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&amp;gt; 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 m relative to these moving axes are known 

 to be 



m(x 2m/ n^x) = X, ^ 



m(y + 2nx-n 2 y)=Y, I .................. (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 l} ^ 2 ,.... If we multiply the above equations by dxjdq S) 

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

 all the particles of the system, we obtain 



/ dx ..dy ..dz\ ^ ^ f . dy . dx 

 \x^- + y-r- + z -3- } + 2n2&amp;lt;m(x-f- -y -,- 

 \ dq s y dq s dq s J \ dq s y dq s 



= mtf + f + + + -... 



dq s \ dq s dq s dqj 



There is a similar equation for each of the generalized coordinates 

 q s - 



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

 the first term in (2) may be replaced by 



dt dq s dq s 

 where ^ = iSm(^ + ^ 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 





+ Y + z _ ......... (4) 



dq s dq s dq s j dq s 

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



* See ante p. 201 (footnote). 



212 



