25, 26] 



D'ALEMBERT'S PRINCIPLE. 



63 



of the d's other than 0, the determinant of the coefficients must 

 vanish 



, m t , , m 2 



x, , ii :' , o , o 



r z 



\J \J Ju& j && 



= 



or reducing, 



(%#! -f m 2 a; 2 ) (iu 2 ^ 3 ojjtfg) = 0. 



The solution that applies is given by the vanishing of the first 

 factor, that is, 



m x + m x =0 



In like manner if we had assumed dx = dx 2 = 0, we should have 

 obtained 



This equation with the preceding gives by the elimination of m lt w 2 , 



Hence the points lie in a vertical plane containing the center of the 

 sphere. The two equations express the fact that a point dividing 

 the line connecting the particles in the inverse ratio of their masses 

 is vertically below the center of the sphere. The azimuth of the 

 plane containing the particles is indeterminate on account of the 

 symmetry about the vertical. 



26. D'Alembert's Principle. The equations of motion of a 

 particle may be written 



dt< 



17) 



Z r m r 



o, 



0. 



Multiplying these equations respectively by the arbitrary quantities 

 dx r , dy r , dz r , adding, and taking the sum for all values of the 

 suffix r y belonging to the different particles of a system, 



This equation may be called the fundamental equation of dynamics, 

 and is the analytical statement of what is known as d'Alembert's 



