56] WORK AND ENERGY. 105 



equation (10) is equivalent to the equation (9), for since the 

 quantities Bx, by, Bz, are arbitrary, if X, Y, Z, are different from 

 zero, we may take Bx, By, Bz 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 Bx, By, Bz, 

 X, Y, Z must vanish. 



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

 <t> 0, So;, by, Bz are not entirely arbitrary, but must satisfy 



(7) 



dx dy ' dz 



Let us multiply this by a quantity X and add it to (10), 

 obtaining 



CD 



We may no longer conclude that the coefficients of Bx, By, Bz 

 must vanish, for Bos, By, Bz are not arbitrary, being connected by 

 the equation (7). Two of them are however arbitrary, say By and 

 Bz, X has not yet been fixed suppose it determined so that 



Then we have + X By + z + X Bz = 0, 



in which By and Bz are perfectly arbitrary, it therefore follows of 

 necessity that the coefficients vanish. 



, 



dy 



By the introduction of the multiplier X we are accordingly 

 enabled to draw the same conclusion as if Bx } By, Bz were arbitrary. 

 Eliminating X from the above equations we get 



dy dz 



Now the direction cosines of the normal to the surface <f> = 



are proportional to -^ , ~, ^ , consequently, the components 



ox oy oz 



X, F, Z being proportional to these direction cosines, the resultant 



