110 Mr. stokes, OiN SOME CASES OF FLUID MOTION. 



are siipposeil not to alter abruptly. Hence, all the requisite conditions are satisfied ; and lience, 

 (Art. 2) the hypothesis of steady motion is correct. 



4. In the case of an incompressible fluid, either of infinite extent, or confined, or interrupted in 

 any manner by any solid bodies, if the motion begin from rest, and if there be none of the cutting 

 motion mentioned in Art. 2, the motion at the time t will be the same as if it were produced instan- 

 taneously by the impulsive motion of the several surfaces which bound the fluid, including among 

 these surfaces those of any solids which may be immersed in it. For let ?<, ii, w, be the velocities at 

 tlie time ^ Then by a known theorem udw -v vdy -v wda will be an exact differential d<^, and 

 d) will not alter abruptly (Art. 2). must also satisfy the equation (£), and the conditions 

 (ff) and (/(). Now if ?/', v' , w , be the velocities on the supposition of an impact, these quantities 

 must be determined by precisely the same conditions as u^ ii and iv. But the problem of finding 

 ?«', v and w , being evidently determinate, it follows that the identical problem of finding m, x> 

 and 10 is also determinate, and therefore the two problems have the same solution ; so that 



u = u' , V = v , w = w . 

 This principle has been mentioned by M. Cauchy, in a memoir entitled Menioire sur la Theorie 

 des Ondes, in the first volume of the memoirs presented to the French Institute, page 14. It 

 will be employed in this paper to simplify the requisite calculations by enabling us to dispense 

 u'ith all consideration of the jtrevious motion, in finding the motion of the fluid at any time 

 in terms of that of the bounding surfaces. One simjjlc deduction from it is that, when all the 

 bounding surfaces come to rest, each element of tiie fluid will come to rest. Another is, that if the 

 velocities of the bounding surfaces are altered in any ratio the value of (p will be altered in the same 

 ratio. 



5. Superposition of different motions. 



In calculating the inital motion of a fluid, corresponding to given initial motions of the bounding 

 surfaces, we may resolve the latter into any number of systems of motions, which when compounded 

 give to each point of each bounding surface a velocity, whicli when resolved along the normal is 

 equal to the given velocity resolved along the same normal, provided that, if the fluid be enclosed 

 on all sides, each system be sucli as not to alter its volume. For let ?t', )'', ?<>', v , a', be the values 

 of «, )•, &c., corresponding to the first system of motions ; u", v", &c., the values of those quantities 

 corresponding to the second system, and so on ; so that 



n = u + u" + ... , V = v' + v" r ..., w = w' + w" + ..., v = v' + r" + ..., <x = <t' + a" + . . . . 



Then since we have by hypothesis u'dw + v'dy + w'dz an exact differential d(p', u'dx + xi'dy 

 + w"d!s an exact differential dcp", and so on, it follows that udx + vdy + wdz is an exact dif- 

 ferential. Again by hypothesis n' = u' , v"= a", &c., whence v = a. Also, if the fluid extend to an 

 infinite distance, u, v, and w must there vanish, since that is the case with each of the systems 

 u , v', iv', &c. Lastly, the quantities d>', d>'\ &c., not altering abruptly, it follows that <p, which 

 is equal to cp' + (p" + ..., will not alter abruptly. Hence the compounded motion will satisfy all 

 the requisite conditions, and therefore, (Art. 2) it is the actual motion. 



It will be observed that the pressure p will not be obtained by adding together the pressures 

 due to each of the above systems of velocities. To find p we must substitute the complete value of 

 <b in equation {D). If, liowever, the motion be very small, so that the square of the velocity is 

 neglected, it will be sufficient to add together the several pressures just mentioned. 



In general the most convenient systems into which to decompose the motion of the bounding 

 surfaces are those formed by considering the motion of each surface, or of a certain portion of each 

 surface, separately. Such a portion may be either finite or infinitesimal. In fact, in some of the 

 cases of motion that will be presently given, where (h is expressed by a double integral with a 

 function under the integral sign expressing the motion of the bounding surfaces, it will be found 

 that each element of the integral gives a value of (h such that, except about the corresponding 



I 



