AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 291 



the element considered moves as a solid might do, these quantities coincide with the angular 

 velocities considered in rigid dynamics. A further reason for this definition will appear in Sect. m. 

 Let us now investigate whether it is possible to determine a;', y , z' so that, considering only 

 the relative velocities U, V, IF, the line joining the points P, P shall have no angular motion. 

 The conditions to be satisfied, in order that this may be the case, are evidently that the incre- 

 ments of the relative co-ordinates x, y, z' of the second point shall be ultimately proportional 

 to those co-ordinates. If e be the rate of extension of the line joining the two points considered, 

 we shall therefore have 



Fx' + hy + g«' = e x' , 



hx + Gy +fz' = ey , 

 + fy + //«'= ex ; J 



.(3) 



gx 



„ du dv dw dv dw d w du du dv 



where F=~, G = — , H ^ -- , 2f = -- + -y , ^g = ^ + T ' ^A = — + — . 

 dx dy dz dz dy dx dz dy dx 



If we eliminate from equations (3) the two ratios which exist between the three quantities 

 x\ y, z\ we get the well known cubic equation 



(e -F){e- G) (e - H) -f (e - F) - g' {e - G) - h' {e - H) - 2fgh = 0, (4) 



which occurs in the investigation of the principal axes of a rigid body, and in various others. 

 As in these investigations, it may be shewn that there are in general three directions, at right 

 angles to each other, in which the point P' may be situated so as to satisfy the required conditions. 

 If two of the roots of (4) are equal, there is one such direction corresponding to the third root, and 

 an infinite number of others situated in a plane perpendicular to the former; and if the three 

 roots of (4) are equal, a line drawn in any direction will satisfy the required conditions. 



The three directions which have just been determined I shall call axes of extension. They 

 will in general vary from one point to another, and from one instant of time to another. If we 

 denote the three roots of (4) by e', e", e'", and if we take new rectangular axes Ox, Oy , Oz , 

 parallel to the axes of extension, and denote by m,, U^, &c. the quantities referred to these 

 axes corresponding to u, U, &c., equations (3) must be satisfied by y'^ = 0, z^ =0, e = e', by a;' = 0, 

 z' = 0, e = e', and by .r ' = 0, y '= 0, e = e", which requires that /^ = 0, g = 0, A = 0, and 

 we have 



e' = F = — ' "= G = ^ e" =//■ = ^ 



' dx^ ' dy^ ' dz^ 



The values of U^, F, W , which correspond to tiie residual motion after the elimination of 

 the motion of rotation corresponding to w, w" and w"\ are 



U^^e'x', V=e"yf, lV=e"'z'. 



The angular velocity of which w', w", u>"' are the components is independent of the arbitrary 

 directions of the co-ordinate axes : the same is true of the directions of the axes of extension, 

 and of the values of the roots of equation (4). This might be proved in various ways; perhaps 

 the following is the simplest. The conditions by which w', w", w" are determined are those which 

 express that the relative velocities U, V, W, which remain after eliminating a certain angular 

 velocity, are such that Udx' + Vdy' + Wdz' is ultimately an exact differential, that is to say 

 when squares and jiroducts of x' , y' and ;:;' are neglected. It appears moreover from the solution 

 that there is only one way in which these conditions can be satisfied for a given system of 

 co-ordinate axes. Let us take new rectangular axes Ox, Oy, Oz, and let U, V, W be the resolved 

 parts along these axes of the velocities U, V, W, and x', y', 7.', the relative co-ordinates of P^ ; then 



1' p 2 



