AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 309 



of the distances. Differentiating the first of equations (27) with respect to y, and the second 



with respect to !c, subtracting, putting for —— and — their values, adding and subtracting 



du dv , . I • ,. 



— — , and employing the notation of Art. 2, we obtain 



dz dz 



Dm" du , dv ,, /du dv\ „. 

 Dt d« dz \dj! dyj 



By treating the first and third, and then the second and third of equations (27) in the same 

 manner, we should obtain two more equations, which may be got at once from that which has 

 just been found by interchanging the requisite quantities. Now for points in the interior of 



the mass the differential coefficients -— , &c. will not be infinite, on account of the continuity 



dz •' 



of the motion, and therefore the three equations just obtained are a particular case of equations (25). 

 If then udx + vdy + wdz is an exact differential for any portion of the fluid when # = 0, 

 that is, if o)', w" and w" are each zero when t = 0, it follows from the lemma of the last 

 article that w, w" and w" will be zero for any value of t, and therefore ud.v + vdy + tvdz 

 will always remain an exact differential. It will be observed that it is for the same portion 

 of fluid, not for the fluid occupying the same portion of space, that this is true, since equations 



(28), ac. contain the differential coefficients &c., and not , &c. 



^ ' Dt dt 



14. The circumstance of udx + vdy + w dz being an exact differential admits of a physical 

 interpretation which may be noticed, as it is well to view a subject of this nature in different 

 lights. 



Conceive an indefinitely small element of a fluid in motion to become suddenly solidified, 

 and the fluid about it to be suddenly destroyed ; let the form of the element be so taken 

 that the resulting solid shall be that which is the simplest with respect to rotatory motion, 

 namely, that which has its three principal moments about axes passing through the centre 

 of gravity equal to each other, and therefore every axis passing through that point a principal 

 axis, and let us enquire what will be the linear and angular motion of this element just 

 after solidification. 



By the instantaneous solidification, velocities will be suddenly generated or destroyed in the 

 different portions of the element, and a set of mutual impulsive forces will be called into 

 action. Let x, y, z be the co-ordinates of the centre of gravity G of the element at the 

 instant of solidification, x + x, y +y', z + z those of any other point in it. Let u, «, w be 

 the velocities of G along the three axes just before solidification, ii , v, w the relative velocities 

 of the point whose relative co-ordinates are .r', y, z . Let m, », w be the velocities of G, u, o , w^ 

 the relative velocities of the point above mentioned, and w , w" , ui" the angular velocities just 

 after solidification. Since all the impulsive forces are internal, we have 



u = u, V = V, w = w. 



We have also, by the principle of the conservation of areas, 



2m \y' {w ^ - w) - z' (u - u')} = 0, &c., 



m denoting an element of the mass of the element considered. But u^ = ui' z' - lu" y\ xd is 



du , du , du , 

 ultimately equal to — — r + -:— V + -;— ^ > and similar expressions hold good for the other 

 ^ dx dy'' dz ' " 



