139] IGNORATION OF COORDINATES. 215 



Hence the equations (1) now take the form 

 d d d 



__ (6)*, 



dt dq. 2 



from which the velocities ^, ^ &amp;gt; ... 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, being now 

 equal to T, with the velocities ^, ; , ... eliminated by means of the 

 relations 



dT d f 



dx 0&amp;gt; *r 



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

 Cf. 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 



&amp;lt;H)=:&amp;lt;H) 2)0 + &amp;lt;H) 1)1 +@ 0)2 ..................... (8), 



where 2j0 is a homogeneous quadratic function of q L , &amp;lt;&amp;gt;,..., 

 without C, C ,...j &amp;lt;H) lfl is a bilinear function of these two sets 

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

 of C, C ,..., without q L&amp;gt; q.,, .... Substituting in (7), we find 



T=e a&amp;gt;0 - 0&amp;gt;3 ........................ (9), 



or, to return to our previous notation, 



T=1& + K ........................... (10), 



where *& and K are homogeneous quadratic functions of q lt q.,, ... 



* Eouth, I, c. 



