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XLVII. On the Applicability of Lagrange's Equations in cer- 

 tain Cases of Fluid-Motion. By John Purser, M.A., Pro- 

 fessor of Mathematics in the Queen s College, Belfast*. 



THE ordinary condition for the applicability of Lagrange's 

 equations to the motion of a system is that the position 

 of all its parts be determined as a function of the generalized 

 coordinates or parameters which enter into these equations. 

 When such is not the case, even though the kinetic energy 

 may be expressible in terms of these coordinates and their dif- 

 ferential coefficients with respect to the time, the Lagrangian 

 equations of motion are known not to be in general valid. 



A familiar illustration of this is afforded by the motion of a 

 rigid body rolling on a plane so rough as to prevent sliding. 

 Here it is evident that the kinetic energy can be expressed in 

 terms of three coordinates defining the angular position of the 

 body and their differential coefficients. It is not, however, 

 possible to express the position of all the points of the body in 

 terms of these coordinates only ; and accordingly the use of 

 Lagrange's equations in terms of these coordinates is known 

 to lead to erroneous results. It becomes, therefore, a matter 

 of considerable interest to inquire into the grounds which 

 justify recent important applications of these equations to 

 sundry problems of hydrokinetics, relating to the motion of 

 rigid bodies in an incompressible frictionless fluid. 



This case, in fact, is in so far similar to that of the rolling 

 body already alluded to, that while the kinetic energy of the 

 system can be expressed (in virtue of Green's theorem) in 

 terms of the limited number of parameters which define the 

 position of the rigid bodies, it is clear from the smallest con- 

 sideration that these parameters do not determine the position 

 of the particles of the fluid. It would certainly seem, then, 

 that we are not entitled prima facie to assume the validity of 

 Lagrange's equations when applied to such problems, and that, 

 if their use be here justifiable, it must be in virtue of special 

 reasons. To endeavour to supply the proof which is thus seen 

 to be requisite is the object of the present communication. 



Given a number of rigid bodies moving in a frictionless in- 

 compressible fluid, whether infinitely extended or enclosed in 

 a rigid envelope ; given also that the motion at one epoch is 

 irrotational, and therefore always so, — then by D'Alembert's 

 principle, 



2,dm{'x8a: + y8y + z8z}+8V = 0, . . (A) 



* Communicated by the Author, having been read before the British 

 Association at Dublin, August 1878. 



