the diameter ; consequently, by the rule, the mean altitude to which the water rises, is h' =.. 0214 -~ . 03 0.713 of an inch, and therefore, the point of highest ascent, is 0.713 4- .02 =z 0.733 of an inch. 543. If the mean altitude of the fluid is given, the radius of the tube can easily be found from equation (315), for it only requires the con- stant number .0214 to be divided by the given altitude ; but when the observed altitude, or the distance between the surface of the fluid in the vessel, and the lowest point of the meniscus is given, the radius can only be determined by the resolution of an adfected quadratic equation ; for by equation (314), we have i**4- Ar-zz.0214, OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 431 which being multiplied by 3, becomes 7- 2 -f3r=.0642. (316). 544. Suppose now, that the observed altitude is 0.703 of an inch; then, .by substituting 0.703 instead of h in equation (315), we obtain consequently, by completing the square, we get r+ 2.11r -4- 1.055 2 1.177225, from which, by extracting the square root, we obtain r + 1. 055 = 1.085; hence, by subtraction, we have r 1.085 1.055 = . 03 of an inch. Now, if one third of the radius just found, be added to the observed altitude, the sum thence arising will express the mean altitude ; and if the whole radius be added to the observed altitude, the sum will express the greatest height to which the fluid rises in the tube. PROBLEM LXX. 545. Two parallel planes of glass or other materials, are placed in a vertical position, with the lower sides immersed in a fluid : It is required to determine how high the fluid rises between them, their distance asunder being very small in comparison to their surfaces. Let ad and be represent the ends or sections of two plates of glass, placed in a position of vertical parallelism, and having their lower edges d and c im- mersed in a fluid of which the surface is IK. Suppose now that a b or cd, the distance between the plates, is very small in com- parison to their extent of surface ; then it is obvious, that the fluid will rise between them as high as e in consequence of the immersion, and from thence it moves upwards by the attractive influence of the glass, and the mutual action of its own particles, until it arrives at rs, where it forms a semi-cylindrical meniscus rsfg, whose diameter is equal to the dis- tance between the planes, and its length the same as their horizontal breadth. 432 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. In this position, the whole weight of the elevated fluid, and the united efforts of the attractive forces, are in equilibrio among them- selves, and the problem requires the height to which the fluid rises, when the powers of gravitation and attraction become equal to one another ; for this purpose, Put b zr the horizontal breadth of the planes by whose attraction the fluid is elevated, d =. ab, the perpendicular distance between the planes, li zz e n, the distance between the lowest point of the meniscus and the surface of the fluid, h' zz ev, the mean altitude of the fluid, or the height at which it would stand, if the meniscus were to fall down and form a level surface, TT zz the ratio of the circumference of a circle to its diameter, and m zz the magnitude of the volume of fluid raised. Then, if the constants <, 0', S and g denote as before, the magnitude of the elevated volume will be found as follows. 546. By the principles of mensuration, the solidity of the fluid parallelopipedon, whose breadth is b, thickness d, and height h, is expressed by bdh; and the solidity of the fluid meniscus whose section is grnsf, is equal to the difference between a semi-cylinder and its circumscribing parallelopipedon, the length being equal to bj and the diameter equal to d, the distance between the attracting planes. Now, the solidity of the circumscribing parallelopipedon is |6d 8 , and the solidity of the semi-cylinder is J6eP?r; consequently, the solidity of the meniscus, is to which if we add the solidity of the uniform solid, the whole magni- uid becomes m bdh -\- \bd\1 TT). tude of the elevated fluid becomes But the periphery of the fluid which is elevated between the planes, is manifestly equal to 2( + d) ; consequently, by substituting this value of the periphery for dir in equation (311); and for m, let its value as determined above be substituted, and we shall obtain {bdh+lbd\1-Tr}}Zg=i1(b+d)( f), and dividing both sides by b$g, it becomes \ vr OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 433 but since d is conceived to be very small in comparison with b, the horizontal breadth of the plates, the fraction may be considered as evanescent, and then we get The solidity of the fluid parallelopipedon corresponding to the mean altitude, is expressed by bdh'i but this is equal to the whole quantity of fluid raised ; therefore we have bdh'=bd{h + Jd(l J*0}, from which, by casting out the common terms, we get A'= A +Jrf(l TT); (318). Now, (1 |TT) is a constant quantity ; hence it appears, that the mean altitude varies inversely as the distance between the planes. 547. Let the symbol for the mean altitude, be substituted in equation (317), instead of its analytical value as expressed in equation (318), and we shall obtain where the value of the constant quantity is the same as before ; hence we have rfA'=.0214. (319). The practical rule which this equation supplies, may be expressed in words, in the following manner. RULE. Divide the constant number 0. 021 4 by the perpen- dicular distance between the planes, and the product will give the mean altitude to which the fluid rises. 548. EXAMPLE. The parallel distance between two very smooth plates of glass, is 0.06 of an inch ; now, supposing the lower edges of the plates to be immersed in a vessel of water ; what is the mean altitude to which the fluid ascends ? Here, by operating as the rule directs, we have h' = 0.0214 -f. 0.06 0.356 of an inch. In this example, the distance between the planes is the same as the diameter of the tube in the preceding case, but the mean altitude of the fluid is only one half of its former quantity ; hence it appears, that if the tube and the planes are of the same nature and substance, and the radius of the one the same as the distance between the VOL. i. 2 F 434 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUID9. other, the fluid will rise lo the same height in them both, if they are placed under the same or similar circumstances. 549. Having given the mean altitude to which the fluid rises, the distance between the plates can easily be ascertained ; for we have only to divide the constant number 0.0214 by the given altitude, and the quotient will give the distance sought ; but if the observed altitude, or the distance between the lowest point of the meniscus and the surface of the fluid in the vessel be given, the operation is more difficult, since it requires the reduction of an adfected quadratic equation. By recurring to equation (317), it appears, that but we have shown, equations (315 and 319), that the constant quantity L ~ , has, from the comparison of experiments, been *9 assumed zz 0.0214 ; hence it is d{ h + \d (1 |TT) = 0.0214 ; now, the value of the parenthetical quantity (1 JTT) is also known, being equal to 1 . 7854 . 2146 ; consequently, by substitution, we have 0.1073d 2 +hd= 0.0214, and dividing both sides by 0.1073, it becomes d 2 -f- 9. 32hd= 0.1994. Let us therefore suppose, that the observed altitude of the fluid, or the value of h is equal to 0.2913 parts of an inch, and on this supposition, the above equation will become rf*+ 2.71473^:= 0.1994, and this equation being reduced according to the rules for quadratics, we finally obtain d = 1.429 1.357 zz 0.072 of an inch. 550. The preceding theory has reference to the phenomena of capil- lary attraction, as they are displayed in cylindrical tubes and parallel plates of glass ; it would however, be no difficult matter to extend the inquiry to figures of other forms, and placed under various cir- cumstances ; but being aware that an extended inquiry would elicit no new principle, we have thought proper to omit it ; the property disclosed in the following problem, is however, of too curious and interesting a character to be passed over without notice, we shall therefore endeavour to draw up the solution in the most concise and intelligible manner which the nature of the subject will permit. OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 435 PROBLEM LXXI. 551. If two smooth plates of glass be inclined to each other at a very small angle, having their lower sides brought in contact with a fluid, to the surface of which the coincident edges are vertical : It is required to determine the nature of the curve which the fluid forms upon the plates, by rising up in virtue of the attraction. Let ABEF and CDEF be the smooth plates of glass, having their edges coinciding in the line EF, and their planes inclined to each other in the angle BED; and suppose the edges BE and DE to be coincident with the fluid, while EF the line in which the plates are brought toge- ther, is perpendicular to its surface, which is represented by the plane BED; then shall FTWOB and FW^D, be curves described by the particles of the fluid upon the surface of the plates. Take any two points t and r in the line DE, and in the plane CDEF, draw tn and rp perpendicular to DE, and meeting the curve FW^D in the points n and p ; the lines tn and rp are therefore parallel to EF the line of coincidence, and perpendicular to BED the surface of the fluid. Again, from the same points t and r, and in the plane BED co- incident with the fluid's surface, draw the straight lines ts and rq perpendicular to DE, and consequently, parallel to each other; then, from the points s and q, in which the lines ts and rq meet BE the lower edge of the plane ABEF, draw sm and qv respectively parallel to tn and rp, and meeting the curve FWOB in the points m and o; these lines are consequently parallel to EF and perpendicular to the plane BED. 552. Since by the supposition, the angle BED which measures the inclination of the planes, is very small, the fluid in each section may be conceived to be elevated by the attraction of parallel planes, and consequently, by an inference under the preceding problem, the altitude of the fluid at any two points, will vary inversely as the distances between the planes at those points ; therefore, we have 2r2 436 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. in : rp : : rq : ts ; but by the property of similar triangles, it is Er : E : : rq : ts; consequently, by the equality of ratios, we obtain Er : E : : tn : rp, and by equating the products of the extreme and mean terms, it is vrXrp = EtXtn. (320). Now, according to the principles of conic sections, we have it, that in the common or Apollonian hyperbola, if the abscissae be estimated from the centre along the asymptote, the corresponding ordinates are to one another inversely as the abscissae ; hence it is manifest, that the curve which the surface of the elevated fluid traces on the plates, is the curve of a hyperbola, whose properties are indicated by equation (320). 553. Such then is the theory of capillary attraction, in so far as it is necessary to pursue it; but we shall just remark in passing, that other fluids, such as alcohol, spirit of turpentine, oil of tartar, spirit of nitre, oil of olives, and the like, are elevated in the same manner as water, but to a less degree ; thereby showing that the affinity of glass to water, is greater than its affinity to any other liquid. Again, on the other hand, some fluids are depressed by the action of the capillary force, such as mercury, melted lead, and indeed all the metals in a state of fusion, are more or less depressed, according to their density or specific gravity ; but an inquiry into the quantity of depression in this place, would lead to nothing new or interesting, and as a subject of practical utility, it is altogether unimportant ; we therefore pass it" over, and hasten to lay before our readers a detail of the experiments performed by the celebrated M. Monge, on the approximation and recession of bodies floating near each other on the surface of a fluid. The following are a few of the principal experiments that have been made on this subject. 554. EXPERIMENT 1. If two light bodies, capable of being wetted with water, are placed one inch asunder on its surface, in a state of perfect quiescence, they will float at rest, and experience no motion but what is derived from the agitation of the air ; but if they are placed apart only a few lines, they will approach each other with an accelerated velocity. Also if the vessel be of glass, or such as is capable of being wetted with water, and if the floating body is placed within a few lines of the edge of the vessel, it will approach to the edge with an accelerated velocity. OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. 