independent variables. Should the three equations (27) be satisfied, the expression (31) will be simplified, becoming du dv ―dp=P2d +Pd +Pd y dy 2 dv du dw dz dw + T2d (du + de) + T, d (du + de) dv dy dz where T, denotes Ty or Ty, and similarly for Ty, T. dz dx The general expressions for the tensions resulting from Cauchy's method are written at length in the equations numbered 17 and 18, pp. 133, 134 of the 4th volume of his Exercices de Mathématiques,' where the normal and tangential tensions, referred to surfaces in the actual state of the medium, are denoted by A, B, C, D, E, F. These expressions contain 21 arbitrary constants, of which six, A, B, C, D, E, F, denote the tensions in the state of equilibrium. If these be for the present omitted, the remaining terms will be wholly small quantities of the first order, and therefore the tensions may be supposed to be referred to a unit of surface in the actual, or in the undisturbed state of the medium indifferently. On substituting now for P, P., P, T, T., T. in (32) the remaining parts of A, B, C, D, E, F (observing 'x'y' that the E, n, in Cauchy's notation are the same as u, v, w), it will be seen that the right-hand member of the equation is a perfect differential, integrable at once by inspection, and giving the arbitrary constant being omitted as unnecessary. We see that this is a homogeneous function of the second degree of the six quantities (24) and (25), but not the most general function of that nature, containing only 15 instead of 21 arbitrary constants. Let us now form the part of the expression for involving the constants which express the pressures in the state of equilibrium. It will be convenient to effect the requisite transformation in the expressions for the tensions by two steps, first referring them to surfaces of the actual extent, but in the original position, and then to surfaces in the original state altogether. yz' Let P', T., &c. denote the tensions estimated with reference to the actual extent but original direction of a surface, so that P', dS, for instance, denotes the component, in a direction parallel to the axis of x, of the tension on an elementary plane passing through the point (x, y, z) in such a direction that in the undisturbed state of the medium the same plane of particles was perpendicular to the axis of x, dS denoting the actual area of the element. Consider the equilibrium of an elementary tetrahedron of the medium, the sides of which are perpendicular to the axes of x, y, z, and the base in the direction of a plane which was perpendicular to the axis of x; and let 1, m, n be the direction-cosines of the base; then P=1A+mF+nE, T=lF+mB÷nD, T'zz=/E+mD+nC; xy (34) substituting in (34), and writing down the other corresponding equations, we have Lastly, since an elementary area dS originally perpendicular to the axis of a dv, dw becomes by extension (1+ + duds, and similarly with regard to y and z, we have dy dz xy' zy xy, dx dy xy, &c. Expressing P, T, &c. in terms of P', T &c. by (36), then P', T', in terms of A, B, C, D, E, F by (35), and lastly substituting for A, B .... F the expressions given by Cauchy, we find Substituting now these expressions in (31) and integrating, we have 2 + dv du du du dv dv dw dw +20 + + dz dy dy dz dy dz dy dz du du du du dv dv dw dw + 2 + { dy + dx + dx dy + dx dy + dx dy}, which is exactly Green's expression*, Green's constants A, B... F answering to Cauchy's A, B... F. The sum of the right-hand members of equations (33) and (38) gives the complete expression for -24 which belongs to Cauchy's formulæ. It contains, as we see, 21 arbitrary constants, and is a particular case of the general form used by Green, which latter contains 27 arbitrary constants. I have been thus particular in deducing the form of Green's function which belongs to Cauchy's expressions, partly because it has been erroneously asserted that Green's function does not apply to a system of attracting and repelling molecules, partly because, when once the function is formed, the short and elegant methods of Green may be applied to obtain the results of Cauchy's theory, and a comparison of the different theories of Green and Cauchy is greatly facilitated. **Cambridge Philosophical Transactions, vol. vii. p. 127. Fourth Report of the Committee on Steamship Performance. Report. CONTENTS. Sheet of indicator diagrams of H.M.S. 'Colossus,' 'Arrogant,' and 'Hansa,' and scale of displacement of the McGregor Laird.' Appendix, Table 1.-Form of Engineers' Pocket Log, issued by the Committee. Table 2.-Return of the particulars of the dimensions of 20 vessels in H.M. Navy, with the results of their trials upon completion for service. Table 3.-Table showing the results of the performances at sea, and when on trial, of H.M.S.'Arrogant,' ' Colossus,' and 'St. George.' Tables 4, 5, and 6.-Results of trials of H.M. screwships, officially tabulated by the Admiralty, in 1850, 1856, and 1861. Steam Transport Service.-Tables Nos. 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16 (the last 5 tables being summaries of the Tables 7 to 11) show the results obtained from vessels employed in transport service during the latter part of the Russian War, showing the respective values of the several steamships, classified according to the nature of the employment, or the special character of the duties required to be performed; and giving, in addition, the cost of moving each ship 1000 miles, &c. Table 17.-Table showing performances of the Royal West India Mail Company's Steamers from June 1861 to July 1862. Table 18.-Summations of the indicator diagrams taken on all the voyages included in Table 17. Table 18 A.-Table showing the manner in which the summaries in preceding table are obtained. Table 19.-Return of the particulars of the dimensions of the Peninsular and Oriental Steam Navigation Company's steamship Mooltan,' with tabulated statement showing the results of her performance as compared with six other vessels in the same service. Table 20.-Table of the results of the performances of 68 vessels of the Austrian Lloyds' Steamship Company. Table 21.