178 V. OSCILLATIONS AND CYCLIC MOTIONS. 



It is to be noticed that since the coordinates q appear in the co- 

 efficients E of equations 139) they are introduced into T' in a way 

 in which they do not appear in T, so that we do not have 



dT dT' 



hut since q enters in T' both explicitly and implicitly through 

 equations 139), we have for s = r -f 1, r + 2, . . . w, 



141) - = 



3<l s d?, 



and since the same may be said of the velocities, 



142) wn 



' ' 



Now if the eliminated velocities with suffixes 1, 2, ... r are cyclic 

 and the corresponding forces vanish, we have 



<^=- ==c 



Accordingly equations 141/142) become 



dT' dT 



143) -; 



dT' dT 



or transposing and differentiating outside of the sign of summation, 

 144) 



We may therefore use with the coordinates whose velocities remain 

 in the equations Lagrange's equations, except that instead of the 

 kinetic energy T we use the function 



and instead of the Lagrangian function L= T W, we use the function 