437 555. EXPERIMENT 2. If the two floating bodies are not capable of being wetted with the fluid, such as two balls of iron in a vessel of mercury, and if they are placed at the distance of a few lines, they will move towards each other with an accelerated velocity; and if the vessel is made of glass, in which the surface of the mercury is always convex, the bodies will move towards the side when they are placed within a few lines of it. 556. EXPERIMENT 3. If one of the bodies is susceptible of being wetted with water, and the other not, such as two globules of cork, one of which has been carbonized with the flame of a taper ; then, if we attempt, by means of a wire or any other small stylus, to make the bodies approach, they will fly or recede from each other as if they were mutually repelled; and if the vessel is of glass, having the carbonized ball of cork placed in it, it will be found impossible to bring the cork in contact with the sides of the vessel. In these experiments it is manifest, that the approximation and recession of the floating bodies, are not produced by any attraction or repulsion between them ; for if the bodies, instead of floating on the fluid, are suspended by slender threads, it will be observed that they have not the slightest tendency either to approach or recede, when they are brought extremely near to each other. From an attentive consideration of the phenomena exhibited in these experiments, we may deduce the following laws. 557. (1.) If two bodies, capable of being wetted by a fluid, are placed upon its surface and brought near to each other, they will approach as if they were mutually attracted. For if two plates of glass AB, CD are brought so near each other, that the point H, where the two curves of elevated fluid meet, is on a level with the rest of the mass, they will remain in a state of perfect equilibrium. If, however, they are brought nearer together, the water will rise between them to the point G ; the water thus raised, attracts the sides of the glass plates, and D 438 OF CAPILLARY ATTRACTION AND THE COHESION OF FLUIDS. causes them to approximate in a horizontal direction, the mass of fluid having always the same effect as a heavy chain attached to the plates. The same thing is true of two floating bodies, when they come within such a distance that the fluid is elevated between them ; for it is obvious that the bodies A and B, being placed at a capillary distance asunder, have the fluid elevated between them, and are therefore brought together by the attractive influence of the fluid upon the sides of the globules. 558. (2.) If two bodies are not susceptible of being wetted by the fluid, they will still approach each other when brought nearly into contact, as if they were mutually attracted. For if the two floating bodies A and B, are not capable of being wetted by the liquid, it will be depressed between them as at H, A. B below its natural level, when they are placed at a capillary distance ; hence it appears, that the two bodies are more pressed inwards by the fluid which surrounds them, than they are pressed outwards by the fluid between them, and in virtue of the difference between these pressures, they mutually approach each other. 559. (3.) If one of the two bodies is susceptible of being wetted by the fluid, and the other not, they will recede from each other as if they were mutually repelled. For if one of the bodies as A, is capable of being wetted, while the other as B is not, the fluid will rise round A and be depressed round B; hence, the depression round B will not be uniform, and therefore, the body B, being placed as it were upon an inclined plane, its equilibrium is destroyed, and it will move towards that side where the pressure is least. These laws, deduced from experiment by M. Monge, have been completely verified by the theory of capillary attraction as developed by La Place ; from his theory it follows, that whatever be the nature of the substances of which the floating bodies are made, the tendency of each of them to a coincidence, is equal to the weight of a prism of HYDROSTATIC PRESSURE EXEMPLIFIED IN SPRINGS, &C. 439 the fluid, whose height is the elevation of the fluid between the bodies, measured to the extreme points of contact of the interior fluid, and minus the elevation of the fluid on the exterior sides. The elevation, however, must be reckoned negative when it changes into a depression, as is the case with mercury and other metals in a state of fusion, as has been observed elsewhere. HYDROSTATIC PRESSURE EXEMPLIFIED IN SPRINGS AND ARTESIAN WELLS. 560. The atmosphere is the uninterrupted source of communication between the sea and the earth ; it is the capillary conductor of water from the ocean to the land. Water ascends in the form of vapour, and descends as dew or rain upon the earth, which however it pene- trates but a small depth, except by fissures and permeable strata, which conduct it to subterranean reservoirs, whence it again issues as in the discharge of springs; or when the earth is bored through, it rises as in wells. Some wells are fed by land springs springs of shallow depth ; others are fed from the percolation of water through strata that act as conduits, conveying a current of the fluid through their permeable texture from one high land to another. Hence it is, that in valleys and champaign districts, very deep wells are dug, in order to arrive at those great feeders, where the hydrostatic pressure sends the water up with amazing force. In some cases we can trace the source of springs ; and with the help of FATHER KIRCHER'S Mundus Subterraneus, a man of a fanciful wit might present the public with a very learned treatise on Natural Hydraulics and Artesian Wells.* 561. As regards rock springs, we know of none that surpass the sources of the Scamander, in Asia Minor, an account of which will be found in some notes accompanying Poems of the Rev. Mr. Carlyle, who saw the stream of the Menderi issuing from a cave surrounded with trees, and tumbling down the crags in a foaming cascade ; for there the cavern that " broods the flood divine," discharges its sacred stores by two large openings in the rock, which leads into the cavern. Upon entering the recess, two other openings, nearly answering to the out- ward ones, like arches in a cloister, present themselves to the sight; and through one of them, in a basin below, the traveller perceives the * From Artois (the ancient Artesium of Gaul), where perpetually flowing artifi- cial fountains are obtained, by boring a small hole through strata destitute of water, into lower strata loaded with subterraneous sheets of this important fluid, which ascends by pipes let down to conduct it to the surface. 440 HYDROSTATIC PRESSURE EXEMPLIFIED just emerged Scamander. The channel that conducts the stream into the basin is a cleft in the rock towards the right, only about four feet wide and nearly twenty in height; it winds inwards in a curve, and is soon lost in darkness ; at its bottom glides the current, which for a few moments seems to repose in the basin beside, and then, by another subterraneous channel, rushes to the mouth, from whence it issues to the day : it here bursts from the precipice, and forms a noble water- fall between forty and fifty feet high, broken, and furnished with every accompaniment that the admirer of picturesque beauty could require : its sides are fringed with pine and brushwood ; below, it is almost hidden from the view by immense fragments of rock that have fallen from the precipice ; and above it hang crags of from two to three hundred feet in height, that jut over the bases in large angular prominences. Such is the spring which flows through the sweet vale of Menderi in many a winding turn. Far above this is the summit of Ida the snowy head of Khasdag, the seat of the immortals from whence the bard of yore could view Mysia, the Propontis, the Hellespont, the JEgean sea, Lydia, Bythynia, and Macedonia. 562. If sea water, which is nauseous to taste, and of perceptible smell, be the constituent condition of the fluid we call water, then rain water, which is without smell and taste, is salt water distilled by the atmosphere; and this is the common quality of rain, river, and spring water, except where accidental varieties of this last occur, distinguished by the physical qualities of taste, odour, colour, and temperature. 563. Two constructions in the physical constitution of the earth contribute to originate springs, which from the same circumstances never cease to flow : one is the adaptation of the atmosphere to transport water from the sea to high lands ; and the other is, the porous beds of sand, and stone, and clay, which exert a capillary influence in conveying the fluids they may be charged with from one elevation to another. Those beds or strata of sand and stone, resemble sponge, paper, or pipes, as conductors of fluids that are heavy and incompressible, as water; clay strata, which are impe- netrable by water, form the great reservoirs or basins in which the treasure of the skies lies hid. Dislocations in the general mass, resulting from fractures, intersect the strata and facilitate the dis- charge from the reservoirs formed by the clay stratum. 564. The water-bearing strata are at various depths, from 50 to 500 feet below the surface, and a sheet of impure, or mineral water, may be perforated till the operation conducts to a stratum containing pure IN SPRINGS AND ARTESIAN WEILS. 441 water; for the pipe let down into the lower stratum will not allow the impure water from above to mix with the pure ascending from below. Water from two different strata may thus be brought to the surface by one borehole of a sufficient size to contain a double pipe, viz. a smaller pipe included within a larger one, with an interval between them for the passage of the water. The smaller pipe may thus discharge the water of the lower, and the larger pipe that of the upper stratum ; for in either case the fluid is but endeavouring to regain the level at its feeding source on the surface of the earth. Fountains of this sort Artesian wells are very well known on the eastern coast of Lincolnshire by the name of blow wells. This district is low, covered by clay between the wolds of chalk near Louth and the sea shore ; and by boring through the clay to the subjacent chalk, a spring is found that yields a perpetual jet, rising several feet above the surface. But wells of this kind are common in many parts of the world ; in the neighbourhood of London ; in Artois, Perpignan, Tours, Roussillon, and Alsace, in France ; in some parts of Germany ; in the duchy of Modena ; in Holland, China, and North America. 565. But whence come those vast issues of fresh water that sometimes rise up in the sea, as in the Mediterranean near Genoa, and in the Persian Gulf, where the ascending volume is so vast as to allow mariners in the one case, and divers in the other, to water ships ? Springs such as these are the issues of subterranean rivers, all of which consist of meteoric water, or that which the atmosphere had transferred to itself from the ocean, distilled and discharged upon the undulating surface of the earth. 566. The annual fall of rain between the tropics is about ten feet in depth ; and estimating this in other countries as nearly propor- tional to the cosine of the latitude, the quantity of moisture ex- haled in a year, over the surface of the globe of our earth, would form a sheet of water five feet deep ; therefore the number of cubic feet of water turned into vapour, and dispersed through the mass of the atmosphere every minute, would be 5x10,424,000,000, or fifty- two thousand one hundred and twenty millions. But this enormous mass Leslie further multiplies by 18,000, the mean height* of the atmosphere in feet, and again by 62 J, the weight in pounds avoir- * In taking 18,000 feet as the mean height of the atmosphere, we have followed Leslie ; but the mean height is 27,800 feet in round numbers, for air is to water as 1 to 1000; therefore we have 1 : 1000 : : 34 : 27,818 feet for the height of the cloud sustaining atmosphere j that is to say, there are no clouds carried higher than five miles. 442 HYDROSTATIC PRESSURE EXEMPLIFIED IN SPRINGS, &C. dupoise of a cubic, foot of water ; and the final measure of effect he therefore takes leave to express by 58,635,000,000 million Ibs. and equal to the labour of 80,000,000 millions of men. Now the whole population of the globe being reckoned at 800 millions, of which only half, or less than that, is incapable of labour, it follows, that the power exerted by Nature, in the mere formation of clouds, to produce rain and make rivers and springs, exceeds by two hundred thousand times the whole accumulated toil of mortals, who, if all employed in carrying the water of the ocean to the mountain tops, for streams, and watering the fields, meadows, and woods, could not rival Nature in her simple process of evaporation, absorption, and distribution. 567. Such is the enormous power exerted in the great laboratory of Nature above the earth. Let us now contemplate her exertions beneath its crust, in the grand hydraulic apparatus of permeable strata the casual introduction of faults and dislocations in imper- vious strata, causing natural vents of water the interposition of syphons, cavities, thermal springs, mineral waters all resulting from the sea co-operating with the atmosphere to irrigate, to fertilize, to bless the habitable earth. 568. The surface of our own island contains 67,243 square miles, which are watered annually by a pool of water about 36 inches deep, of which, if one-sixth flow to the sea, there is still 2| feet depth left to fertilize the land, to feed the permeable strata, and afford to each individual the most abundant supply of this inestimable blessing. 569. If a vertical section of Hertfordshire, Middlesex, Kent, and Surrey, be taken, we shall have a pretty fair type of the sources of Artesian or any other wells. Below the London clay we have plastic clay, then chalk, then fire-stone, then gault clay, and below that woborn sand. It is sufficient to bore through the tenderest plastic clay into the chalk, to obtain the finest fresh water in the world. Kent and Surrey abound in chalk, which dips deeply below the plastic clay stratum, and makes its appearance at St. Alban's and Dunstable. The woborn sand met with at Sevenoaks sinks below the fire-stone and gault clay, and re-appears at Leighton Buzzard. Any one may for himself sketch a perpendicular section of these districts, and a few perpendiculars let fall through the London clay, to penetrate into the chalk, by passing entirely through the plastic clay, will exhibit the exact position of the borer in searching for water ; or the reader will find it done to his hand in Dr. Buckland's Geology. CHAPTER XVI. MISCELLANEOUS HYDROSTATIC QUESTIONS, WITH THEIR SOLUTIONS. 