352 Mr Basset, On the Steady Motion [May 30, 



which is consequently a linear function of the ac's, the modified 

 Lagrangian function is 



L = %+Z(®0)-®-V (1), 



where V is the potential energy, measured from a configuration of 

 stable equilibrium*. 



The equations of motion of the system are accordingly 



ddX d® d% *(d®A\,d$ dV , 



dtTe + ~dt~^e~ \dd V i + dd + W "° {Z) - 



From this equation it appears, that a steady motion may 

 usually be obtained by assigning constant values to the coordinates 

 ; whence the equations of steady motion are 



de + d0~ l) {3) ' 



where the number of equations of the type (3) is equal to the 

 number of coordinates 6. 



2. Let there be to coordinates of the type 0, and n ignored 

 coordinates of the type % ; then we have three cases to consider, 

 according as to is equal to, less than, or greater than n. 



Case I. to = n. 



In this case, the number of equations of the type (3) is equal 

 to the number of momenta k ; hence these equations are sufficient 

 to determine these momenta. Accordingly the conditions of steady 

 motion are, that it should be possible, without violating the con- 

 nections of the system, to assign constant values to the 0's, such 

 that the values of the to momenta k, furnished by the solution of 

 (3), should be real. 



Case II. m<n. 



In this case, the number of equations of the type (3) is less 

 than the number of momenta k. It will therefore be usually 

 possible, when the values of the 0's are given, for the momenta to 

 possess a series of arbitrary values, which lie between certain 

 limits. 



Case III. m>n. 



In this case, the number of equations of the type (3) is greater 

 than the number of momenta k; it will therefore be possible to 

 eliminate the momenta from (3) in one or more ways. Hence in 

 order that steady motion may be possible, it will be necessary that 



* Proc. Gamb. Phil. Soc, vol. vi. p. 117; Basset, Hydrodynamics, vol. i. p. 174; 

 where, in equation (36), the sign of V ought to be changed. 



