133-134] LAGRANUE S EQUATIONS. 201 



are still arbitrary, their coefficients must separately vanish. We 

 thus obtain Lagrange s equations 



dT 



dt 



(17)* 



134. Proceeding now to the hydrodynamical problem, let 

 &amp;lt;?i % be a system of generalised coordinates which serve to 

 specify the configuration of the solids. We will suppose, for the 

 present, that the motion of the fluid is entirely due to that of the 

 solids, and is therefore irrotational and acyclic. 



In this case the velocity-potential at any instant will be of the 

 form 



(1), 



where &amp;lt;j&amp;gt; lt &amp;lt;f&amp;gt; 2) ... are determined in a manner analogous to that of 

 Art. 115. The formula for the kinetic energy of the fluid is then 



... + 2A ]2 g 1 g 2 + ......... (2), 



where, for example, 





the integrations extending over the instantaneous positions of the 

 bounding surfaces of the fluid. The identity of the two forms of 

 A 12 follows from Green s Theorem. The coefficients A n , A 12 ,. . . will, 

 of course, be in general functions of the coordinates q lt q. 2) .... 



* The above sketch is introduced with the view of rendering more intelligible 

 the hydrodynamical investigations which follow. Lagrange s proof, directly from 

 the variational equation of Art. 132 (2), is reproduced in most treatises on 

 Dynamics. Another proof, by direct transformation of coordinates, not involving 

 the method of variations, was given in the first instance by Hamilton, Phil. Trans. 

 1835, p. 96 ; the same method was employed by Jacobi, Vorlesungen iiber Dynamik 

 (ed. Clebsch), Berlin, 1864, p. 64, Werke, Supplementband, p. 64; by.Bertrand in the 

 notes to his edition of the Mecanique Analytique, Paris, 1853 ; and more recently by 

 Thomson and Tait, Natural Philosophy, (2nd ed.) Art. 318. 



