142 MR. GWYTHER ON THE LAGRANGIAN FORM OF 



IV. 



The velocity can also be written in a form of simple ap- 

 pearance, thus. 



since 



whence 



Btf, = (p,-{SaV<pi)=o, 



Bt(r= -Dif,~-Dtip,^-J)if,^ 



(■ d • d ■ d\ 



But 



. • . c?iO- = - Dfcpj g - Di9^ jj - D^fp J ^, 



and the condition for steady motion is Di(p = o, showing a 

 close resemblance in form between the velocity in this case 

 and the rotation in a perfect fluid ■^. 



We may at any time, by elimination of t, find two sur- 

 faces of the nature (p for which (p = o ; and since in steady 

 motion Di(p = o, the property will continue in the surface _ 

 Taking the two such surfaces for c^^ and (Pp they will act as 

 fixed boundaries, their intersections will be the stream- 

 lines, and the form of g will be 



<^=-|'«- • (2) 



* [Note that ^j is the flux through the side ^j of the element of the fluid 

 under consideration, and remains unaltered if the motion is steady. If also 

 Svo- = o, <Py has properties similar to those of 2,.] 



