320 VII. DYNAMOS OF ROTATING BODIES. 



The resultant J, often known as the acceleration of Coriolis, is 

 evidently perpendicular to the relative velocity whose components 



are -jrr> ~> -^ and to the axis of jp, q, r and is equal to twice the 



vector-product of the angular velocity of the axes and the relative 

 velocity of the particle. It is interesting to notice that the accel- 

 eration of Coriolis arises from the presence of linear terms in the 



velocities, -^y ^| > -^ in the kinetic energy, the effect of which in 



introducing gyroscopic terms was explained in 50. Thus a particle 

 may be arranged to represent hy its motions relatively to a uniformly 

 revolving body, such as the earth, the motions of a system containing 

 a gyrostat. This remark is due to Thomson and Tait. 



1O4. Motion relatively to the Earth. Let us suppose the 

 axes chosen are taken fixed in the earth, the origin at the center, 

 the #-axis the axis of rotation. Let the earth rotate with the constant 

 angular velocity <, which expressed in seconds is 



and is very small. Then p = q = 0, r = &. The centripetal accel- 

 eration of transportation is then 



Accordingly for a point at rest on the earth we may consider the 

 earth at rest, provided we add to other applied forces a centrifugal 

 force whose components are m& 2 x, m& 2 y. This centrifugal force is 



239) ro^yS^Tp = m& 2 Rco8 y, 



where E is the radius of the earth and q) is the latitude. This is a 

 subtractive part of g, the acceleration of gravity, which is consequently 

 greatest at the poles, least at the equator. The vertical part of the 

 centrifugal force is m& 2 Rcos 2 cp. This acceleration is common to 

 all bodies at rest on the earth, and hence is included with gravity 

 in our ordinary experiments. It need not then be further noticed. 

 There is however to be considered the apparent compound centrifugal 

 force, m J x , mJ y , mJ z , which acts on bodies in motion rela- 

 tively to the earth. 



- mJ x = 



24 ) ; 



- m J, = 0. 