570. QUESTION 1. How deep will a cube of oak sink in fresh water, each side of the cube being 15 inches, and its specific gravity 0.925, that of the water in which it is immersed being expressed by unity ? The solution of this question is extremely simple, for by art. 311, page 257, it is announced as an established hydrostatical principle, that the magnitude of the whole body is to the magnitude of the immersed part, as the specific gravity of the fluid is to the specific gravity of the solid. But since the base of the whole solid and that of the immersed portion are the same, it follows from the principles of mensuration, that the magnitudes are as the altitudes, and conse- quently, the altitudes are as the specific gravities ; hence we have 1 : 0.925 : : 15 : 13.875 inches, the depth required. 571. QUESTION 2. If a cube of wood floating in fresh water, have three inches of it dry, or standing above the surface of the fluid, and 3J-I-3 inches dry when in sea water; it is required to determine the magnitude of the cube, and what sort of wood it is made of? This question may be resolved on the same principles as the last ; for if we put a; ~ the side of the cube in inches, and s the specific gravity of the wood ; then, by art. 311, page 257, we have 1000 : s : : x : TT the part immersed in fresh water, and 1026 : s : : x : SX the part immersed in sea water; but the part immersed and the part extant, together make up the whole altitude or side of the cube ; hence we have 444 MISCELLANEOUS HYDROSTATIC QUESTIONS, WITH THEIR SOLUTIONS. -f- 3 :rz x, in the case of fresh water, 1000 O -V* and - + 3fH = * in the case of sea therefore, if one of these equations be subtracted from the other, we shall have 533520 26xzz ' , or arm 40 inches, the side of the cube required ; 51 o hence, the altitude of the immersed part, as referred to fresh water, is 40 3 = 37 inches ; and the altitude as referred to sea water, is 36 yVy inches ; and from either of these, the specific gravity of the wood is found by the proposition referred to above ; for we have 40 : 37 : : 1000 : siz:925; indicating the specific gravity of oak, when that of fresh water is expressed by 1000. 572. QUESTION 3. If a cube of wood floating in sea water be | below the plane of floatation, and it sinks VV of an inch deeper in fresh water ; what is its magnitude, and what is its specific gravity ? This question at first sight would appear to be the same as the last ; it may indeed be resolved by the same principles ; but since the im- mersed parts are given in this instance, instead of the extant parts, as was the case in the preceding question, this circumstance suggests a simpler and a better mode of solution ; for by the inference in art. 317, page 261, it appears that the parts immersed below the surface of the different fluids, are to each other inversely as the specific gravities of the fluids ; hence, if x denote the side of the cube in inches, then by the question, is the altitude of the part immersed below the sur- 3ar 3 3Qx -4- 12 face of sea water, and 4- TZ ~ IT; is the altitude of the 4 ' 10 40 part immersed below the surface of fresh water ; consequently, by the inference above cited, we obtain 1000 : 1026 : : : 30 * ."*" 12 ; 4 40 and from this, by equating the products of the extreme and mean terms, we get 78x1= 1200, or arzr 15 T S T inches, the side of the cube required. Having thus determined the side of the cube, the specific gravity of the material will be found as in the last question, for we have 15 T T : IX 15 T T : : 1026 : 769, the specific gravity sought. VHlVERSiTY MISCELLANEOUS HYDROSTATIC QUESTIONS, WITH THEIR SOLUTIONS. 445 Mr. Dalby makes the side of the cube equal to 13| inches, and the specific gravity 772 ; but this only shows that he has employed a higher number for the specific gravity of sea water; 1030 brings out his results. 573. QUESTION 4. How deep will a globe of oak sink in fresh water, the diameter being 12 inches and the specific gravity 925, that of water being 1000? By the rules for the mensuration of solids, the solidity of the globe is expressed by the cube of its diameter multiplied by the decimal .5236 ; consequently, we have 1728 X .5236 = 904.7808 cubic inches for the solidity of the globe; therefore, according to art. 311, page 257, we get 1000 : 925 : : 904.7808 : 836.923 cubic inches, the solidity of the immersed segment. Now, according to the principles of mensuration, as applied to the segment of a sphere, if x be put to denote the height of the segment, then its solidity is expressed by .5236 (36ar 2z s ), and this must be equal to the solidity of the segment found by the above analogy ; hence we get 1 So; 2 x 3 = 799.2. In order to reduce this equation, let the signs of all the terms be changed, and put a;zr z -f- 6 ; then, by substitution, we have x a = z s -f 18z 2 + 108z + 216, and 18**=:* 18s 2 216z 648; hence, by summation, we obtain 3 8 108zr= 367.2, and from this equation, the value of z is found to be 3.9867 very nearly ; but by the supposition, x z -f- 6, and consequently, it is x =. 3.9867 4- 6 = 9.9867 inches very nearly, for the height of the segment, or the depth to which a globe of oak descends in fresh water, the diameter being 12 inches, and the specific gravity 925. This result agrees with that obtained by Dr. Hutton, in the second volume of his Course of Mathematics. 574. QUESTION 5. If a sphere of wood 9 inches in diameter, sinks by means of its own gravity, to the depth of 6 inches in fresh water ; what is its weight, and also its specific gravity ? By the corollary to the third proposition, art. 233, page 212 214, it is manifest, that the weight of the body is the same as the weight of the fluid displaced by its immersion ; that is, the weight of the entire sphere, is equal to the weight of as much fluid as is represented by the solidity of the immersed segment ; but by the principles of 446 MISCELLANEOUS HYDROSTATIC QUESTIONS, WITH THEIR SOLUTIONS. mensuration, the solidity of the segment is (9x3 12 X 2) x 36 x .5236 zn 282.744 cubic inches ; consequently, the whole weight of the body, is 282.744X0.03617 10.226 Ibs., the decimal fraction 0.3617 being the number of Ibs. in a cubic inch of fresh water. (See note to art. 329, page 268.) Having thus determined the weight of the globe, the specific gravity of the material may be found in various ways ; but we shall here determine it by the principle of Proposition VII. art. 311, page 257; from which we have the following process, viz. 9 8 : (27 12)X36 : : 1000 : 740|f, the specific gravity sought. 575. QUESTION 6. An irregular piece of lead ore, weighs in air 12 ounces, but in water only 7 ; and another piece of the same material, weighs in air 14| ounces, but in water only 9 : it is required to compare their densities or specific gravities ? This question may be very simply resolved, by the principle stated in Proposition V. art. 264, page 229 ; which is the same as the principle employed by Dr. Hutton for the same purpose; from it we have 12 7 : 12 : : 1000 : 2400, the specific gravity of the lightest fragment ; and again, we have 14.5 9 : 14.5 : : 1000 : 2636.36, the specific gravity of the heavier piece. The specific gravities are therefore to one another, as the numbers 2400 and 2636.36. Dr. Hutton makes the ratio as 145 to 132. (See question 52, page 298, vol. ii. 10th ed. Course, 1831 ;) his formulae will be found in arts. 250 or 251. The above solution however, is not correct, for the weight of the body in air is not its real weight, as it would be exhibited in vacuo ; the correct specific gravity will therefore be obtained by equation (186), art. 270, page 234; and the operation is as follows, the specific gravity of air being 1|-, that of water being 1000. of the lighter fragment ; and for the specific gravity of the heavier, we have 14.5X1000-9X1* . d 14.5 _ 9 34 - J6 - By the results of our solution, it appears that the heavier fragment is also the densest; by Dr. Hutton's solution, exactly the reverse is the case. MISCELLANEOUS HYDROSTATIC QUESTIONS, WITH THEIR SOLUTIONS. 447 576. QUESTION 7. An irregular fragment of glass, weighs in air 171 grains, but in water it weighs only 120 grains; what is its real weight ; that is, what would it weigh in vacuo ? The answer to this question is obtained by equation (185), art. 267, page 232 ; and the operation as there indicated is simply as follows. 171X1000 120XH lry the real weight of the glass in vacuo. 577. QUESTION 8. A fragment of magnet weighs 102 grains in air, and in water it weighs only 79 grains ; what is its real weight, or what does it weigh in vacuo ? The solution of this question is effected exactly in the same manner as the preceding, the conditions from which the data are obtained being precisely the same; that is, the body is weighed in air and in water ; consequently, the operation is as under. 102X1000 79X1|- ' the real weight of the magnet in vacuo. From the real weights of these materials, as determined in the above examples, the absolute specific gravities can be found by the principle of Proposition V. page 229 ; for the weights lost, are to the whole weights, as the specific gravity of water, is to the specific gravities of the substances in question ; hence we have 120 : 171-gVVs- : : 1000 : 3350^, the specific gravity of the glass. 79 : 102 y V : : 100 : 443 Hf the specific gravity of the magnet. Dr. Hutton makes the specific gravities of the glass and the magnet, respectively equal to 3933 and 5202, and says that the ratio is very nearly as 10 to 13; our own numbers give the same ratio. 578. QUESTION 9. Taking the specific gravity of glass equal to 3350, suppose that a globe is found to weigh 10 Ibs. avoirdupoise ; what is its diameter ? The cubic inch of glass of the given specific gravity weighs 0.1211695 of alb.; therefore, according to the equation 187, page 235, we have / Z-T in 5.402 inches nearly. .5236X0.1211695 448 MISCELLANEOUS HYDROSTATIC QUESTIONS, WITH THEIIl SOLUTIONS. 579. QUESTION 10. Supposing the same piece of glass to weigh 9.996 Ibs. in air, but in water only 7.015 Ibs. ; what is its diameter? The specific gravity of water, when reduced to pounds per cubic inch, is 0.03617, and that of air is 0.000045 ; therefore, by equation (188), page 236, we have 9.996 7.015 . 5.402 inches, the same as before. .5236 (0.03617 0.000045) 580. QUESTION 1 1 . The lock of a canal is 40 feet wide, and the lock gates being rectangular planes, stand 16 feet above the sill, with their upper edges on a level with the surface of the water ; now, supposing that the gates are found to meet each other in an angle of 141 35'; what is the amount of pressure which they sustain, a cubic foot of water weighing 62 \ Ibs. avoirdu poise ? Here the lock gates meet each other in an angle of 141 35' ; which, according to Barlow, is the situation in which, with a given section of timber, they obtain the greatest strength. But by the principles of Plane Trigonometry, the length of each gate is 20Xsec.l9 25'=21.206 feet; and by the question, the depth is 16 feet ; therefore, the whole surface exposed to the pressure of the water, is 2 1. 206 X 32 = 678. 592 square feet. Now the centre of gravity of each gate is 8 feet below the surface of the water, the specific gravity of which is unity ; conse- quently, by equation (8), page 19, the entire pressure upon the gates, is p=: 8X678.592 5428.736 cubic feet of water; or when reduced to Ibs. it is 5428.736X62J z= 339296 Ibs., or 151 tons 9 cwt. qrs. 48 Ibs. 581. QUESTION 12. If the diameter of a cylindrical vessel be 20 inches; required its depth, so that when filled with a fluid, the pres- sure on the bottom and sides may be equal to each other ? This question is resolved by the equations (59 and 60), page 100, where it is manifest, from the construction of the equations, that Bzz: \d, and consequently, by substitution, we obtain .7854D 2 dzz 3.1416Ddx Jd, and this expression is equivalent to .7854D 2 d 1.5708D particles at E, will, in consequence of their gravity, continually descend towards the lower parts at F. Again, the greater pressure which obtains among the particles under E, and the lesser under F, will obviously cause the particles at E to descend, and those at F to ascend ; and thus the higher parts of the fluid at E, descending and spreading themselves over the lower parts at F, which at the same time are ascending ; it is obvious, that the surface will at last be reduced to the horizontal position A B ; and having attained that position, it must continually remain in it, for then there is no part higher than another, and consequently, there is no tendency to descend in one part more than in another, and therefore the fluid must rest in a horizontal position. Art. 7. If two fluids that do not mix, are poured into the same vessel, and suf- fered to subside, their common surface is parallel to the horizon. Let A B D c be the vessel containing the two fluids which do not mix, and let E F denote the common surface, or that in which the fluids come in contact. A a L ft The upper surface A B of the lighter fluid is horizontal by art. 6; therefore, let P and Q be two contiguous particles of the heavier ] fluid, equally distant from a horizontal plane, and consequently, equally distant from A B ; if they are not also equally distant from E F the common surface, the vertical pressures upon them will be unequal, for this pressure is made up of the weights of two columns, containing different quantities of fluid matter, viz. PC, qd of the heavier fluid, and ca, db of the lighter; consequently, the pressures in opposite directions will be unequal, and motion must take place, which is contrary to the supposition. The particles P and Q are therefore equally distant from E F the common surface of the fluids; and the same being true for every other two contiguous particles in the same horizontal plane, it follows, that E F must also be horizontal. Art. 8. The particles of fluid situated at the same perpendicular depth below the surface, are equally pressed. This is almost self-evident, but nevertheless it may be thus demonstrated ; for let the plane passing through E F, be parallel to the surface AB; then, since the height of the fluid is the same at all the points of E F, it is manifest that the weights of the fluid columns standing upon any equal parts of it, must also be equal, and consequently, the pressure on all the points of the plane passing through EF is the same, since they are all situated at equal depths below the surface AB. Art. 9. When a fluid is in a state of rest, the pressure upon any of its constituent elements, wheresoever situated, varies as the perpendicular depth of the particle or element pressed. The demonstration of this principle is evident from that to article 8; for the pressure depends upon the weight of the superincumbent column, and the weight of this column manifestly varies directly as its height ; hence, the pressure upon any particle varies as its perpendicular depth below the surface of the fluid. NOTES. 