193-195] MOTION RELATIVE TO A ROTATING SOLID. 323 



in Art. 165. It is therefore worth while to devote a little space 

 to it before entering on the consideration of special problems. 



Let us take a set of rectangular axes x, y, z, fixed relatively to 

 the solid, of which the axis of z coincides with the axis of rotation, 

 and let n be the angular velocity of the rotation. The equations 

 of motion of a particle ra relative to these moving axes are known 

 to be 



m(x 2ny rtfac) = X, } 



m(y + ^-n*y)=Y,\ .................. (1), 



mz = Z } 



where X, F, Z are the impressed forces on the particle. Let us 

 now suppose that the relative coordinates (x, y, z) of any particle 

 can be expressed in terms of a certain number of independent quan- 

 tities q lt q 2 , .... If we multiply the above equations by dx/dq s , 

 dyjdq s , dz/dq s , and add, and denote by S a summation embracing 

 all the particles of the system, we obtain 



/..dec ..dy ..dz\ ^ /. dy . dx\ 

 (x^ r - + y-f L + z j- +2n2mla?-j-y;v- 

 V dq s y dq t dq s J \ dq s y dq s j 



+ + -... 



dq s dq s dqj 



There is a similar equation for each of the generalized coordinates 

 fc. 



Now, exactly as in Hamilton's proof* of Lagrange's equations, 

 the first term in (2) may be replaced by 



d 



dt dq s dq s ' 



where = ^Sm ( e + ^ 2 + ^ 2 ) .................. (3), 



i.e. denotes the energy of the relative motion, supposed expressed 

 in terms of the generalized coordinates q s , and the generalized 

 velocities q s . Again, we may write 



/ &, F ^ +Z d 1 \_dv ......... 



V dq s dq s dqj dq s 

 where Fis the potential energy, and Q s is the generalized com- 



.* See ante p. 201 (footnote). 



212 



