360 ON PHYSICAL DATA 



In both of the above cases of motion the centre 

 of gravity will move uniformly, so that the rela- 

 tion of cT' to ,v, will be of the form .r* — N + N^ a? 

 N and N' being constant quantities. 



Let us now examine equation (18) in order to 

 find the values of <r which will make m f(<2') > 

 m} $(ci^). At every value of x, which is a root 

 of the equation m f(,v) — m^ ^(^), there will be a 

 change in the effective attractive and repulsive 

 forces take place. Therefore the difficulty will 

 be to assign the roots of the equation m f(.r) n 

 m' 0{x). This equation, in the general form 

 which we have put it, cannot have its roots de- 

 termined only in particular cases. 



We shall proceed, therefore, to investigate the 

 preceding equations by giving particular values 

 to the functions f(cv) and ^(a?) — 



Let f(a^) = P + P, ^ + Pao;' and <i>(.r) = P3 + 

 ViiV + Ps.^;^ where mV^zz m> P4 and w P2 = m^ Pj 

 P, ; P2 &c. &c. are all constants. 



Then from equation (21) and (23) we shall have 



m i{x) <fl m' *(a;) m P to P, p /m^ 'm>'P^ 

 5 ^= — 5~ ~r -|- Wl Xj ■" I ^ 1" „ 