451 NOTE B. PROPOSITION I. CHAPTER I. In this proposition, and the several laws and consequences deduced from it, the effect of the atmospheric pressure is entirely disregarded. It may however be proper to remark, that in numerous delicate hydrostatical inquiries, the pressure thus excited must be taken into the account: it is equal to the pressure of a column of water 34 feet in perpendicular height, NOTE C. CHAPTER VI. Experiment 7. Page 160. Since these experiments were selected and inserted in this work, a living eel has been killed in the cylinder of the hydrostatic press, in which also an egg has been broken. But the eel would have been killed by suffocation, if no pressure had been applied to the fluid, and the fracture of the egg was due to the air it contained between the white and the shell, or to the different densities of the shell, the white, the yolk, and the water. Thus we can easily conceive, as Mr. Tredgold remarks, that the trial of an experiment may be the means of condemning a very useful principle, merely through inattention to the proportions and the mode of action. We may still affirm, that fishes will endure a very high degree of fluid pressure, provided they be allowed to breathe ; indeed it is recorded, that a whale in the arctic seas, being struck by a harpoon, descended perpendicularly by the line about 900 fathoms, before it returned to the surface to respire ; it was then under a pressure of nearly 164 atmospheres, or 2,460 Ibs. upon a square inch of its surface; now if the living animal could sustain this natural pressure without inconvenience, we are at liberty to conclude that it could sustain an equal degree of artificial pressure. It is manifest, that fishes which do not come to the surface, breathe the air with which the water is impregnated, at whatever depth they may be found. Moreover, if an eel were killed by pressure, we suppose it would be crushed, or burst asunder. In short, we require evidence of the death by pressure, to remove our belief in death by suffo- cation. Air, which is invisible, by squeezing the heat out of it by strong pressure, may be compressed into water ; but the contraction which water suffers at every increase of pressure, exceeds not the twenty thousandth part of what air would undergo in like circumstances ; and fishes are at their ease in a depth of water, where the pressure around will instantly break or burst inwards almost the strongest empty vessel that can be let down. We are perfectly aware of the experiments of Mr. Canton, in 1760, which established incontestably the compression of water. Indeed the theory of com- pression extends to all bodies : Dr. Young says that steel would be compressed into one-fourth, and stone into one-eighth of its bulk at the earth's centre ; but a density so extreme is not borne out by astronomical observation. And the late Sir John Leslie, who suggests the idea that the ocean may rest upon a subaqueous bed of compressed air,* says that water at the depth of 93 miles would be com- pressed into half its bulk at the surface of the earth ; and at the depth of 362.5 miles it would acquire the ordinary density of quicksilver.t Practical men, m reply to all this physical science, may justly reply, " We are seldom called upon * Article " Meteorology," in the Supplement to the Encyclopaedia Britannica. t See Leslie's Elements of Natural Philosophy, vol. i. 2G 2 452 NOTES. to execute undertakings much below the level of low water, and those investigations suit us best, which are confined to the depth of a few fathoms, where we know that water is, to all intents and purposes in our business, wholly unaltered by com- pression." NOTE D. CHAPTER X. The principle of fluid support, and the doctrine of specific gravity, which we are now considering, explain many curious facts that daily pass unobserved. Thus, a stone which two men on land can h ardly lift, may be borne along by one man in water ; and in diving, a dog will bring to the surface a human body, which the strongest of his species could not lift on land : hence the ease also with which a bucket is lifted from the bottom of a well to the surface of the water. And as the human body in an ordinary healthy state, with the chest full of air, is lighter than its equal bulk of water, a man naturally floats with about half the head extant ; " having," as Dr. Arnott says, " then no more tendency to sink than a log of fir." When a swimmer floats on his back, with merely his face above water, in which position he can breathe freely, he exhibits the true position of floatation, in which the human body is lighter than water, for its specific gravity is one ninth less than that of water, being about 0.891. In some cases however, the bodies of men are heavier: thus, a person who weighs 135 Ibs. would be 12 Ibs. heavier than two cubic feet of river water, and would require a float of cork equal to 4 Ibs. to keep him from sinking ; for 123 + 4# 135 -\-x, where x represents the weight of cork; consequently, 123 + 3a;=135; therefore, 3x= 135 123=12; whence x =4=. When a solid specifically heavier than a fluid, is immersed to a depth which is to its thickness, as the specific gravity of the solid to that of the fluid, and the pressure of the fluid from above is removed, the body will be sustained in the fluid ; for the pressure from above being removed, the body is in the same state with respect to the contrary pressure, as if the same weight filled the whole space to the surface of the fluid; which means, as if its specific gravity and that of the fluid icere equal. The principle here enunciated helps the philosophers in their explanation of the common experiment of making lead to swim, in consequence of being fitted to the bottom of a glass tube. In the case cited above, of solid bodies being lighter in water than in air that is to say, being more easily moved in the water than on dry land the meaning of the proposition is, that all bodies, when immersed in a fluid, lose the weight of an equal volume of that fluid. Thus, in raising a bucket of water from the bottom of a well, so long as the bucket is under the water, we do not perceive it to have any additional weight beyond the wood it is made of; but the moment we raise the bucket to the surface, and suspend it in air, then we feel the additional weight of the water, which if equal to 6 gallons, or to one cubic foot, will add nearly 62 pounds, or 1000 ounces avoirdupois weight to the bucket. Now all this weight existed in the bucket when under the surface of the water, being supported by an equal bulk, or 62J pounds. The weights thus gained or lost by immersing the same body in different fluids, are as the specific gravities of the fluids ; hence we affirm that all bodies of equal weight, but of different volume, lose in the same fluid, weights which are reciprocally as the specific gravities of the bodies, or directly as their volumes. In the salt sea it will be one thirty-fifth lighter than i n fresh water. NOTES. 453 NOTE E. CHAPTER XII. It will not, however, be out of place to remark, that the weight of the whole solid, and that of the portion immersed below the plane of floatation which corresponds to the magnitude of the fluid displaced are very appropriately represented by the areas drawn into the respective specific gravities of the solid and the fluid on which it floats. But the most cursory observation shows, that a solid may be immersed in a fluid in numberless different ways, so that the part immersed, shall be to the whole magnitude in the given proportion of the specific gravities, and yet the solid shall not rest permanently in any of these positions. The reason is obvious : the floating body is forced down by its own weight, and borne up by the pressure of the fluid ; it descends in the direction of a vertical line passing through its centre of gravity ; it is pushed up in the direction of a vertical line passing through the centre of gravity of the part immersed, or the displaced fluid. Unless therefore, these two lines are coincident, or that the two centres of gravity shall be in the same vertical line, it is evident that the solid thus impelled, must revolve on an axis until it finds a position in which the equilibrium of floating will be permanent. To ascertain therefore, the positions in which the solid floats permanently, we must have given the specific gravity of the floating body, in order to fix the proportion of the part immersed to the whole; and then, by geometrical or analytical methods, determine in what positions the solid can be placed on the surface of the fluid, so that the centre of gravity of the floating body, and that of the part immersed may be situated in the same vertical line, while a given proportion of the whole volume is immersed beneath the surface of the fluid. The incumbent weight may be considered as collected in the centre of gravity of the floating body, and the sustaining efforts as united in the centre of buoyancy, which, as we have already said, is the same as the centre of gravity of the water displaced, or of the immersed portion of the uniform solid. To these two points therefore, the antagonist forces are directed, and the line which joins them, called the line of support, will have constantly a vertical position in the case of equilibrium. The centre of gravity of the whole mass, about which it turns in the water, must evidently continue invariable ; * but the centre of buoyancy will change its relative place, according to the situation of the immersed portion of the solid. If these two centres should coincide, the body will float indifferently in any position of stability. It will therefore float, as often as a vertical line, drawn from the centre of buoyancy, shall pass through the centre of gravity. But this will obtain when- ever the line of support becomes perpendicular to the horizon. The equilibrium may, however, be either permanent or instable. It is permanent, if on pulling the body a little aside it has a tendency to redress itself, or to recover its original position ; it is instable, when the body, on being slightly inclined, tumbles over in the liquid and assumes a new situation. These opposite conditions will occur in a body of irregular form, when the centre of gravity occupies the highest or the lowest possible position, (when the centre of gravity is the lowest possible, the situation is that of maximum stability) for though the volume of immersion remains the same, the solid will evidently be less or more depressed in the fluid medium, according * This is not strictly true, but it causes no difference in the theory that it is otherwise. 