SOLID* ON SllUWCES. 15 I 



When the surfaces are considered as perfectly unooth, we 

 have seen that there are as many equations of condition as 

 there are surfaces of support to be taken in conjunction with 

 the general dynamical equation. Multiplying each of these 

 equations by an indeterminate coefficient and equating to 

 nought the sums of the coefficients of the variations, there re- 

 sults (26) 



d>^-+-SXI)»i-hZ«L = o. 

 ,/ r -+- SYDm -+- %\)M = o . 

 d $ .-t-SZUm + zofy -= o; 



L' + SiZy — V: )Dm -\-ze(N,y—Mz.) = o. 

 V -h S( Xz —Z x XDm -+- 20(L, z , — Nx) = o , 

 W-^S(Yx l — Xij)Dm^rZQ{M i x—L i y) = o: 



where 2 denotes the sum of similar quantities. 6 one of the 

 indeterminate coefficients, the mass at the same time being 

 put equal to unity. 



These equations are evidently the same as those which 

 would have been obtained immediately by substituting in 

 place of the surfaces unknown forces acting constantly in the 

 direction of the normals at the variable points of contact, and 

 then considering the system as free. The equations of con- 

 dition however would still have been indispensable, in order 

 to supply the number of equations lost in the elimination of 

 the unknown forces of reaction. I should also on other ac- 

 counts have preferred investigating these equations by the 

 preceding method; because it furnishes a variety of formulas 

 useful in the analytical geometry of touching surfaces, and 

 extremely convenient in the determination of the motions ol 

 bodies subject to a friction producing some assignable relation 

 between their sliding and their rolling motions. 



If we restrict ourselves to the examination of the motion 

 on a single surface, the body being acted on by common gra- 

 vity g, the preceding formulas become, (reckoning the positive 