-Return of experiments with H.M.S. 'Stork,' 'Shannon,' and 'Psyche,' with different kinds of screw propellers. Table 22,-Seven logs of voyages of the 'Great Eastern' for 1861-62. Table 23.-Statement showing the summary of the performances of the Pacific Steam Navigation Company's new vessels Peru' and 'Talca.' Table 24.-Abstract log of, and notes upon, the performance of the African Royal Mail Company's steamship 'McGregor Laird.' Table 25.-Notes on the performance of the North German Lloyds' Company's steamship Hansa.' Table 26.-Log of the Earl of Durham's sailing-yacht 'Beatrix,' on her recent Mediterranean voyage. REPORT. ["The object of the Committee is to make public such recorded facts through the medium of the Association, and being accessible to the public in that manner, to bring the greatest amount of science to the solution of the difficulties now existing to the scientific improvement of the forms of vessels and the qualities of marine engines. They will especially endeavour to guard against information so furnished to them being used in any other way, and they trust they may look for the cooperation of members of Yacht Clubs having steam-yachts, of shipowners, as well as of steamship-builders and engineers."Third Report, 1861, p. 16.] Ar the meeting of the British Association held at Manchester in September 1861, the Committee were reappointed in the following terms:"That the Committee on Steamship Performance be reappointed. "That the attention of the Committee be also directed to the obtaining of information regarding the performance of vessels under sail, with a view to comparing the result of the two powers of wind and steam, in order to their more effectual and economical combination; with £150 at their disposal." The following noblemen and gentlemen were nominated to serve on the Committee: The Duke of Sutherland. W. Fairbairn, Esq., LL.D., F.R.S. The Hon. Capt. Egerton, R.N. R. Roberts, Esq., C.E. Henry Wright, Esq., Secretary. With power to add to their number. The following noblemen and gentlemen, having consented to assist your Committee, were, during the present year, elected as corresponding members: Lord C. Paget, M.P., C.B. The Earl of Durham. The Marquis of Hartington, M.P. Viscount Hill. Lord John Hay. Admiral Elliott. Captain Hope, R.N. Captain Ryder, R.N. Robert Dalglish, Esq., M.P. Captain Robertson, R.N. T. R. Tufnell, Esq. Wm. Froude, Esq. Your Committee have the pleasure of stating that, at the unanimous request of the members of the Committee, his Grace the Duke of Sutherland undertook the office of Chairman. The Committee have, since February last, held monthly meetings, and intermediate meetings of a sub-Committee. Your Committee have pleasure in reporting very satisfactory progress, and that they have had an increasing amount of useful information placed at their disposal. Much greater interest is now taken in the objects of the inquiry, and a still increasing number of observers have adopted the forms of the Committee, for recording the performances of vessels. The importance of the information collected by your Committee is attracting the attention of steamship-owners, as well as scientific investigators; and it is hoped the result of greater efficiency and economy in the application of steam, as well as improvements in the construction of steam-vessels, will be the result of these Reports; and your Committee have reason to believe that considerable advantages have already been derived from their labours by steamship-owners. The Royal Navy.-Your Committee, in their Third Annual Report, stated the results of their communications with the Admiralty, and have now to report that the objects of your Committee continue to meet with the approval of the Lords Commissioners of the Admiralty, and of the intelligent scientific officers in that branch of Her Majesty's service; that your Committee have been furnished from time to time with accurate returns of the performances of the more important steamships in Her Majesty's service which have been tried at the measured mile during the last twelve months, and also some similar returns, received too late for insertion in the Report of last year. In the Appendix will be found a selection from these returns, preference having been given to the returns of vessels of which the future steam performances at sea have been promised. Your Committee have received several returns of performances of Her Majesty's ships at sea, the publication of which, owing to their being incomplete in some important particulars, and to the lateness of the time at which they were received, is necessarily postponed. Your Committee call attention to the selection they have made, which will be found in the Appendix. As numerous inquiries have, from time to time, been made of your Committee as to the particulars of certain of Her Majesty's steamships, the performances of which were noticed in previous Reports, your Committee, with a view to avoid unnecessary correspondence, and to give the required information more fully than can be done by written communications, determined to include in the present Report three sets of tables of trials of H.M.'s ships, which were officially tabulated by the Admiralty, but not issued by them to the public. The reprinting of those tables, and the textual information accompanying them, in the Appendix to the present Report will now supply those who possess the previous Reports of your Committee with the means of comparing the results obtained upon the trials of nearly the entire of the steamships of war composing the British Navy, and will also enable them to compare with the results of such trials the performances whilst at sea of very many of the vessels included in the complete and extensive lists to be found in the three Reports previously published, and in the present Report of your Committee, without the necessity, which before existed, of searching elsewhere for the information. The publication of the three Admiralty Tables will also render it un |