139] IGNORATION OF COORDINATES. 215 



Hence the equations (1) now take the form 

 d d!O d ~ 



c^d_d&_ ..................... (6)*, 



dtdq, %~ | 



from which the velocities %, %, ... corresponding to the ' ignored ' 

 coordinates ^, ^', ... have been eliminated. 



In the particular case where 



0=0, C" = 0,..., 



these equations are of the ordinary Lagrangian form, O being now 

 equal to T, with the velocities %, %', eliminated by means of the 

 relations 



so that (B) is now a homogeneous quadratic function of q lt q. 2 , 

 Of. Art. 134 (4). 



In the general case we proceed as follows. If we substitute in 

 (3) from the last line of (5) we obtain 



Now, remembering the composition of , we may write, for a 

 moment 



where @ 2>0 is a homogeneous quadratic function of (ji> <?2>"-> 

 without C, C',...\ B lfl is a bilinear function of these two sets 

 of quantities; and 0)2 is a homogeneous quadratic function 

 of (7, C',..., without q 1} q.,,.... Substituting in (7), we find 



or, to return to our previous notation, 



T= + K ........................... (10), 



where *& and K are homogeneous quadratic functions of q l} q 2 , ... 



* Bouth, L c. 



