25] PRINCIPLE OF VIRTUAL WORK. 59 



If we multiply these equations respectively by the arbitrary small 

 quantities dx, dy, dz, and add, we get 



10) Xdx + Ydy + Zdz = 0, 



which states that the work done in an infinitesimal displacement of 

 a point from its position of equilibrium vanishes. The equation 10) 

 is equivalent to the equations 9), for since the quantities dx, dy, dz 

 are arbitrary, if X, Y, Z are different from zero, we may take 

 dx, dy, dz respectively of the same sign as X, Y, Z, -- each product 

 will then be positive, and the sum will not vanish. If the sum is 

 to vanish for all possible choices of dx, dy, dz, X, Y, Z must vanish. 

 If the particle is not free, but constrained to lie on a surface 

 cp = 0, dx, dy, dz are not entirely arbitrary, but must satisfy 



Let us multiply this by a quantity A and add it to 10), obtaining 



We may no longer conclude that the coefficients of dx, dy, dz 

 must vanish, for dx, dy, dz are not arbitrary, being connected by 

 the equation 7). Two of them are however arbitrary, say dy and dz, 

 I has not yet been fixed - - suppose it determined so that 



Then we have 



in which dy and dz are perfectly arbitrary, it therefore follows of 

 necessity that the coefficients vanish. 



By the introduction of the multiplier A we are accordingly 

 enabled to draw the same conclusion as if dx, dy, dz were arbitrary. 

 If X, Y, Z refer to the resultant of the impressed forces only, not 

 including the reaction, equations 9) do not hold, but if we suppose 

 10) to hold, we shall obtain the conditions for equilibrium. Elimin- 

 ating A from the above three equations we get 



X Y Z 



dtp drp 



dx dy dz 



Now the direction cosines of the normal to the surface qp = 



O r\ r\ 



are proportional to **r-> 7p? -^-j consequently, the components X, Y, Z 

 being proportional to these direction cosines, the resultant is in the 



