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independent variables. Should the three equations (27) bo satisfied, the expression (31) will be simplified, becoming
where Tx denotes Tyl or T^, and similarly for Tg, Tt.
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 ' Exerciccs dc Mathematiqucs,' 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, % J3, C, S, <£, df, 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 bo 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»> pc> T*> T,> Tr "» (32)the remaining parts of A, B, C, D, E, F (observing that the £, »j, £ in Cauchy's notation are the same as M, 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 <p 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.
Let P'j., T"yr, &c. denote the tensions estimated with reference to the actual extent but original direction of a surface, so that P'x rfS, for instance, denotes the component, in a direction parallel to the axis of x, of tho tension on. an
elementary piano 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, rfS denoting tho actual area of the element. Consider the equilibrium of an elementary tetrahedron of the medium, the sides of which are perpendicular to tho axes of x, y, z, and the base in tho direction of a plane which was perpendicular to the axis of x; and let 7, m, n be the direction-cosines of the base; then
FI=ZA-r-«iF+«E, T^lF+mB+nD, T'„=7E+roD+«C; (34)
but to the first order of small quantities
i i du
substituting in (34), and writing down the other corresponding equations, we have
Lastly, since an elementary area c7S originally perpendicular to the axis of x becomes by extension (*"r'^'r'^r) cTS, and similarly with regard to y and z,
dx dy dx dy .
which is exactly Green's expression*, Green's constants A, B . .. F answering to Cauchy's 9, 33 . . . jf. The sum of the right-hand members of equations (33) and (38) gives the complete expression for —2tj> which belongs to Cauchy's formula?. 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 <p 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.
_ . Contents. Report.
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, 6, and 6.—Results of trials of H.M. sorewships, officially tabulated by the Admiralty, in 1850, 1856, and 1861.
Steam Transport Service.—Tables Nos. 7, 8, £>, 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 tho several steamships, classified according to the nature of the employment, or the special character of the duties required to be performed j and giving, m 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.