48, 49] EXAMPLES OF CONCEALED MOTIONS. 181 



happens to bodies at rest relatively to tlie surface of the earth. To 

 the unsophisticated mind they seem at rest, the constant rotation of 

 the earth being concealed. The rotation however produces an effect 

 in altering the weight of the body by the vertical component of the 

 force just found , which is known as the centrifugal force. (In the 

 case of the rotation of the earth it is to be observed that it is <p' 

 that is constant rather than p^, nevertheless , if in the case above 

 r is constant ; <p' will be constant at the same time with jp y .) The 

 effect of the motion of the earth will be treated in detail in 104. 



49. Example. Three Degrees of Freedom. General Case. 



As a further example of the process of ignoration of coordinates we 

 will work out the case of a system with three degrees of freedom, 

 one of whose coordinates is cyclic, this being the simplest example 

 which typifies the behavior of the general system. TVe have then, 



152) T= Q n q>* + &&" * 



If q B is the cyclic coordinate, all the Q's are independent of g 3 , and 

 if the corresponding force P 3 vanishes, we have the constant momentum, 



153) p B = Cis &' + fts &' + 33 &' - ^3 ; 



From this we determine the cyclic velocity, 



.!) ft ,_^-ft./-ft. & ', 



inserting which in the kinetic energy gives, on combining terms, 

 155) T'= 



+ -- 



It is noticeable that the linear terms in #/, q 2 ' have cancelled each 

 other. It will be proved below that this always happens. But when 

 we form the kinetic potential, which is to be used instead, they 

 reappear. We have 



156) <Z> = T'- 



) 



V33 



, c s Q 13 , c s Q, , 1 c 3 2 



I O bfl~i/0 b/2 ^/O 



V33 VSS ? V33 



Thus the effect of the cyclic motion, which may itself be concealed 

 from us, is made evident to our observation by the presence of the 

 fourth and fifth terms, which are linear in #/, q 2 r . The apparent 

 coefficients of inertia, that is the cofficients of q t ' 2 , q 2 ' 2 , #/ g 2 '; are 



