388 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



passes the area m 2 , and V and p the velocity and density of the fluid 

 which simultaneously passes the other area, then assuming these quan- 

 tities (as is permitted) to be uniform during the small time St, and con- 

 sidering the velocity positive when directed from the focal lines, the 

 increment of matter between the two areas in the time It is 



, (r + Sr) (r' + Sr) , v , u . —1 



— m . '— i o V ct + nr . pVSt, 



rr r r 



or, - m 2 St{ d -^~ f p V (J t ?)}ir ultimately. 



And this quantity is also equal to rrfSpSr ultimately. Hence 



m*S P Sr + wtMrf^f + pV(\ + £)} = 0. , 



And passing from differences to differentials, 

 dp d.pV T ,i\ 1\ 



which coincides with equation (5). 



15. To complete our investigation it will now be requisite to ob- 

 tain the partial differential equation containing the variables cp, x, y, 

 z and t, <p being the principal variable. This is readily done in the 



case of an incompressible fluid. For substituting N -J- for u, N -— for v, 



ax ay 



and N -f for w, in the equation -=— + -*- + -j— = 0, the result is, 

 as, * ax ay d% 



dN d<j> dN d<p dN d(p /eP0 d^p eP<p \ 



dx ' dx dy ' dy d% ' d% Kdx 2 dy 2 dz 2 ) 



And eliminating JV by means of the equation 



dt + M [dx 2 + dy 2 + ~dz~ 2 ) °' 



