KINETICS OF A PARTICLE. [212. 



so that x can be regarded as a function of t, q, and q = dq/dt. 

 Hence we have 



dq dqdt dq 2 ' dq dq 

 In the former of these expressions, the right-hand member can 



be put into the more compact form =i, as is easily verified 



dt dq 



by carrying out the indicated differentiation with respect to t. 

 As similar results hold for y and z, we have 



/\ 

 dq~dtdq dq~dtdq 



__ 



' ~~ 



' 



dq dq dq dq dq dq 



212. Let us now add the equations of motion (i) after multi- 

 plying them by dfjdq, df^/dq, dfjdq. The coefficient of X in 

 the resulting equation, viz. 



^_,^,^ 



dx dq dy dq dz dq' 



is equal to zero, since it is evidently proportional to the cosine 

 of the angle made at a given time by the tangent to the curve 

 (3) with the normal to the surface </>=o. For a similar reason, 

 the coefficient of /j, vanishes ; and the resulting equation is 



m (x d A + y d A + z S A\=Q, (6) 



V dq dq dqj 

 if, as in Arts. 180, 181, we put for shortness 



. 



dq dq dq 



This quantity Q can evidently be expressed as a function of 

 q, and q. 



The equation (6) can also be written in the form 



m 