454 NOTES. to the width of its section or water lines. We have a curious proof of this in the construction of the French ship of the line, of 74 guns, called Le Scipion, fitted for sea at Rochfort, in 1779; but she wanted stability, which, after various fruitless attempts, was achieved by applying a bandage or sheathing of light wood to the exterior sides of the vessel. This cushion, bandage, or sheathing, was from one foot to four inches in thickness, extending throughout the whole length of the water line, and ten feet beneath that line. We are left to infer Le Scipion then floated with permanent stability. If the centre of buoyancy stand higher than the centre of gravity, the floating body will, in every declination maintain its stability, and regain its perpendicular position ; for though made to lean towards either side, the vertical pressure exerted against that variable point will soon bring it back again into the line of support. But the elevation of the centre of buoyancy above that of gravity, is by no means an essential requisite to the stability of floatation; on the contrary, it falls in most cases considerably below the centre of gravity about which the body rolls. The buoyant efforts may be considered as acting upon any point in the vertical line, and consequently, as united in the point where the line crosses the axis of the floating body. If the point of concourse, thus assigned, should stand above the centre of gravity, the body will float firmly, and will right itself after any small derangement. If it coincide with the centre of gravity of the homogeneous body, this will continue indifferent with regard to position; but if the vertical line should meet the axis below the centre of gravity, the body will be pushed forwards, its declination always increasing till it finally oversets. Thus, a sphere of uniform consistence floated in water, will sink till the weight of the fluid displaced by the immersed portion shall be equal to its own load. The centre of gravity of this body is the centre of the sphere itself; but the centre of buoyaney must be the centre of gravity of the volume of immersion, which there- fore lies below the centre of gravity of the body, in an axis perpendicular to the water line, or line of floatation. The ball is hence pressed down by its own weight collected in its centre of gravity, and pushed up in the opposite direction by an equal force combined in the centre of buoyancy ; both of the forces, however, concurring in the centre of gravity of the immersed sphere. Wherefore, being always held in equilibrium by those antagonist forces, it will remain still in any position which it may happen to occupy. But this indifference to floating will obtain only when the sphere is perfectly homogeneous, and its centre of gravity coincides with the centre of magnitude, for otherwise, the former descending as low as possible, would always assume a determinate position. A cylinder will, according to its density, and the proportion of its diameter and altitude, exhibit the three features of a floating body, in indifference, instability, or permanence of equilibrium. For example, a cylinder, the specific gravity of which is to that of the fluid in which it floats, as 3 to 4, its axis being to the diameter of the base as 2 to 1, if placed on the fluid with its axis vertical^ will sink to a depth equal to a diameter and a half of the base ; and as long as the axis is sustained in a vertical position by external force,, the centre of gravity of the solid and the centre of the immersed part will be situated in the same vertical line ; but the solid will not float permanently in that position, for as soon as the external force is removed, it will overset and float with its axis horizontal. But a cylinder whose axis is one half, instead of twice the diameter of the base, being placed in a fluid with its axis vertical, will sink to the depth of three fourths of a diameter^ and will NOTES. 455 float' permanently in that position. Incline it as you may, on being left to itself it will ultimately settle permanently, with its axis perpendicular to the horizon. The differences of the phenomena in this case, arise from the change which takes place in the position of the line of support; and what is true of the cylinder is true also of other figures ; for when a solid changes its position, by revolving on an axis on the surface of a fluid, any position of equilibrium is always succeeded by a position of equilibrium which is of a contrary description. A segment of a sphere floating in water, will have its centre of gravity below the centre of the sphere, when the segment floats with its vertex downwards, arid in an axis at right angles to its base ; but the centre of buoyancy, or the centre of gravity of the immersed segment, must, in every situation of the floating mass, occur in a perpendicular bisecting the water line, and conse- quently passing through the centre of the sphere. In the case of equilibrium this perpendicular must have a vertical position, or the involved base of the segment must form a horizontal plane. If this body be now drawn aside, into a position which shall incline ,jts base in any angle with the water line, it will be pressed down by its own weight, collected in the centre of gravity of this body, and pushed upwards by an equal buoyant power exerted at the centre of buoyancy. This force may now be conceived to act upon any point in the line connecting the centres of gravity and buoyancy, and therefore at the concourse of the axis in the case of equi- librium, and of the vertical line when the body is drawn aside. The buoyancy trans- mitted to this point pushes the axis of inclination obliquely, the greater part of it bearing the point of concourse in the direction of the axis of permanent floatation, while another small part of this force, pressing perpendicular to the axis of inclina- tion, makes the body turn about its centre of gravity, from the higher or lower point of inclination of its upper surface, till it ultimately coincides with the water line. Every derangement is thus corrected by a restoring energy which maintains a permanent equilibrium. An oblate homogeneous spheroid will sink in a manner similar to the segment of the sphere, and carry the centre of buoyancy in a like position. The declination of its axis, by drawing the body aside from the position of permanent equilibrium, is restored to its vertical position by the effort of buoyancy exerted at a point above the centre of gravity of the spheroid, which tends to redress the floating body and secure its stable equilibrium. On the other hand, a prolate spheroid will have its centre of buoyancy and plane of floatation, each the same height as in a sphere described on the longer axis of the spheroid. But the shifting of its centre of buoyancy will be diminished in proportion to the narrowness of the spheroid. The vertical will meet the principal axis below the centre of gravity of the solid, and will push it aside more and more till the spheroid falls, and extends its longer diameter in a horizontal position. It may then roll indifferently upon that line, as the sphere turns about its diameter. A solid of any form, not abruptly irregular, set to float in water, will be divided into correspondent equal portions by its principal axis, which will cross the plane of floatation at right angles. If the body be inclined, its centre of buoyancy will shift its place as the inclination varies, until the antagonist forces meet in a point in the axis, where the effort of the body to redress itself remains unaltered, like the centre of gravity itself. That characteristic point standing always above the centre of gravity of the mass, and limiting its greatest elevation in the case of permanent stability, has been called the metacentre. If the floating body be a homogeneous parallelepiped placed vertically in the 456 NOTES. fluid, it will evidently sink till the immersed part shall be to its whole height, as its density is to that of the fluid. The centre of buoyancy will be below the centre of gravity, but both will be in the axis of the solid ; the former midway between the base of the parallelepiped and the water line ; the latter halfway between the base and summit of the body. If the solid be inclined to one side, its water line will shift its position on the body; the centre of buoyancy will make a corresponding change, describing a small arc of a circle, till it be raised in relation to the altitude of the centre of gravity of the extant triangle, as the area of the adjacent rectangular figure is to that of the triangle, while the moveable centre of buoyancy is carried laterally in the same ratio. And when the metacentre coincides with the centre of gravity, the solid floats passively and indifferent to its position. If the paral- lelopiped become a cube, then its breadth and length being equal, the two densities of indifferent floatation are expressed in the numbers ^ and |. Between these limits there can be no stability, but above and below them the floating body acquires permanence. Both experiment and calculation prove, that a parallelepiped of half the density of water, and having 9 inches for its altitude, and 11 inches for the side of its square base, will float indifferently ; but it will gain stability if its density be either increased or diminished. With a density two thirds that of water, the metacentre will stand ^ parts of an inch above the centre of gravity ; and $ parts of an inch above it if the density be reduced to one third. With such proportions, a paral- lelopiped might therefore in every case continue erect ; and copper or sheet-iron tanks, with such proportions, would float safely as pontoons for flying bridges. But this is not all : we can prove, that if the parallelepiped be set upon water, with one of its solid angles uppermost, the stability will be limited within the densities of 3 9 5 and ||. j n a W ord, let the specific gravity be greater than ^ or less than |, the solid would permanently float in that position : but were the specific gravity either less than the former, or greater than the latter, the body would overset. Were the parallelepiped thus set on water, with one of its diagonals immersed and the other vertical, its equal side being 18 inches, then it would sink about 14^ inches on the side ; 9% inches of the diagonal would be immersed, and nearly 16 extant; supposing the specific gravity of the solid to be 0.326, that of the fluid being equal to unity. In short, the determination of the positions of equilibrium of a solid body, floating on a fluid of a given density greater than itself, is reducible to a problem of pure geometry, which may be better expressed as follows : To cut any proposed "body by a plane, so that the volume of one of the segments may be to that of the whole body in a given ratio ; and such that the centre of gravity of the whole body, and that of one of its segments, may be both found in a line perpendicular to the cutting line. In order to the complete solution of this problem, it is necessary in each parti- cular case, to express the two conditions of equilibrium by means of equations, the solutions of which will make known all the directions that can be given to the cut- ting plane, and whence necessarily result all the positions of equilibrium of the body. This is precisely the plan we have pursued, and all our investigations proceed to ascertain these two conditions of equilibrium; and from the resulting or final equations, to draw up a geometrical construction of the positions so determined ; for calculation is here an instrument of necessity, and not a vain exhibition of analytical formulae, difficult to follow and still more difficult to apply. NOTES. 457 NOTE F. CHAPTER XIII. The investigations pursued in this and the previous chapter, explain the cause of the oversetting of the large icebergs which sometimes float within the limits of the temperate zone. These enormous blocks of frozen fresh water assume various forms i some are columnar, others approach the parallelepiped in their outline, others again resemble mis-shapen cylinders; but all evidently different in form below the plane of floatation to what they exhibit in their extant volume. The action of the atmosphere as the summer advances, slowly thaws the upper surface ; the under side likewise melts at first, but becomes soon protected by a pool of fresh water of the same temperature, consisting of the dissolved portion of the ice which is upheld by the superior density of the surrounding medium. The principal waste of the icy mass taking place along its immersed sides, the current of melted water continually rises upwards, and leaves a new surface to the attack of a warmer current. Whenever therefore, the breadth of the vast column becomes so reduced that it approaches to three fourths of its altitude, the icy parallelepiped will overset, and present a new position of equilibrium. Thus, if the whole height of the mass were 1000 feet, 890 feet would be submerged in the ocean, and 110 feet would be extant, towering amidst the waves. In this case, the elevation of the centre of gravity beyond that of buoyancy would be 35 feet, which is the limit of the metacentre after the base of the column has been reduced to a breadth of 766 feet. An iceberg of a cylindrical form 1000 feet high, would sink 889 feet in the ocean ; but when the diameter of its base was reduced to nearly the same dimen- sions, say 885, it would overset and take a new position. The instability of the cylinder takes place earlier than that of the parallelepiped, or when the width below becomes eight ninths, instead of three fourths of the whole height. Since then the extant portion wastes more slowly than the immersed portion, the greater the extension of the summit, the more it will hasten the change of position by over- whelming the icy mass. And if the block be wasted and rounded below into the shape of a parabolic conoid, it will suffer a total inversion the moment its base is reduced to Its depth in the ratio of about 11 to 20, and its lowest point will become the summit of the extant mass. This form of a body of ice would therefore suffer a greater previous waste ; but its balance is in the end more effectually destroyed. In every case, stability becomes precarious after the breadth of the block is inferior to its depth. NOTE G. CHAPTER XIV. Upon the pressure, cohesion, and capillary attraction of fluids that are heavy, depends their transmission through fissures of the earth and between its strata, which are pervious to the percolation of water. We can penetrate but a small distance, say 500 fathoms, in digging for coal ; a less depth suffices for some ores, and water is found at all depths, from a few feet to three hundred, as in the neigh- bourhood of London. In the great coal area of Britain, extending lengthwise 260 miles, and in breadth about 150 miles, in a diagonal line from Hull to Bristol, in England, and from the river Tay to the Clyde, in Scotland, we find a great variety 458 NOTES. of rocks or strata, piled up at a small angle with the horizon, though in some instances, like the primitive, nearly vertical. These strata consist of sand-stone, clay-slate, bituminous slate, indurated argillaceous earth, or fireclay, argillaceous ironstone, and greenstone or blue whinstone. And to possess the valuable treasures concealed among these rocks, we employ a vast capital in money, and tax all the ability of the human mind in the science of engineering. To bring