122 PARTICULAR CASES OF EQUILIBRIUM. 



141. ANY Two GIVEN MASSES. Let there now be two masses 

 m and m' situate at A and A' (Fig. 28) ; they form a system of 

 revolution in reference to the straight line joining them. The total 

 flow which traverses any zone of revolution whose semi-arc is PP', 

 is the sum of the flows which correspond to the angles to and to' for 

 the two masses separately; that is to say n + n'=N. 



For the same reason as above, the sheet which corresponds to 

 the flow of the value N, and which passes through the point P is 

 tangential at A to the cone whose angle is 20, which comprises the 

 same flow for the mass m taken separately. 



This sheet is also an asymptote to a cone, the apex of which is 

 the centre of gravity O of the two masses. 



The equation of the line of force AP, that is to say 72 + #'=N, 

 gives 



2irm (i - cos o>) + 2Trm'(i cos a/) = N = 27rm(i cos 0), 

 whence 



(8) m cos to + m' cos a/ = m' + m cos 9. 



If the two angles in this equation to and <o' are made equal, we 

 have the angle of the asymptote with the axis ; we thus get 



(9) (m + m')cosa = m' + mcosO. 



By eliminating the ratio , between the equations (8) and (9) we 

 have the equation of the line of force in functions of 6 and of a : 



I COS to' I COS a 



COS to COS COS a COS 



This is still a general method, and may be applied to any number 

 of centres situate on the same right line. 



The equation of a line of force starting from the mass m is 



m cos to + m' cos w' + m" cos w" = m' + m" + + m cos 0, 



and that of the asymptote 



(m + m' + m" . . . . .) cos a = m' + m" + m' + m cos 0. 



