APPLICABLE TO MATHEMATICS, &C. 359 



There are several values of x, which are the 

 roots of the equation m f(<2?) := m' ^(oj), that will 

 satisfy the equation (22), and consequently for 

 each of these values of x there will be an equi- 

 librium between the two forces of attraction and 

 repulsion. 



The equations of motion of the two particles 

 A and B will be, 



M--^-^= — ^ ^^i_^y -, for A s motion (24) 



,„ (fa;' wiffa;'— x)— TO'0(a;'— a;) n „, ,. /c,e\ 



-W-j^=^ j^rz^ -» for B s motion (25) 



for the attractions, where x and x^ are the co- 

 ordinates of A and B respectively at any time t 

 and measured from a fixed origin in the line 

 joining A and B. 



To the equations of repulsive motion we have 



d'x m^ (t-iw^ —x) —7n{(a;^ —x) o ,^ 

 M-j^ = ^ — (x'-xY ' *°^ *"® motion 



of A from B (26) 



-M' -j-^ ^ (^i_^). -, for the motion 



of B from A (27) 



