101, 102, 103] MOVING AXES. 317 



producing accelerations equal and opposite to the actual accelerations 

 of the origin of the moving axes. 



As an example, let us consider the problem of two bodies ( 16), 

 which is important in the practical case of the sun and a planet, 

 neglecting the action of the other planets. We have seen in 32 

 that the center of mass of the two bodies remains at rest, while the 

 sun moves about it, in practice however we are interested in the 

 motion of the planets with respect to the sun. We will therefore, 

 in order to consider the sun as at rest, apply to the whole system 

 an acceleration equal and opposite to that possessed by the sun. Let 

 us call the mass of the sun _M~, its coordinates with respect to fixed 

 axes , r}, , the mass of the planet m, its coordinates with^ respect 

 to the fixed axes x',y',&, with respect to parallel axes through the 

 sun x, y, z. We then have by the equations of 16, 



230) 



while by combining these with 229) we obtain for the relative motion, 



d'g' 



We accordingly find that the differential equations for the relative 

 motion are the same as those for the absolute motion, except for 

 the factor M + m on the right instead of M . Thus if the sun be 

 considered to be at rest, the first two of Kepler's laws are still valid, 

 while the third needs the slight correction that the ratio of the 

 cubes of the semi -axes to the squares of the times of revolution are 

 not absolutely constant for all the planets, but proportional to the 

 sums of masses of sun and planet. As even in the case of Jupiter, 

 the largest, m is less than one -thousandth of Jf, the correction 

 is slight. 



1O3. Rotating Axes. Theorem of Coriolis. Suppose now 

 that the origin is fixed, but that the moving axes revolve with an 

 angular velocity whose projections upon their own instantaneous 

 directions are p, q, r. Then we have found in 77, 128) for the 



