ON THE FUNDAMENTAL FORMULA OF DYNAMICS. 3 



frictionless surface of a body (which it cannot penetrate, but which 

 it may leave), and is acted on by given forces. For simplicity, we 

 may suppose that the normal to the surface, drawn outward from the 

 moving point at the moment considered, is parallel to the axis of X 

 and in the positive direction. The only restriction on the values of 



Sx, 6V Sz is that -* ^ 



&e=0. 



Formula (7) will therefore give 



X Y Z 



=, y~y z= 



m m m 



The condition that the point shall not penetrate the body gives 

 another condition for the value of x. If the point remains upon the 

 surface, x must have a certain value N, determined by the form of 

 the surface and the velocity of the point. If the value of x is less 

 than this, the point must penetrate the body. Therefore, 



x^N. 



But this does not suffice to determine the acceleration of the point. 



Let us now apply formula (6) to the same problem. Since x cannot 

 be less than N, ;f jg = _ y> ^Q, 



This is the only restriction on the value of Sx, for if x > N, the 

 value of Sx is entirely arbitrary. Formula (6), therefore, requires 



that Y 



c -\T -^ -Ji- 

 ll x=N, x> ; 

 ~m 



X 



but if x> N, x = : 

 m 



that is (since x cannot be less than N), that x shall be equal to the 

 greater c 

 and that 



greater of the quantities N and ;-, or to both, if they are equal, 



.. Y .. Z 



y = . z = . 

 m m 



The values of x, y, z are therefore entirely determined by this 

 formula in connection with the conditions afforded by the constraints 

 of the system.* 



The following considerations will show that what is true in this 

 case is also true in general, when the conditions to which the system 



* The failure of the formula (7) in this case is rather apparent than real ; for, 

 although the formula apparently allows to x, at the instant considered, a value 



v- 



exceeding both N and , it does not allow this for any interval, however short. For 



m 

 if x<N, the point will immediately leave the surface, and then the formula requires 



that x = 

 m 



