MECHANICS 697 nstants similarly for the other relative coordinates. Hence the strain in the the immediate vicinity of the point x, y, z is given by ent dx __ dx dx dij ay dy dz 1 + dz oncli- an iat .lame lin gered. If the differential coefficients are all small quantities, whose squares and products two and two may be neglected, i.e., if the strain is slight, we have for the ratio in which the volume is in creased dx dy dz Hence the condition of no change of volume is dx dy dz To examine this case more closely, let us suppose that it consists of a pure strain as in 93 (B), superposed on a rotation w x , u y , ta z about the axes of x, y, and z as in 77. Let these be so small as not to interfere with one another. That compound strain would be Comparing with the above, we find or, if we put e for the "elongation," so that = rf 2 with similar expressions for 5 and -i . dy dz These give ~r~.+ j ! ~r~ = i + f > + ft dx dy dz * i Aain we have with other two of the same kind, eter- Also we have three equations of the form lina d( dr, - on of 2o> x = -^ -T- lerota- dl J dz ous. _ _d^ d y dz dx )ndition These expressions show, simply, that when there is no elementary no rotation the quantity is the complete differential of a function of three independent variables. If we combine the condition that there shall be no change of volume with those that there shall be no rotation, we can eliminate {, 17, ; and we arrive at Laplace s equation This shows at once how a graphical representation of stationary distributions of temperature, electric potential, &c., may be given by means of a strain. If dS be an element of a surface at the point x, y, z, and I, m, n the direction cosines of its normal, the rotation about the normal is obviously lux + mw y + n<a z The integral of double of this over a finite portion of surface is JJ dy This, as seems to have been first pointed out by Stokes, can be expressed as a simple integral in the form extended round the boundary of the surface. Hence the double integral has the same value for all finite surfaces having the same boundary ; and, as a consequence, it vanishes when taken over a closed simply-connected surface. Hence we see at once that it vanishes for multiply-connected surfaces also. The proof of the equality of the single and double integral has only to be established for a mere surface element. For, when that is done, the common boundary of each pair of elements gives equal portions, with opposite signs, iu the single integral. Directly connected with the displacements of a group of points, So-calle<] we have the question, What is the mathematical expression of the equation fact that the number of points is not altered ? There are many ways of con- of answering this ; but the following, which is immediately tiuuity. deducible from our recent investigation, seems sufficiently simple. If I, m, n be the direction cosines of the normal to an element o?S of a singly-connected closed surface, the number of points which pass through the element in the time St in consequence of the displacement 8t, -nSt, St at the point x, y, z is where p is the number of points per unit volume at x, y, z. But at every point inside the closed surface the density is altered from p to p + pSt. It will be noticed that , rj, now stand for the x, y, z components of velocity. Hence, if the excess of the number of points passing into the surface over those escaping be equated to the increase of the number of points included in the closed space, which is calculated from the change of density inside, we have
- tff(l + m-r] + n)pdS = Ufffpdxdydz.
If we take for S an elementary rectangular parallelepiped, with edges Sx, Si/, 82, this becomes at once dp ~ d(pfi I -- r = dz ~r 5 -- T dt dx dy If the arrangement is incompressible this becomes, as above, ^H-^-O. dx dy dz In any one of the last four forms the expression is called the " equation of continuity," another of the preposterously ill-chosen terms which have been introduced with only too great success into the nomenclature of our subject. 95. In the strains which we have hitherto considered Changes all parts of a figure were regarded as capable of changing of figure their form and volume; and the strain of any element, ? f . a , when not identical with that of a proximate element, wasc;y steil supposed to differ only infinitesimally from it. But there of rigid is another class of changes of form, for which this restric- parts. tion does not hold. The most important case, and the only one we can here consider, is that of "link-work." Here each finite piece is treated as incapable of change of form, and the change of form of the whole depends merely upon the relative motions of the parts. We will further restrict ourselves by the condition that the link-work is such that its form is determinate when the relative position of two of its parts is assigned. Thus, a jointed parallelo gram is completely determined inform if the angle between two of its sides is assigned. Instead of an angle, we m:iy assign the length of a diagonal ; then the fact that the sum of the squares of the diagonals is equal to that of the squares of the sides determines the other diagonal. This gives us the kinematics of the more complex arrangement called "lazy-tongs." The most important applications of this Lazy- branch of our subject are to what is called " Mechanism." tongs?. One important practical problem in that branch was suggested by a stationary steam-engine, in which it was required to connect, by link-work of some kind, a point (of the piston-rod), which had a to-and-fro motion in a straight line, with another point (of the beam) which had a to-and-fro motion in a circular arc. Watt s original Watt s solution of the problem depends ultimately upon the near parallel approach to rectilinearity of the motion of any point of a mi rod whose extremities move in two circles in the same plane. Thus, if OP, PQ, QO (fig. 31) be three jjars jointed together at P and Q, having and O fixed, and
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