the subject matter of capillary attraction, as regards Artesian wells, springs, mountainous marsh lands, or bogs, fairly before the reader in a very brief manner, we shall avail ourselves of a vertical section of the strata in Derbyshire, selecting our materials from the valuable work of Mr. Whitehurst, " On the original State and Formation of the Earth." If the reader conceive the alluvial covering to be removed, the strata will at once appear on the upper surface, as in the external contour of the country between .Grange Mill and Darley Moor, in Derbyshire. Let now the numbers 1, 2, 3, 4, &c. represent the strata in their vertical position, bassetting towards 8, with the river Derwent running over a fissure filled with rubble in the centre. Then, the upper stratum, or No. 1, at Darley Moor, is Millstone Grit, a rough sandstone, 120 yards deep, composed of granulated quartz and quartz pebbles, without any trace of the animal or vegetable kingdoms. The next stratum, called No. 2, which is found on both sides the Derwent, is a bed of Shale, or Shiver, 120 yards deep, being a black laminated clay, much indu- rated, without either animal or vegetable impressions. It contains ironstone in nodules, and the springs issuing from it are chalybeate, as that at Buxton Bridge, or that at Quarndon, and another near Matlock Bridge, towards Chatsworth. Next in- succession we have No. 3, Limestone, 50 yards thick, productive of lead ore, the ore of zinc, calamine, pyrites, spar, fluor, cauk, and chert. This stratum is full of marine debris, as anomince bivalves, not known to exist in the British seas ; also coralloids, entrochi or screw stones; and amphibious animals of the Saurian, or lizard and crocodile tribe, some of which, in a fossil state, are of enormous size. Following this we have No. 4, a bed of Toadstone, 16 yards thick, but in some instances varying in depth from 6 feet to 600 feet. It is a blackish substance, resembling lava, very hard, with bladder holes, like the scoria of metals or Iceland lava. This stratum is known by different names in different parts of Derbyshire. At Matlock and Winster it is loadstone and blackstone ; at Moneyash and Tidswell it is called channel} at Castleton, cat-dirt; and at Ashover, black-clay. " This toadstone, channel, cat-dirt, and black-clay, is actually lava, and flowed originally from a volcano, whose funnel or shaft did not approach the open air, but which disgorged its contents between the adjacent strata in all directions," at a period when the limestone strata and the incumbent beds of millstone-grit, shale, argilla- NOTES. 459 ceous stone, clay, and coal, had an uniform arrangement concentric to the centre of the earth. Beneath all these we have No. 5, a Limestone formation, 50 yards thick, and similar to No. 3; that is to say, laminated, containing minerals and figured stones. It is productive of marble ; it abounds with entrochi and marine exuviae : it was thence at one time the bed of a primaeval ocean. No. 6 is Toadstone, 40 yards deep, and similar to No. 4, but yet more solid, showing that the fluid metal was much more intensely heated and combined than No. 4. No. 7, Limestone, very white, 60 yards deep ; laminated like No. 3 and 5, and like them it contains minerals and figured stones, and was either a continuation of Nos. 3 and 5, the entire mass having been split at different depths by the expansive power of the boiling lara. No. 8, is Toadstone, 22 yards deep, similar to No. 6, but yet more solid. No. 9, Limestone, resembling Nos. 3, 5, and 7. To this enumeration of the Derbyshire strata we must now add six other strata f too minute to be expressed in the same scale, but which are in fact the capillary strata, which we may liken to the glass plates referred to hi Problem LXXI. Miners call these minute parallel strata, clays, OT way-boards : in general they are not more than four, five, or six feet thick, and in some instances not more than one foot. They are the channels for water, and all the springs flowing from them are warm y like those at Buxton and Matlock Bath. The first stratum of clay separates Nos. 3 and 4 ; the second, Nos. 4 and 5; the third, Nos. 5 and 6; the fourth, Nos. 6 and 7 ; the fifth, Nos. 7 and 8 ; the sixth, Nos. 8 and 9 : and what is very remarkable, by these clays the thickness of the other strata may be ascertained, which would otherwise be difficult, as the limestone beds consist of various lamina. There are several circumstances illustrative of this capillary attraction, which receive illustration from the diagram before us ; to these we shall now address- ourselves ; and, first, it is observable that all the parallel strata basset or shoot towards the surface, occasioning thereby a diversity of soil ; and as the beds or layers of rock, c. contain fossil remains, we may expect to meet with shells, corals,, bones, plants, trees, &c. on or near the surface. All these rocks ranged in beds or layers, whether perfectly horizontal or shooting up at any angle, are called strati- fied ; while abrupt masses of granite,* having none of this masonic appearance, are said to be unstratified. It is obvious, from what has been observed above, that the stratified parts of the globe are those in which we must look for capillary veins and sheets of water. In the diagram before us all the strata are distinctly marked with their various dislocations and fissures. The river Derwent is supposed to flow over a vast fissure, R ; the letters A, A, A indicate lesser fissures ; G, G, G do the same, and all these fissures are hi the limestone strata. Hence it appears that the toadstone or lava * Granite consists of distinct aggregations of quartz, felspar, mica, and hornblende, each in a crystalline form. Felspar is of a whitish, sometimes of a reddish colour, quite opaque, aad occasion- ally crystallized in a rhomboidal form ; quartz is less abundant, somewhat transparent, and of a glassy appearance ; mica is dispersed throughout in small glistening plates, the colour is dark and the appearance metallic ; hornblende imparts a deep green colour to rocks called greenstone and basalt. 460 NOTES. strata are attended with many peculiar circumstances, very different from their associates, 3, 5, 7, 9. These peculiarities are : 1. Toadstone is similar to Iceland Lava both in its appearance and chemical qualities. 2. It is extremely variable in thickness. 3. It is not universal. 4. It has no fissures corresponding to those in limestone. 5. It frequently fills up the fissures in the stratum underneath it, as at H, and the bottom of the shaft s, which enters a fissure of toadstone that in a liquid state has flowed into the limestone stratum, numbered 9. Throughout the limestone strata of Derbyshire the fissures we have marked correspond ; and in these fissures, and between their lamina, the minerals are found. The mines in the fissures are called rdke-wor ks ; the mines in the laminae are called pipe-works. Thus in the stratum No. 3, we find Yatestoop mine ; the Portaway and Placket mines are in No. 5. ; in No. 7. we have Mosey- meer, and in No. 9 Gorseydale mines ; Hangworm mine is on Bonsai Moor. The stratum No. 5 bassets and forms the surface of the earth at Foolow and Bonsai Moor. No. 8 bassets and becomes the base of the land called Grange Mill. No. 3 again bassets and becomes the districts Trogues Pasture to the right, and Wensley to the left of the great shaft sunk at o and trending below ground to the fissure G in No. 5. Here we have a beautiful illustration of the genius of geological engineering. A spring occurs at I in the fissure G, No. 3, too powerful to be overcome, or too expensive to be kept under ; accordingly a shaft is sunk at o higher up the acclivity. The miners pioneer to a, descend to the fissure G by driving a gallery or gate, as they term this tunnel, and this is a common practice, and never fails in producing dry work in the stratum No. 5, for the close texture of the toadstone will not allow the water in the seam between 3 and 4 to percolate its impervious mass, although the pool may accumulate from 10 to 15 fathoms in No. 3. If the water in 3 rise not to the horizontal level L L, it can never incom- mode the shaft o a. The grand geological fact elicited here is, as regards capillary attraction, that toadstone turns water, is free from fissures, nay, sometimes fills up fissures, as at s and H, which the miners call troughing. In the Slack and Salterivay mines on Bonsai Moor, some forty years ago, these cros.-ralte fissures were noticed by Mr. Whitehurst. Their occurrence in other mines need not astonish geological engineers. In other districts in Britain, we find that the coal formations sometimes repeat, in precisely the same order, arid in nearly the same thickness, the following earths and minerals : sandstone, bituminous shale, slate clay, clay iron, stone, coal ; or the coal is covered with slate, trap, or limestone, or rests upon these rocks. The strata generally follows every irregularity of the fundamental rock on which they rest j but in some instances their directions appear independent, both of the surface of the rock, and of the cavity or hollow in which they are contained, and in general take a waved outline, seldom rising greatly above the level of the sea. We have now, however, merely represented the general arrangement of the strata ; not all the particular circumstances accompanying them, with respect to their several fractures, dislocations, &c. ; but it will enable us to reason upon the chemical effects of water upon limestone and gypsum rocks, where we meet with caverns, caves, and extensive fissures, that reach sometimes to the surface, some- times dip to a greater or less distance, and afford channels for great springs and subterranean rivers. These caves in the gypsum and chalk formations vary in magnitude from a few yards to many fathoms in extent, forming upon the surface of the ground, when their superincumbent roofs give way, those funnel-shaped NOTES. 461 h6llows of such frequent occurrence in gypsum districts. The limestone strata, besides being " loaded with the exuviae of innumerable generations of organic beings," says Dr. Buckland, " afford strong proofs of the lapse of long periods of time, wherein the animals from which they have been derived, lived, and multiplied and died, at the bottom of seas which once occupied the site of our present con- tinents and islands." * With how much reason then may we not suppose those formations to have held large beds of rock salt, which the percolation of water, in the lapse of ages, removed, and left the chambers empty, or the receptacles of meteoric water. The percolation of water through felspar rocks, must of necessity wash away the alkaline ingredient, which combining with iron will form hydrate, or by its decomposition oxidate the metallic substance. Hence result chalybeate, acidulous, sulphureous, and saline springs, all the result of capillary attraction in the strata of the earth, and the disintegration by water of the various ingredients which the universal solvent holds in a state of fluidity. Supposing these cavities, to which we have just referred, to have been freed from their original salt deposits, by water percolating the fissures leading to and from the masses of salt, we trace the operation of salt springs. For in all cases in which water holds any mineral in solution, it acts by combination, but where it simply destroys the mineral aggregation, the mineral falls into small pieces with an audible noise, as is observed in bole ; or it falls without noise into small pieces, which are soon diffused through the fluid, without either dissolving in it or becoming plastic, as in Fuller's earth, and some minerals, as unctuous clay ; it renders plastic other minerals, absorbs water in greater or less quantity, by which their transparency, and also their colour, are changed. The toadstone, which intersects mineral veins, totally cuts off all communication between the upper and lower fissures, and by the closeness of its texture permits not the water in the clay strata, or way-boards, to filtrate. Hence toadstone is said to be capable of turning water, as we have shown in the shaft and gallery, o a G G. Sandstone strata, of an open porous texture, becomes a great feeder of water. Several of the sandstones are, however, impervious to water, and almost all the beds of light-coloured argillaceous schistus, or fine clays, are particularly so, being very close in their texture. But the percolation of water at the beds or partings of two strata is an occurrence so general, that our wonder ceases when examining parts of the country where the strata basset or shoot to the surface in an acute angle, to find the alluvial covering in places swampy, marshy, and overrun with puddles, springs, and all that species of soil, which, being damp and cold, subjects its inhabitants to rheumatism, agues, and a train of diseases, unknown in regions that are not incumbent on the extremities of way-boards and capillary strata. The source or feeder of these subterranean capillaries receiving a constant supply, keeps up the train of human ills from one generation to another, while local interests or associations bind the natives to their hereditary doom. Capillary attraction and cohesion, besides expounding the phenomena of fluid ascent in strata of earth, direct us in penetrating those troublesome quicksands and beds of mud, which in the winnings of collieries are met with in mining, and where * Dr. Buckland's Bridgewater Treatise, pp. 112-116, 1st ed. vol. i. 462 NOTES. cast-iron tubbing is employed to support the sand or mud bed, and carry the water down to the bottom of the pit. Water stands higher in narrow than in wide glass tubes, but quicksilver mounts higher if the inside of the tube be lined with bees-wax