108 Mb. stokes, on SOME CASES OF FLUID MOTION. 



Lastly, if Mo, v^, w„, be the initial velocities, subject of course to satisfy equations (S) and 

 (a), we niust have 



u = u^, V = v^, w = w^, when t = (c). 



In the most general case the equations which u, v and tv are to satisfy at every point of the 

 mass and at every time are (S) and the three equations 



dTH", dSTs d-STi dSTj rfTZTa d-ZJT, 



dy da; dss dy ' dx dz 



These equations being satisfied, the quantity TtT^dx + -sr-idy + sr^dx will be an exact differen- 

 tial, whence p may be determined by integrating the value of dp given by equations (.4). Thus 

 the condition that these latter equations shall be satisfied is equivalent to the condition that the 

 equations (C) shall be satisfied. 



In nearly all the cases considered in this paper, and in all those of which the complete solution 

 is given, the motion is such that tidx + vdy + wdx is an exact differential d(p. This being the 

 case, the equations (C) are, as it is well known, always satisfied, the value of p being given by the 

 equation 



'r'^^'^-t-mp(i)*m] <°'- 



■v// (t) being an arbitrary function of t, which may if we please be included in <^. In this case, 

 therefore, the single condition which has to be satisfied at all times, and at every point of the mass 

 is (fi), which becomes in this case 



d'd) d'(b d'(b 



:r^ + J-T+ T^ = o (-E)- 



ax- ay rftr 



In the case of impulsive motion, if u^, v^, iv„, be the velocities just before impact, «, v, w, 

 the velocities just after, and q the impulsive pressure, the equations (A) are replaced by the equations 



1 do 1 dq I do ,„^ 



- 3— = - !' + "„5 --:-=-'• + "„, — = - tv + tir (F) i 



p dw p dy P dx: 



and in order that these equations may be satisfied it is necessary and sufficient that (u-u„)dx 

 + (« - v^)dy + (w - iVg)dz be an exact differential dcp, which gives 



q = C - p<l). 

 The only equation which must be satisfied at every point of the mass is (5), which is equivalent to 

 (E), since by hypothesis iif,, «)„, and w^ satisfy (B). The conditions (a) and (6) remain the 

 same as before. 



One observation however is necessary here. The values of v, v and iv are always sujjposed to 

 alter continuously from one point in the interior of a fluid mass to another. At the extreme boun- 

 daries of the fluid they may however alter abruptly. Suppose now values of tt, v and iv to have 

 been assigned, which do not alter abruptly, which satisfy equations {B) and (C) as well as the con- 

 ditions (a), (6) and (c), or, to take a particular case, values which do not alter abruptly, which 

 satisfy the equation (B) and the same conditions, and which render jidx + vdy + wd.c an exact 



differential. Then the values of -— , -— and —- will alter continuously from one point to another, 



ax dy ds: ■' '^ 



but it does not follow that the value of p itself cannot alter abruptly. Similarly in impulsive 



motion the value of q may alter abruptly, although those of — i- , — and — alter continuously. 



dx dy dz 



Such abrupt alterations are, however, inadmissible ; whence it follows as an additional condition to 



be satisfied, 



that the value of p or q, obtained by integrating equations {A) or {F), shall ") 



not alter abruptly from one point of the fluid to another. j ^ '' 



