360 ON THE MOTION OF 



CP(dL, -hMdR — NdQ) 

 d.(LF-+-M,Q + N,R) = \ Q (dM, + NdP — L dR) 



(R{dN, + L,dQ — MdP). 



These equations are true of all supporting and supported sur- 

 faces whatever. It might easily be shown that this last equa- 

 tion is capable of being derived from the principle that the 

 rate of increase of the sum of all the areas projected on the 

 plane tangent to the point of variable contact is momentarily 

 constant, the tangent plane being supposed to remain for a 

 moment fixed while the body passes on to its consecutive po- 

 sition on the surface of support. 



When the sustaining surface is an inclined plane. L. L. L 

 become constant, and the right member of the last equation 

 will vanish on the substitution of the values which Z ; , M„ N 

 acquire in such a case, so that the equation becomes integrable 

 with respect to time, and we obtain 



LP ^-MQ+NR = /, 



/ being an arbitrary constant. 



Again, if Ave add together the second triplet (27), after 

 multiplying the three equations respectively by dP, dQ. dR. 

 and reduce by means of the equation of condition (22). we 

 obtain 



Udv -t- vdQ-h mm — {Ldi + L'dg -+- LW) = o . 



Substituting for L, L, L" their values (20), and performing 

 the operation indicated in the first three terms of this equa- 

 tion, there will result 



dT-t- d^Vi + dgd'g + dZ'd V — gd%" = o . 

 an equation whose integral gives us the principle of living for- 



