Application of Analysis to the Motion of Fluids. 4-79 



of fluid passes the two areas at the same time. On this prm- 

 ciple I have obtained the equation (A.) in the Cambridge 

 Philosophical Transactions (vol. v. part ii. p. 182). 1 he 

 mathematical reasoning is too long to be inserted. 1 have 

 recently obtained the corresponding equation for compressible 

 fluids by a similar process. , . , , , 



The above method of obtaining the equation (A.) has the 

 advantage of making known the condition that is introduced 

 when udx -^vdy^iadz is assumed to be an exact diffe- 

 rential of a function of x, y, and z. For if V be the velocity 

 at the point j;j/^ distant by r from a local line, and H, »] be 

 two anMes fixing the direction of the normal through that 

 point, we may put V =/(^, r,) x ^ (;), the latter factor ex- 

 pressintr the law of the variation of V at a given instant 

 throucrh a small space resulting from the convergence or ci- 

 vero-ence of the normals. Hence if «, /3, 7 be the coordi- 

 nates of the intersection of the normal with the focal line, we 

 shall have 

 n =/(^, >,) . ^ (r) f-^-.v =f{e, .). <^ (r) .^-^ ; 



K)=/(6>, T)).4)(0-^; andM(/a;-K vdy + iadz 



Now since r' = (a—«)- + (.?/-/3)^ + {^-7)% the above 

 quantity is an exact differential of a function of x, y, and z, 

 if «, /3,V» ^5 a"f^ 1 ^^^ constant; that is, \nhe variation with 

 respect to space be in the direction of the motion. This condi- 

 tion which has hitherto been unnoticed, is independent of the 

 particular mode in which the fluid is disturbed, and being 

 introduced when f/<tJ is substituted for udx + vdy -\-wdz, 

 must be attended to in all the subsequent calculations m 

 which <f is involved. 



The preceding reasoning fails if the lines ot motion be not 

 convercrent, or the motion be such as the fluid might take if 

 it were^either wholly solid, or consisted of solid disconnected 

 parts: for instance, when the fluid mass revolves about a fixed 

 axis and the velocity is a function of the distance from the 

 axis. In fact, for this instance, udx +-vdy + wd^ is not 

 an exact differential. ... , . 



As it appears from the above reasoning that, 111 general, at 

 the same time that the equation (A.) is transformed into one 

 consisting of partial differential coefficients of a principal va- 

 riable, tlfe condition is introduced, that the variation with re- 



