82 KINETICS OF A PARTICLE. [154. 



154. The integration of the equations (i) would give the 

 absolute path of the planet. But the constants could not be 

 determined, because the absolute initial position and velocity 

 of the planet are, of course, not known. The same holds for 

 the absolute path of the sun. All we can do is to determine 

 the relative motion, and we proceed to find the motion of the 

 planet relative to the sun. 



Taking the sun's centre as new origin for parallel axes, we 

 have for the co-ordinates f, 77, f of the planet in this new 

 system, 



Now, dividing the equations (i) by m, the equations (2) by 

 M, and subtracting the equations of set (2) from the corre- 

 sponding equations of set (i), we find for the relative accelera- 

 tions of the planet 



</ 2 f_ M+m 



7^> ^ 9 * ~> 



dfi r* r 



^ I' <3> 



M+m ? 



f 



The form of these equations shows that the relative motion of 

 the planet with respect to the sun is the same as if the sun were 

 fixed and contained the mass M + m. Thus the problem is 

 reduced to that of a fixed centre, the only modification being 

 that the mass of the centre M should be increased by that of 

 the attracted particle m. 



155. This result can also be obtained by the following simple con- 

 sideration. 'The relative motion of the planet with respect to the sun 

 would obviously not be altered if geometrically equal accelerations were 

 applied to both. Let us, therefore, subject each body to an additional 

 acceleration equal and opposite to the actual acceleration of the sun 

 (whose components are obtained by dividing the equations (2) by M). 



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