60, 61] HAMILTON'S PRINCIPLE. 117 



Differentiating the above by t, 



daci _ ctai dqi 8#i d<fr 9^ dq m 



dt " dq, dt + dq, dt^ *" dq m dt 



dz l 9zi , dz, , 



Since the #, ?/, <z's are given as functions of q lt q 2 , ... q m alone, 



doc 

 every =-^ is given also as a function of the ^'s. Hence every 



velocity-component is a linear function of the q"s, whose coeffi- 

 cients are certain functions of the ^'s. The ^"s are called the 

 velocities corresponding to the coordinates q. 



Squaring, adding, and summing, we get 







a homogeneous quadratic function of the #"s, whose coefficients 

 are certain functions of the g's, so that we may write 



(3) T=l Q n ?i /2 + J Q 22 g 2 /2 + ...... + Q u2l V+ ...... , 



where 



- P ^ n , Pvpfap.typfyptepd 



U/ rg 4* 1 1 in -\ ^: - ;r -f ^ - ^ -- P ^ - ^ 



P =i (Sgv 3g s 9^ 9g s 9$r 9 



Performing the operation of variation upon the integral occur- 

 ring in Hamilton's Principle, we obtain 



W) ,}~] ,, 



~ ** \\ dt = a 



and since 



we may integrate the second term by parts. Since the initial and 

 final configuration of the system is supposed given, the Sq's 



