the Stability of the Flow of Fluids. 63 



Thus, du w du dQ ^ 



dt +yy dz dr' K } 



dt + w dz rdtr { } 



dW , dw , TT7 dw dQ /o\ 



u lF + lu +W Tz = lh> ■ ■ ■ • (3) 



which, with the " equation of continuity," 



*£ + »+'£-* (i) 



determine the motion. 



The next step is to introduce the supposition that as func- 

 tions of t,z,6, the variables w, v,w, and Q are proportional to 



e i(nt+kz+s6) t 



We S et dO s 



i(n + W)u=j^, ( n + kTN)v = fa . . . (5) 



dW 

 U ~aV + *(* + *W)w = *7:Q, . ... (6) 



— (n«) +isu + ikvw = (7) 



dr 



From these equations three of the variables may be elimi- 

 nated, so as to obtain an equation in which the fourth is 

 isolated. The simplest result is that in which Q is retained. 

 It is 



J$ + !fL- Q £ +J A-_ » igS.a . (8) 



dr* r dr \r J n + kW dr dr v ' 



But the equation in u lends itself more readily to the impo- 

 sition of boundary conditions. If 5 = 0, that is in the case of 

 symmetrical disturbances, the equation in u is obtained at 

 once by differentiation of (8), and substitution of u from 

 (5). After reduction it becomes 



(»+*^"{£ + £-*-*•} 



\ dr 2 r dr J ~~ ' ' ^ ' 



If the undisturbed motion be that of a highly viscous fluid 

 in a circular tube, W is of the form A + B?* 2 , and the second 



