696 MECHANICS Hence when we apply (2) to a figure previously strained by the reciprocal of (1) the result is the strain (! ). Hence we verify that the latter is the conjugate of (1) ; as it possesses the properties described in 90, all of which are thus established. Analysis To analyse a strain in the simplest manner, we must find the axes of strain, of the strain ellipsoid (88), as well as the original radii of the unit sphere which were distorted into them. It is more direct, however, to consider the ellipsoid which becomes a unit sphere in consequence of the strain. .The equation of that ellipsoid is (ax + by + czf + (dx + ey +fz) z 4(gx + hy + iz) 2 = I ; or, with the notation employed in (2) above, But the square of any radius-vector is o; 2 + 2/ 2 + 2 2 =r 2 , suppose. The maximum radius-vector, therefore, of the ellipsoid is found from the two equations (ax + 5y + yz}dx + (Sx + y + ftz)dy + (yx + &y + ajdz = 0, xdx + ydy + zdz . Hence, p being a numerical quantity to be found, ax + Sy + yz =px, Axes of the strain ellipsoid. One line, at least, unaltered in direc tion. Multiply respectively by x, y, z, add, and take account of the two preceding undifferentiated equations. We thus have l=pr-, or p is the reciprocal of the square of the maximum semi-axis required. But, if we eliminate x, y, and z simultaneously from the preceding linear equations, we have 5 7 -P /3 /3 i-p This equation is known to have three real positive roots, because the determinant is symmetrical. The roots are the squared reciprocals of the semi-axes of the ellipsoid, i.e., they are the squares of the semi-axes of the strain ellipsoid. When the three values of p have been found from this equation, any two of the equations (4) give in an unambiguous form the corresponding values of the ratios x : y : z for each of them. Thus we know the original positions of the lines which become the axes of the strain ellipsoid. Their final positions are found from these by means of (A). And, since we thus know the original and final positions of the rectangular system, the method of Eodrigues enables us to calculate the axis and amount of the rotation. In homogeneous strain, one direction at least is unchanged. This is an addition to, or extension of, the singular result of 75. For, if x, y, z be shifted to a point on its radius-vector, we must have ax + by + cz = ex ~ dx + ey +fz = so that d = 0, - f f g h i- a cubic equation, which must have one real root. Strain a When the figure is rigid, the strain must be a rotation only, mere Hence in the formulae (A ) above we have e l ^e. 2 = e 3 =1. Thus the rotation, last written equation becomes = 0; or (by the properties of the direction cosines of a set of rectangular axes) l-(? 1 + ?n 2 + M 3 )(-e 2 )-6 3 = 0. This has, of course, the real root e = 1. But we also have This cannot have real roots if the coefficient of e lie between the limits 2 or - 2. But these are its greatest and its least possible values. For, first, 1 1} m v n 3 may be each = 1 simultaneously. Here we have Or two of them may be each = - 1, but then (to avoid perversion) the third must be =1. Then we have (l + e) 2 = 0. In the first case the figure has no rotation. In the second it rotates through an angle ir about the axis of e = l. The proposition that two pure strains succeeding one another Comp > usually give a rotational strain is proved at once by analysis. Let tion o the pure strains be such that two pn strains and x" = a x + d y + c z , y" = d x + e y + b z , z" = c x + b y + i z . Then, writing only the second term of x" and the first of y" in terms of x, y, z, we have x" = ....... + (a d + d c + c b)y + . . . y" = (d a + e d + b c}x + ...... + . . . It is clear that, in general, this is not a pure strain. But it is also clear that a third pure strain can be found whose application in succession to the other two will give a pure strain. For let the last equations be written x" = a"x + b"y + c"z , y" = d"x + e"y +f"z , and let us apply further the pure strain x" = ax" + Sy" + yz" , Then we have
- " = ........ + (ab" + Se" + yh")y +
y " = (Sa" + ed" + Pg")x + . ........ + There are but three conditions to satisfy, that this strain may be pure. But we may accomplish this in an infinite number of ways, for we have five disposable quantities, viz., the ratios of any five of a, e, i, /8, y, S to the remaining one. In a precisely similar manner we may show that three pure strains can be found, such that their resultant is a mere rotation. In fact, all we have to do, since two pure strains in general produce a distortion accompanied by rotation, is to apply a pure strain to annihilate the distortion, which can of course always be done. 94. In general when a figure is continuously strained, Heterc which, is usually the case in physical applications, at least geneou until cracks occur, the strain is not homogeneous. But, on straiu - account of the continuity of the strain, portions indefinitely near one another are strained indefinitely nearly alike. Hence we may treat such a case by the ordinary process for homogeneous strain, so long as we confine our attention to small regions of the figure strained. When there is discontinuity in the motion of a fluid, it is the common practice to treat the motion as continuous by the fiction of an infinitely thin vortex-sheet separating the two discontinuously moving portions. This is, in all likelihood, physically true in ordinary fluids ; but, so far as the imaginary frictionless fluid of the mathematicians is con cerned, it is a mere analytical artifice to enable us to carry out the investigation. See ATOM and HYDROMECHANICS, in which the mathematical theory of " vortex motion " is very fully considered. Suppose space to be uniformly occupied by points which are dis- Displai placed in a continuous manner. Let , rj, be the rectangular nients i components of the displacement of a point originally situated at system x, y, z. The continuity of the displacement requires merely that points. the differential coefficients, of all orders, of , TJ, with respect to x > lit z (and any combination of them) shall be finite. That being assumed, the displacement, parallel to x, of the point whose initial coordinates were x + Sx, y + Sy, z+Sz (where Sx, Sy, Sz are indefi nitely small quantities of the first order) is necessarily expressed by . dx dy dz Hen^e the relative coordinate of the second point with regard to the first is changed from Sx to 8x + -5x + -^S + -^Sz. And - dx -
dy