2 G\\'\'TnFAK,Mofw7i oftJie Fluid Particles in Steady Motion. 



for in the problem, this gives a convenient measure of the 

 quantity which we are neglecting. 



The point to which I wish to call attention is that in 

 these cases, the solution of the Lagrangian equations and 

 also the expression for the pressure take the same form as 

 the corresponding quantities derived from the Eulerian 

 equations, and that, in form, only a quadrature is necessary 

 to complete the solution for the motion of the particles. 



The first part of this statement will be accepted as 

 obvious on a little consideration, but I will give a formal 

 proof based on the Lagrangian equations. The functional 

 solution of these equations, in irrotational motion, is that 



d 



shall be a function of .r-zj' and of /, and in our case of 

 steady motion a function of x—iy only. 

 Writing 



^^(x + /v)=f(^--/» 



we get 



0' (^ + 6') 7^ Gv + /)') = (/>' (:v + iy)(i>'{x - iy) 



= a real quantity, 

 and therefore 



(/; (.V + iy) ^ !/ + //', 



where n is real and b is independent of /. 



Supposing now that this can be conveniently reversed, 



we write 



x + iy=f{u + i/>) (i), 



and again 



-(.v + /^)=/(« + ;/.)-. 



This expression is to be a function of x - iy and therefore 

 oi u- ib, and can only take the form 



c 

 f\u - ib) ' 

 when c is an absolute constant. 



