108 KINETICS OF A PARTICLE. [200. 



position of the surface at a given time t. In other words, to 

 obtain a displacement Ss compatible with the condition (8), 

 its components &r, By, Sz should satisfy the condition 



<k&T+<k^ + <k&? = o, (9) 



obtained by differentiating the equation (8) with respect to 

 the co-ordinates. 



200. By means of the relation (9), one of the displacements 

 &r, By, Sz can be eliminated from the general equation of 

 motion (2); the two remaining displacements will then be 

 independent, and their coefficients can therefore be equated 

 to zero separately. 



The elimination is again conveniently effected by the method 

 of indeterminate multipliers. Multiplying equation (9) by an 

 indeterminate factor X, and adding it to equation (2), we find 

 the single equation 



in which the arbitrary quantity X can be so selected as to make 

 the coefficient of one of the three displacements vanish. The 

 other two displacements being arbitrary, their coefficients must 

 also vanish. The last equation can therefore be replaced by 

 the following three : 



my = Y+\$ y , mz = Z+\<}> t , (10) 



which, in connection with the given condition (8), fully deter- 

 mine the problem ; for they are sufficient for finding x, y, z, and 

 X as functions of t. 



201. Just as in Art. 195, it follows that the components of 

 the reaction N of the surface are 



whence 7V r = xV<#> x 2 + ^ 2 + <#> z 2 . (n) 