or tallow. We can easily conceive that the lateral action may yet cause the perpendicular ascent j for it is a fundamental property in fluids, that any force impressed in one direction may be propagated equally in every other direction. Hence the affinity of the fluid to the internal surface producing the vertical ascent. A drop of water let fall on a clean plate of glass spreads over the whole surface, in as far as there is liquid to cover the glass, the remoter particles extending the film, yet adhering with the closest union. The adhesiveness of fluids is still more clearly shown in their projection through the pores of minerals, plants, animals, gravel, earth, and sand. Water rises through successive strata of gravel, coarse sand, fine sand, loam, and even clay : and hence, on the sea-coast, those quicksands, which have engulfed armies and ships, the pressure and elevation of the ocean at flood tide sending its advanced column up in the sand to a level with its surface far out at sea. Gravel divided into spaces of the hundredth part of an inch, will allow water to ascend above four inches ; it would mount up through a bed of sixteen inches of this material, supposing sea gravel to be the 500th part of an inch. Fine sand, in which the interstices are the 2,500th part of an inch, allow the humidity to ascend seven feet through a new stratum; and if the pores of the loam were only the 10,000th part of an inch, it would gain the further height of 25f feet through the soft mass ; thence originate v astce syrtes. The clay would retain the moisture at a greater altitude ; but the extreme subdivisions of the clay, which enable it to carry water to almost any elevation, yet make it the most efficient material in puddling or choking up the interstices of masonry. The ascent of water in a glass tube is due chiefly, we think, to the excess of the attractive power of the glass above the cohesive power of the fluid mass over itself. Were the attractive and cohesive forces equal, the fluid would remain balanced at a common level. Mercury hence sinks, by reason of the strong cohesive power of its own particles. Hence we account for mercury closing over a ball of crude platinum, which nevertheless, being gently laid on the mercury, will float, although its specific gravity is above that of mercury. It is however the province of chemistry, rather than of mechanics, to measure the cohesive power possessed by different fluids, or by the same fluid under different degrees of temperature. The suspension of water in any stratum through which it can percolate, must depend entirely upon the smallness of the upper orifice, or superficial extent of the deflection with which the stratum slopes off horizontally above ground, and upon the relative elevation of the extremities of the impervious stratum. Thus, suppose a and b to be two extremities of a stratum pervious to water ; the central column of water at c is pressed with the whole weight of the space be, and this pressure upon c a pushes the fluid out at a by the excess of force in be above that in a; and therefore, while the ground or land at b is generally dry, that at a is per- haps boggy; at all events it will exhibit springs at its surface, be cold, damp, and its inhabitants subject to rheumatism or NOTES. 463 agues. A column of water of this description may occupy a space of many miles extent between b and a ; and c may be many hundred feet deep below the hori- zontal level of a. In digging for water at d, we should find it at e. The cohesion of the particles of water, and its extreme facility to obey any impression, fit it admirably for percolating through fissures of the earth, when in the tenderest filaments it is detached from the general fluid mass, and penetrates only by the laws of capillary attraction from one point to another in an extensive stratum of clay, precisely as if it flowed through a pipe in passing from one hill to another. Hence the certainty with which we meet with water in boring to a proper depth in the earth, and hence also the origin of Artesian wells, which finely expound the varied phenomena of a retreating and subsiding column towards the body of the fluid, as if an equal and opposite pressure from the sides of a capillary tube had come into action. We may hence infer, that in strata pervious to water, the capillary ascension, however much it may be accelerated or retarded by the parallel sides of the stratum and the material of which it is composed, is governed by these three principles which we have fully discussed, pressure from above, cohesion subsisting among the particles of the liquid, and attraction of the parallel sides of the stratum. Were this attraction equal to the antagonist cohesion, the fluid would remain at rest, balanced at a common level, till overcome by the weight of the contents in the longer branch of the fluid column forcing the contents of the shorter column out at the discharging orifice. All the springs which are below the London clay, at the depth of 150, 200, 250, or 300 feet, are fed by sources con- siderably elevated above the Hampstead level. With what ease then might the metropolis be provided in every street with spring water from an Artesian Well ! Any of our readers who may be desirous of acquiring a practical and thorough knowledge of geology, must chiefly prosecute his studies by laborious researches in the great field of nature, and must there explore for himself the various phenomena presented to his view. His first step must be to understand by reading the leading facts and principles of the science ; he must learn to recognise at once the principal simple minerals, entering into the composition of rocks, and also the various metallic ores and other minerals which usually occur in veins. He must likewise be acquainted, the more minutely the better, with at least the more common forms of fossil organization, and with the general mode of their distribution in the rocky masses constituting the crust of the globe. Some preliminary knowledge of chemistry, although not perhaps essential, will form a very desirable addition to the qualifications already named. Thus provided with the knowledge requisite to decipher the instructive pages on which nature has recorded, in her own language, the history and revolutions of our planet, the student may now com- mence the most valuable, but far the most laborious part of his career. He must visit the deep recesses of our mines, which, although too much neglected, afford the finest examples of many of the most important facts on which the science of geology is built. He must observe the strata as laid open in our quarries, and as displayed in the deep cuttings of our roads, railways, and canals. Every excavation will indeed present something worthy of notice to Ms view; but not contented with observing merely these spots, where the labour of man has penetrated into the interior of the earth, he must wander around the base of the lofty cliffs which overhang the ocean, and observe the grand and instructive sections which nature herself presents, and of which our own islands afford such numerous and admirable examples. He must pursue the course of rivers into the interior, and observe the 464 NOTES. strata laid open by the excavations of their currents; but his most instructive studies will be found, when he has arrived far inland at the mountains, where they take their rise. Here he will find that nature has revealed the structure of the globe on the grandest scale ; here the marks of ancient revolutions will be found imprinted in characters not to be mistaken, and the truth both of facts and theories, before known only by description, will at once be impressed on his mind. By researches of this kind, extended over considerable tracts of countries, so as to embrace all the great series of geological formations, and by careful study and comparison of all the phenomena presented to his view, both as regards the mineral structure of the globe, the forms of organized bodies peculiar to each species of rocks, and the physical changes now taking place on the earth's surface, the student wifl at length become a practical geologist, and be enabled by his own observations to improve and advance the science he has been studying. But the course which has here been pointed out, although essential to a practical and thorough knowledge of the subject, can only be pursued by few; and a general idea of its most important facts, and the practical consequences arising from them, is of comparatively easy attainment. The great principles of geology have been most ably brought together in various publications ; and where only a general knowledge is required, geological maps and sections may be made in some measure to supply the place of travelling and observation. A few words then on these important documents, which are the medium of expressing some of the most important practical results of the labours of geologists in the field, may not be misplaced. A map which combines with the geographical and physical features of a country, a view of its internal structure, supposes all wood and vegetation to be absent, and that every species of superficial soil and covering removed, so that the actual rocks and strata which compose the solid crust of the globe beneath shall be perfectly exposed and laid open to our view. The space occupied at the surface by these rocks and strata is then distinctly shown by different tints of colour, in the same manner as territorial divisions are indicated on ordinary geographical maps. But although we thus obtain a perfect view of the surface distribution of the solid materials of the globe, it is evidently essential to know in what manner they are arranged below, and what relations they bear to each other in the internal parts of the elobe. This object is accomplished by means of geological sections, the nature of which will acquire but little explanation. A geological section supposes, that on any given line the internal structure of the earth is laid open in the direction of a vertical plane, as in our section between Barley Moor and Grange Mill in Derby- shire. It therefore merely represents, although generally on a much more extended plane, the same thing which we see in many artificial excavations, and which nature herself exhibits to our view in cliffs and precipices. Geological sections are indeed merely a combination of sections of this kind, in which they bear the same relation as the map of a large country would do, to the smaller plans and sketches from which it was compiled, especially connected with the Mechanics of Fluids. Such a map is that of Messrs. J. and C. WALKER. It appears that the present annual value of the mineral produce of Great Britain, may be estimated at somewhere about 20,000,000?. sterling; independent of any subsequent process of manufacture, and not including the cost of carriage on coal. Burr's " STUDY OF GEOLOGY," London, 1836. A TABLE THE SPECIFIC GRAVITIES OF DIFFERENT BODIES. In consulting this Table of Specific Gravities, it must be borne in mind that water is taken as the unit of measure for solids and liquids ; and atmospheric air as the unit of measure for the different gases. Water at the common temperature is 1,000, and mercury 13.568 ; whence we conclude that mercury is 13 times heavier than water. We mean that a cubic foot of water weighs 1000 ounces ; therefore a foot of mercury weighs 1 3,568 ounces, and a cubic foot of bar iron 7788 ounces; a cubic foot of vermilion 4230, of Portland stone 2496, of indigo 0,769, and of cork 0,240 ounces. METALS. Antimony, crude ' ^ .. .- .,. 4064 glass of . ,-.:-. 4946 molten . . . 6702 Arsenic, glass of, natural . . 3594 molten vi>?;i;i , 5763 native orpiment . . 5452 Bismuth, molten ... 9823 native . . . 9020 ore of, in plumes . 437 1 Brass, cast, not hammered . 8396 ditto, wire-drawn . . 8544 cast, common . . 7824 Cobalt, molten . . . 7812 blue, glass of ,.,^ ,' . 2441 Copper, not hammered . . 7788 the same wire-drawn . 8878 ore of soft copper, or natural verdigris &ggjy. 3572 Gold, pure, of 24 carats, melted, but not hammered . . 19258 the same hammered . . 19362 Parisian Standard, 22 car. not hammered . 17486 VOL. I. Gold, hammered . ,,/';./! 17589 guineaofGeo.il. . . 17150 guinea of Geo. III. . .17629 Spanish gold coin . . 17655 Holland ducats . . 19352 trinket standard, 20 carats not hammered . . 15709 the same hammered . . 15775 Iron, cast ;[,';,!;:, 7207 cast at Carron . ' , . 7248 ditto at Rotherham . .7157 bar, either hardened or not 7788 Steel, neither tempered nor hard- ened .... 7833 hardened,butnot tempered 7840 tempered and hardened . 7818 . ditto, not hardened . . 7816 Iron, ore prismatic . . . 7355 ditto specular . . . 5218 ditto, lenticular . . .5012 Lead, molten .... 11352 ore of, cubic . . . 7587 ditto horned . . . 6072 ore of black lead v . 6745 2 H 466 A TABLE OF THE SPECIFIC GRAVITIES OF D1FFEUENT BODIES. Lead, ore of white lead ditto ditto vitreous . ditto red lead . ditto saturnite . Manganese striated . ,. Molybdena *'> v .. Mercury, solid or congealed fluent . .! natural calyx of . precipitate, per se precipitate, red . brown cinabar . red cinabar - , Nickel, molten . . --* ,> ore of, called Kupfer- nickel of Saxe Kupfer-nickel of Bohemia 6007 Platina, crude, in grains . purified, not ham- mered ditto hammered . ditto wire-drawn . ditto rolled . Silver, virgin, 12 deniers, fine, not hammered ditto, hammered . Paris standard shilling of Geo. II. . shilling of Geo. III. French coin . Tin, pure Cornish, melted, and not hardened . the same hardened . of Malacca, not hardened . the same hardened . ore of, red ore of, black . - ore of, white . " *. ' ;: ' : r i ' Tungsten ; . .. V . ij "",- Uranium .... Wolfram .... Zinc, molten .... PRECIOUS STONES. Beryl, or aqua-marine oriental . ditto, occidental Chrysolite of the jewellers 4059 Chrysolite of Brazil . 