132 MR. GWYTHER ON THE LAGRANGIAN FORM OF 



say, and the similar quantities |3 and y, so that 



V^7= -Sv<P,VfzV'Pj • VPi^Hvfi, say. 



The length of an edge will be given by the thickness 

 divided by the cosine o£ the angle between the correspond- 

 ing normal and the edge. Thus length of the edge a is 



and tlie vector edge is 



-|a; (!) 



whence the area of the face <Pj is 



and the vector of the face 



. nil. 



V<Pi (2) 



Vz^r 



H 

 From either (i) or (2) we may get volume of element 



= ^^-^^fr sa/3y=-^^-^^. . . (3) 



The condition of continuity is therefore that D<(«?H~') = 

 o^ where d denotes the density ; and this condition in an 

 incompressible fluid becomes DiH = o. Generally </= Hip 

 where 'Di<p = o. 



Before proceeding to investigate any forms for a-, the 

 velocity, we will find some expressions from the action of 

 Dt on the expressions just found. 



Remembering that 



