TWO UNEQUAL MASSES OF THE SAME SIGNS. 125 



The equation of the lines of force is 



m cos o) + m' cos to' = m f + m cos 0. 



They always form two distinct systems ; the surface which separates 

 them corresponds to = 7r, and its equation is 



m cos w + m' cos w' = m' m. 

 If we put = i + e, the equation becomes 



COS w + (i + e) COS to' = e, 



or in rectangular co-ordinates, the origin being taken in the middle of 

 the distance 20, 



x-a x + a 



jy2 + ( x - a y + ( I+ > jjr + ( x + a )2 = ' 



This equation represents a surface of the sixth degree, which passes 

 through the point of equilibrium, and which has some analogy with 

 the sheet of a hyperboloid. Its meridian section, like all the other 

 lines of force, has an asymptote which passes through the centre of 

 gravity of the two masses. The equation to this asymptote is 







COS a = 



2+e 



The force makes, with the radius vectors, angles /3 and /?', 

 defined by the ratio 



sin/2 r' 2 m'r 2 w'sin 2 w' 

 sh^~~m s= mr r * = msm*<a j 



from which we get, for the value of the force, 



mcos/3 w'cos/5' 



144. Two EQUAL MASSES OF OPPOSITE SIGNS. We shall 

 proceed to examine in greater detail the case of two equal masses 

 of opposite signs, as it presents several important applications. 