2692 6558 Crystal, pure rock of Madagascar 2653 6027 of Brazil . ...'<*/ 2653 5925 European 2655 4756 rose-coloured . , 2670 4738 yellow .... 2654 15632 violet, or amethyst ,-, * 2654 13568 white amethyst 2651 9230 Carthaginian - . . v 2657 10871 hlirlr Qflf>4 1UO 1 I 8399 Diamond, white oriental . \JiJ i 3521 10218 rose-coloured oriental 3531 6902 orange ditto 3550 7807 , , 1-4.4. 3524 blue ditto . * v ***' 3525 6648 Brazilian . 3444 L 6607 -^,.11 nrr . 3519 yeiiow ... 15602 Emerald of Peru 2775 Garnet of Bohemia . > r 4189 19500 of Syria 4000 20337 dodecaedral . 4063 21042 volcanic, 24 faces . 2468 22069 Girasol 4000 Hyacinth, common . . "'." 3687 10474 Jargon of Ceylon 4416 10511 Quartz, crystallized . 2655 10175 in the mass . ^"v^W 2647 10000 brown, crystallized 2647 10534 ^Fftflfflfe 2640 10408 milky .... 2652 fat, or greasy 2646 7291 Ruby, oriental .... 4283 7299 fivtivhAH 3760 7296 Ballas .... 3646 7307 Brazilian .... 3531 6935 Sapphire, oriental 3994 6901 sJi4-4-jt v'liifn 3991 6008 of Puys 4077 6066 Brazilian . 3131 6440 Spar, white sparkling 2595 7119 red ditto .... 2438 7191 green ditto 2704 blue sparkling . 2693 green and white ditto 3105 transparent ditto 2564 adamantine 3873 3549 Topaz, oriental .... 4011 2723 pistachio ditto . 4061 2782 Brazilian l ; - . 3536 A TABLE OF THE SPECIFIC GRAVITIES OF DIFFERENT BODIES. 467 Topaz of Saxe . . x . white ditto 3564 3554 4230 2590 2638 2625 2607 2667 2632 2616 2664 2606 2665 2587 2615 2630 2612 2623 2623 2591 2613 2594 2582 2612 2565 2950 2966 2983 2359 2681 2661 2691 2710 2711 2735 2696 2816 2750 2628 2114 2684 2664 2654 2609 2612 Pebble, stained .... Prasium ..... Sardonyx, pure . . . speckled . veined . . blackish . Schorl, black prism, hexaedral . octaedral tourmalin of Ceylon 2587 2581 2603 2606 2622 2595 2595 2628 3364 3226 3054 2923 3156 3286 2416 2111 2143 2484 &c. 2730 2762 2699 2744 2691 2693 2699 2713 2638 2876 1078 926 909 2313 2578 3073 2864 1104 2000 2790 2727 2784 2168 2306 2274 2311 2312 SILICIOUS STONES. Agate, oriental . >(:& . .. ..._. cloudy . , ;.. ifutt -.*>}, speckled veined . . %>;<.;',:- Chalcedony, common :.-.* ,'>t;*'-> Stone, paving . . . > tfai cutlers' . :f?',;HfiJifJ-jr~ * grind . . . . ~~ _ -mm veined . . H-JK' reddish . r j hliifch onyx <^df< - veined , ^ .,.>; stalactite . ft&dftt/l VARIOUS STONES, EARTHS, Alabaster, oriental white . id -*- do. semi-transparent . Flint, white . ? -.. ->... ' awittr. - black . ..'.-ray Ira*;' veined . Ltf ttvf'J lu jt >A < Egyptian . i i^i'l .huMlfo Jade, white . *>O Mfiiit^iiltfb green . . U-*iteV. olive . . -tifM -,t$ Jasper, clear green . * j .*& brownish green . < .-us brown . . . V> '*> stained brown . -V v* veined . ) "V-^'w $- of Piedmont . -3 -. of Malta . o&9o8-tf- of Valencia *I*tfoIv~r- of Malaga . . >! -.*-- Amber, yellow transparent '*.-.* Ambergris . . ,/T^ -. Amianthus, long . . sstfiv * -. Asbestos, ripe .... starry. Basaltes from Giants' Causeway Bitumen of Judea Brick .. ., 'astral.. Chalk, Spanish . . .-'.' violet .... cloudy .... - veined . . ..'-.*& onyx . . . ^hJ red and yellow Opal British .... Gypsum, opaque . . - . semi-transparent Pearl, virgin oriental Pebble, onyx .... English .. '.:*a!* rhomboidal 468 A TABLE OF THE SPECIFIC GRAVITIES OF DIFFERENT BODIES. Gypsum, cuneiform crystallized Glass, green . . 2306 2642 2892 2733 3189 3329 2654 2876 2894 3054 4360 5000 2500 2270 3179 3156 3182 2742 2724 2717 2838 2726 2695 2650 2700 2710 2705 2678 2755 2708 2858 2728 2718 2668 2714 2649 2348 1329 1714 2146 2341 2385 2765 2676 2793 2754 2728 4954 3900 Pyrites, ferruginous, round ditto, of St. Domingo . Serpentine, opaque, green Italian ditto, veined black and olive 4101 3440 2430 2594 2627 2586 3000 2669 2672 2854 2186 2766 2324 2478 915 2722 2415 2945 2771 2520 2510 2049 2496 2470 1981 2460 2122 2357 2274 2378 2034 2201 2033 1991 2792 2089 2246 2655 2900 2704 1841 2125 1271 1580 bottle .... . Leith crystal . . --*- - fluid Granite, red Egyptian . > ~v Hone, white razor .- 'A I 1 . i - -r Lapis nephriticus . Ti7ii1i ditto, fibrous ditto, from Dauphiny Slate, common .... Judaicus .... Manati . ."rr,Vf-t-- ;.-> i- Limestone ;*- * ~W& f j :; 't white fluor . VsuJu* Marble, green, Campanian Ife red . . > . .:''. white Carrara . i* flesh polished .- Iv^-.- Stalactite, transparent opaque . . - -; Stone, pumice . fcOBtoy -rS prismatic basaltes . - . ~ tOUCh . . i"--54 --; Siberian blue . $**' -nH oriental ditto . -sin * ,s-s*>- - Bristol - Portland ff/Ji** - - Castilian . i-ralfcv- .-- Valencian . infctfa *~ white Grenadan . -r-dfr Siennien . i5I io *- Roman violet & '!. - African . . *- rotten . . . '''., i a _ hard paving rockofChatillon . clicard, from Brachet ditto, from Ouchain v . r " Norwegian . !!?<> -nr- St. Maur . . . 9?&* St Cloud : "5 VEV"> 5 * green Egyptian . Switzerland . French .... Obsidian stone . .. > Peat, hard . . rvm^' Phosphorus . ... . Porcelaine, Sevres . :*!s-i 'i- ew Sulphur, native .... molten Talc, of Muscovy . :'. rfi*- black crayon ditto German . **lbi7: *- yellow . -'-. black .... white LIQUORS, OILS, &c. Acid, sulphuric .... ditto, highly concentrated nitric .... ditto, highly concentrated . Porphyry, red .... green .- red, from Dauphiny . .- red, from Cordoue - - green, from ditto Pyrites, coppery . - ferruginous cubic . A TABLE OF THE SPECIFIC GRAVITIES OF DIFFERENT BODIES. 469 Ac-id, muriatic . . 1104 Oil, of walnuts .... 0923 red, acetous . jih >;.'* . 1025 of whale .... 0923 white acetous . . 1014 of hernpseed 0926 distilled ditto . . 1010 of poppies . 0924 fluoric . 1500 rapeseed 0919 acetic . . . 1063 Spirit of wine. See Alcohol. 0837 phosphoric . . '-. > . 1558 Turpentine, liquid . . .- 0991 formic . . i . ; . 0994 Urine, human . 1011 Alcohol, commercial . . 0837 nooq Water, rain c Atr.-i.ivi^A 1000 T Ai"lA mixed with water uo^y sea (average) . . ; lUUU 1026 15-16ths alcohol . 0853 of Dead sea . 1240 14-1 6ths ditto . 0867 Wine, Burgundy . . fi:u;';j 0992 13-16ths ditto . 0882 Bourdeaux . . . 0994 12-16ths ditto . 0895 Madeira .... 1038 ll-16ths ditto . 0908 Port . . . - \t 0997 10-lGths ditto . 0920 .T*AftAW** 1033 9-16ths ditto . 0932 8-16ths ditto . 0943 7-16ths ditto . 0952 RESINS, GUMS, AND ANIMAL 6-16ths ditto . 0960 SUBSTANCES, &c. 5-16ths ditto . 0967 4-16ths ditto . 0973 Aloes, socotrine . . ^-Hji'-I 1380 3-l6ths ditto . 0979 hepatic . . . 1359 2-16ths ditto 0985 Asafcetida . i. 1328 l-16th ditto . 0992 Bees-wax, yellow 0965 Ammoniac, liquid . < o >; . 0897 white 0969 Beer, pale . . . 1023 Bone of an ox . . . '. 1656 _ - brown . 1034 Butter 0942 Cider .,,,..-. . 1018 Calculus humanus 1700 Ether, sulphuric . 0739 flQftQ ditto .... l)<|44>.n 1240 T Af\A muriatic . ,, ;r'5 Li uyuy . 0730 Camphor ..... 14d4 0989 acetic . 0866 Copal, opaque . . . .'. *i 1149 Milk, woman's . . . 1020 Madagascar . 1060 cow's . 1032 Chinese . . . .;.... 1063 ass's ' . r .. ,' V , ( ,, . 1036 Crassamentum, human blood 1126 ewe's .... . 1041 Dragon's blood .... 1205 goat's .. ".". ' . ". ' . 1035 Elemi 1018 mare's . . . .!& . 1034 Fat, beef 0923 cow's clarified . fc : < . 1019 hog's wj--. 0937 Oil, essential, of turpentine . 0870 mutton . . . . .. 0924 psscTiti3/l of* IsvcndGr 0894 0934 , , (lit to of cloves 1036 1212 j - - - (Jit/to of cin.nLinoii 1044 1222 of olives . 0915 Gum, ammoniac . . , . ^ 1207 of sweet almonds . 0917 Arabic .... 1452 of filberts . ' ~. V 7^ . 0916 Euphorbia 1124 linseed . 0940 seraphic . - 1201 470 A TABLE OF THE SPECIFIC GRAVITIES OF DIFFERENT BODIES. Gum, tragacanth . 1316 Cedar, Indian . . . ':&.'* . 1315 bdellium . . 1372 American i^^fAt^ . 0561 Scammony of Smyrna . 1274 Citron . . ;.--- - . 0726 ditto of Aleppo w , , j, . 1235 Cocoa-wood . 1040 Gunpowder, shaken . fi&& . 0932 Cherry-tree . . . .** . . 0715 in i lnn~p TIPIH . 0836 Cork . . - . - , Srt . 0240 nstlS'l . 1745 Cypress, Spanish - . *. -.1 ->**r . 0644 Honey ,(,_,., r . , nstiu . 1450 Ebony, American . i H . 1331 Indigo . . . . in . 0769 Indian . ^:|i|;n.-rs r " . 1209 Ivory . .,. . ... !i: : > . 1826 Elder-tree . . . '. ; - . 0695 Juice of liquorice ' *.:" v;i.) i . 1723 Elm, trunk of . iJ w hv.* . 0671 of Acacia . -...- 't i'f . 1515 Filbert-tree .t-U Uti; l-.I. . 0600 Labdanum , ;+ . . ..'^fcao*; 1 . 1186 Fir, male . . >-.;'.M-t-. . 0550 Lard , . . .', s ; h<>. 0948 female 0498 Mastic . . . i-iKj 1074 Hazel . -' : . 0600 Myrrh . . . ., . 1360 Jasmine, Spanish . ' -1 '.. . 0770 Opium . . . -fftfii . 1336 Juniper-tree . . # ' . . 0556 Scammony. See Gum. Lemon-tree . (i . 0943 Linden-tree . . - V . . 0604 Storax .... . 1110 Logwood. See Campechy. . 0913 Tallow . . . * 0942 Mastick-tree . . -^ . . 0849 Terra Japonica . .... a (??. . 1398 Mahogany iMti& & i -f- . . 1063 Tragacanth. See Gum. . . 1316 Maple . '-'.U -'):*'' . . 0750 Wax. See Bees-wax. . 0965 Medlar . **$ if . T --v . 0944 shoemakers' . > r ia '. . 0897 Mulberry, Spanish . -I . . 0897 Oak, heart of, 60 years old . 1170 Olive-tree ... . . . 0927 WOODS. Orange-tree . . *; . 0705 Pear-tree ..<.* ., . .'.-. . 0661 Alder ....,; ... ^.,- ^ . 0800 Pomegranate-tree . -*';;: . 1354 Apple-tree ..,.,.. <-v^ . 0793 Poplar . -: A . . .a' . 0383 Ash, the trunk . . . . . 0845 white Spanish '. - H . 0529 Bay-tree . , , -..*, '<'*;** . 0822 Plum-tree . .- . v . ## . 0785 Beech . . . 0852 Quince-tree . . - s'iiu . 0705 Box, French . . . . 0912 0482 Dutch . ; f 4 :.-i-- Htfj. 1328 Vine .... . 1327 Brazilian red . ^ V; **# 1031 Walnut . , . . * . 0671 Campechy-wood . 0913 waiow . . ... ; . 0585 Cedar, wild . . ..,," . : *.; 0596 Yew, Dutch . 0788 Palestine , . .. . . . - -,y; 0613 Spanish . 0807 471 WEIGHT AND SPECIFIC GRAVITY OF DIFFERENT GASES. Fahrenheit's Thermom. 55 Barometer 30 inches. Spec. Gray. Wt. Cub. Foot. Atmospheric air . 1.2 . 525.0 grs. Hydrogen gas . 0.1 43.75 Oxygen gas . ; 1.435 . 627.812 Azotic gas . . 1.182 . 517.125 Nitrous gas . v< fj 1.4544 . 636.333 Ammoniac gas. ""*.-.. .7311 . 319.832 Sulphureous acid gas . 2.7611 . 1207.978 In this table the weights and specific gravities of the principal gases are given, as they correspond to a state of the barometer and thermometer which may be chosen for a medium. The specific gravity of any one gas to that of another will not exactly conform to the same ratio under different degrees of heat and other pressures of the atmosphere. And if common air, the standard, be taken at unity (1); chlorine oxy muriatic acid will be 2.500 ; and hydrogen 0.069 ; whence we conclude that chlorine is 2 times heavier than hydrogen, and this last is 14 times lighter than common air. For, to arrive at the absolute weight of the gases, we have only to assume 100 cubic inches of atmospheric air to weigh 30.5 grains, and as there are 1728 cubical inches in a cubic foot, the simple proportion 100 : 30.5 grains : : 1728 : 527.04 grains, the weight of a cubic foot of common air. And for any other gas, it is only necessary to observe its specific gravity in relation to that of common ah-; for example, chlorine has a specific gravity of 2.5; hence a cubic foot of chlorine will weigh 2 times as much as a cubic foot of common air ; for 527 .04 x2 =1317.6 grains, the weight of a cubic foot of chlorine. To determine the weight of any gas lighter than common air, we also compare their specific gravities : thus, the specific gravity of ammoniacal gas is 0.5, and that of atmospheric air being =1, we have 1 : 0.5 : : 1728 : 864.0, or simply 1728-J-2=864 grains, for the weight of a cubic foot of ammoniacal gas; and so on for all the other gaseous bodies, as they are arranged in the following table. 472 WEIGHT AND SPECIFIC GRAVITY OF DIFFERENT GASES. Tf atmospheric air be taken at unity (1), then the various gases will stand as under : Atmospheric air . . Ammoniacal gas Carbonic acid . Carbonic oxide . Carburetted hydrogen Chlorine . Chlorecarbonous acid Chloroprussic acid ,. Cyanogon Euchlorine Fluoboric acid . Fluosilicie acid . Hydriodie acid . 1.000 0.500 1.527 0.972 0.972 2.500 3.472 2.152 1.805 2.440 2.371 3.632 4.346 Hydrogen .... 0-69 Muriatic acid .... 1.284 Nitric oxide .... 1.041 Nitrogen 0.972 Nitrous acid .... 2.638 Nitrous oxide .... 1 .527 Oxygen 1.111 Phosphuretted hydrogen . . 0.902 Prussic acid .... 0.937 Subcarburetted hydrogen . . 0.555 Subphosphuretted ditto . . 0.972 Sulphuretted ditto . . .1.180 Sulphureous acid ... . 2.222 CONCLUSION. THE reader will have seen in this volume how the road to abstract science may be smoothed ; but he may rest assured that any popular version of Hydrostatics is quite illusory, for no portion of sound know- ledge was ever acquired without some corresponding exertion of mind. It is one of the improvements to be made in our systems of education for the various professions, and in books written to retrieve the de- clining taste for science, that students in Mechanics should devote themselves methodically to the profitable but toilsome drudgery of computation ; and, in their geometrical constructions, be as clever with their hands as ingenious with their heads. Science and know- ledge are subject, in their extension and increase, to this law of progression : the further we advance, instead of anticipating the ex- haustion of their treasures, the larger the field becomes the greater power it bestows upon its cultivators to add new measures to its rapidly-expanding dominions. It is the science of calculation which has grasped the mighty masses of the universe, and reduced their wanderings to fixed laws ; which prepares its fetters to chain the flood, to bind the ethereal fluid ; and which must ultimately govern the whole application of Hydrostatics to the Arts of Life. London : J. Rider, Printer, 14, Bartholomew Close. 0^ CALIFORNIA NGV 5 1947 One dotton- S ev REC'D LD JUL Ibl e h day overdue. RECEIVED NOV27'67-1QP ewe I 28Ju1'59AJ - 10 Om.l2,'46(A 2 012sl6)4120 UNIVERSITY OF CALIFORNIA LIBRARY