356 Prof. J. Purser on the Applicability of Lagrange's 



dT 

 then, since this latter expression = Bq x -=—, we get at once 



Lagrange's equation, 



dt \dqj dq x dq x ~~ 



It remains only to consider why the above assumption is 

 legitimate. As far as the integral applies to the particles of 



the rigid bodies, since -=■ 8x = 8~ for each particle, the trans- 

 formation is obviously justified. For the fluid, on the other 

 hand, it is evident that, for an individual particle, -=- 8x is not 



dx 



dt 



=zh~. (This may be seen at once by considering the case 



of the motion of a flat piece of cardboard in a fluid, and sup- 

 posing the time displacement perpendicular to its plane and 

 the arbitrary displacement in its plane.) 



To examine the meaning of the differences ( — 8x — 8 — ) dt, 



\ctt etc / 



&c, let us suppose that the generalized coordinates are so 

 taken that one only of the coordinates, say q 2j alters with the 

 time, so that the actual time displacement may be treated in 

 the same way as a possible displacement 8q 2 . 



The above may accordingly be written B^x — B^x, where 

 Sj, 8 2 correspond to the variations 8q l7 8q 2 . 



Consider any point of the fluid A. 

 Suppose the dispacement 8q x fol- 

 lowed by the displacement 8q 2 , and 

 let the answering positions of A be 

 B and C. Again, suppose the dis- 

 placement 8q 2 followed by 8q 1} and 

 let the answering positions of A 

 be D and E. E will not coincide 

 with C, but the displacement is that -^- 

 which has for its projections the expressions above. 



For, projection of EC=projection of DE — projection of AB 

 — (projection of BC— projection of AD) 



= 8 2 $ix — 8iB 2 x. 



The bodies, however, after these two compound displacements 

 are in identical positions ; and consequently the displacements 

 of the fluid (S 2 ^i — &A)#j & c *> correspond to irrotational dis- 

 placements of the fluid compatible with a position of the 



