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XLIX. On the Applicability o^ Lagrange's Equations of Motion 

 in a General Class of Problems ; ivith especial reference to 

 the Motion of a Perforated Solid in a Liquid. By Chaeles 

 Y. Burton, D.Sc* 



1. ~r ET yjr y <f),... be some only of the coordinates of a 

 J~^ material system, so that when the values of -yjr, cf), . . . 

 are given the whole configuration is not completely determi- 

 nate. But suppose it known that the kinetic energy T can 



be expressed as a homogeneous quadratic function of yjr, <£, . . . 

 only ; so that we may write 



2T=(ff)f 3 +2(^)^ + ... 1 _ 



(yfryfr), (^4>), . . . are functions of ty, (£,... only J 



We also suppose it kno^n that (1) continues to hold good so 

 long as the only (generalized) forces and impulses acting are of 

 types corresponding to 



f,4>, (?) 



2. Suppose, now, that such impulses of these types were to 



act on the system that yjr, <f>, . . were all reduced to zero ; the 

 expression for the kinetic energy would accordingly vanish, 

 and the system would be at rest. By supposing the last 

 operation to be reversed, we see that the motion at any instant 

 could be produced from rest by impulses of the types corre- 

 sponding to 



'f > 4>, • • • only (3j 



3. Let x, y, z be the Cartesian coordinates at time r of a 

 mass-element m referred to fixed axes, and let T be the kinetic 

 energy of the system at the same instant. Further, let A be 

 the " action " when the system moves without additional con- 

 straint from one configuration to another, and A + SA the 

 action when by workless constraints the path is slightly 

 modified, so that in place of the coordinates x, y, z we have 

 x + $x, y + By, z + &z. Then f 



SA = {%m [xhx + yhy + z$z) \ — [%m(xSx -f y8y + zBz) ] 



+ a term which necessarily vanishes ; . . (4) 



where [] and {} denote the values of the quantities enclosed 

 at the beginning and end of the motion considered. 



Suppose further that, both at the beginning and at the end, 

 the values of i/r, (p, . . . are the same for the one motion as for 



* Communicated by the Physical Society : read March 10, 1893. 

 t Thomson and Tait's l Natural Philosophy,' 2nd edit. Part I. § 327. 



