I0 6 KINETICS OF A PARTICLE. [195. 



These equations (4), in connection with the two conditions (i), 

 are sufficient to determine the five quantities x, y, z, X, /JL as 

 functions of the time; the values of x, y, z so found give the 

 position of the particle at any time, while X, p can be shown 

 to determine the pressure on the curve. 



195. To find the reaction A^ of the curve, let us compare the 

 equations (4) with the equations of Art. 161. It appears at 

 once that the forces X\ V, Z' that would replace the condi- 

 tions (i), i.e. the components of the reaction A 7 " of the constrain- 

 ing curve, are 



whence 



. (5)- 



These equations determine the magnitude and direction of the 

 reaction N, as soon as X and /JL are found. 



196. Let us now combine the equations (4) according to the 

 principle of kinetic energy ; that is, multi ply them by dx, dy, 

 dz, and add. The left-hand member becomes, of course, the 

 exact differential d(^mv^). The right-hand member, 



will in general contain terms depending on the reaction of the 

 surface; in other words, in the actrial displacement ds = (d^ + 

 .dy^+dz*)^ of the particle the reaction of the moving curve will 

 in general do work. 



197. Only in the particular case when the curve is fixed will 1 

 the work of the reaction be zero; for in this case the condi- 

 tional equations (i) do not contain the time explicitly, and their 1 

 complete differentiation gives the relations 



z = o, r x dx + dy + ^r z dz = O, 



