A FLUID UNDER CONSERVATIVE FORCES. 173 



5. To investigate the energy within a surface drawn 

 within a fluid we get^ by continually using a modification 

 of Green's Theorem and omitting Thomson's correction, 



- 2T =^a^dv =^S {A(j> + k\7f) adv =JSo- v</)^u +^Ska^fdv 



= —^Sya-(f>dv +^Sa(j)vd^ —^S'^ka-yfrdv + JS^'i^o-v</S 



= - JS V<T{cf) + kf) dv -^^f^kadv + JS (</) + Jc^fr) avdS 



= -^S^|rykadv+^S{(f>-{-l:^|r)avdS^,. .... (V.) 



where v stands for the unit normal to the surface, and 

 where the last term vanishes if there is no flow across the 

 bounding surface. 



We may then use the equation of continuity to give 

 other forms to the volume integral. Thus 



forms which will allow us to use Green's theorem again. 



From the rate of change of the circulation in a closed 

 circuit moving with the fluid we get 



DijSo-c?T=o, where dr is an element 

 of the circuit, or 



JSDiO-^T +jjSad . BtT = JSDiO-c?T +JSo-f/o- = o. 



The second is zero over every closed circuit. In order 

 that the first may be so we must have Vv .T>t(r = o, or 

 referring to 



V{vDi^'.V^ + V^V-D^^}=o, . . . (VI.) 



a further justification for our assumption. 



The simplest way of finding D^/j is as follows : — We have 

 seen that 



D,cr=o-— (S v) ('■ = o- — 2Vcrp — -^yo-*. 



