15-17] COMPARISON OF METHODS. 17 



(# , y , #0) and whose edges correspond to variations 8a, Sb, Sc of 

 the parameters, a, b, c, is 



d(a,b,c) 

 so that we have 



d (oc, y, z) _ d Qo, y , z ) 

 p d(a,b,c)~ p<) d(a,b,c) 



or, for an incompressible fluid, 



d(x, y, z) d(a! 0y y , z 

 d(a,b,c) d(a,b,c) 



17. If we compare the two forms of the fundamental equations to which 

 we have been led, we notice that the Eulerian equations of motion are linear 

 and of the first order, whilst the Lagrangian equations are of the second order, 

 and also contain products of differential coefficients. In Weber's transfor- 

 mation the latter are replaced by a system of equations of the first order, and 

 of the second degree. The Eulerian equation of continuity is also much 

 simpler than the Lagrangian, especially in the case of liquids. In these 

 respects, therefore, the Eulerian forms of the equations possess great ad- 

 vantages. Again, the form in which the solution of the Eulerian equations 

 appears corresponds, in many cases, more nearly to what we wish to know 

 as to the motion of a fluid, our object being, in general, to gain a knowledge 

 of the state of motion of the fluid mass at any instant, rather than to trace 

 the career of individual particles. 



On the other hand, whenever the fluid is bounded by a moving surface, 

 the Lagrangian method possesses certain theoretical advantages. In the 

 Eulerian method the functions %, v, w have no existence beyond this surface, 

 and hence the range of values of x, y, z for which these functions exist varies 

 in consequence of the motion which is itself the subject of investigation. In 

 the other method, on the contrary, the range of the independent variables 

 a, 5, c is given once for all by the initial conditions*. 



The difficulty, however, of integrating the Lagrangian equations has 

 hitherto prevented their application except in certain very special cases. 

 Accordingly in this treatise we deal almost exclusively with the Eulerian 

 forms. The simplification and integration of these in certain cases form the 

 subject of the following chapter. 



* H. Weber, I.e. 



