42 II. PARTICULAR MOTIONS OF A POINT. 



OKX ^ _ dy _ dcpdx dcpdy .8cp dz dcp 



= ~ 



d*^ (d#\* ^ /^jA 2 . <^_ /^\ 2 , <PV 



dx*\dt) 1r dy* \dt) 1f ~ dz* \dt) 1 " a* 8 



, o __^ 

 ' 



cp dy 



__^ 



dxdydtdt' dydz dt dt 

 , g 2 qp <?a; , g 2 qp c?y , 



^ H ^^ 



If we put the unknown forces, X 1; Y 1; Z 1; equal to an unknown 

 function A multiplied by certain known functions, by inserting the values 



d*x d*v d*z o^\ o^\ i 



01 -^y -gg) -^jj from o4) in ob) we obtaui an equation, linear in A, 



permitting us to find its value in terms of x, y, 2, t, -^t .> ^~.- 



If the surface is smooth, it is evident that it cannot affect a 

 motion of the particle which would naturally take place on the 

 surface. Consequently the reaction has no component tangential to 

 the surface, but is in the direction of the normal. This is otherwise 

 a definition of a smooth or frictionless surface. The components of 

 the reaction X lf Y^ Z are accordingly proportional to the direction 

 cosines of the normal to the surface cp = 0, so that we may write 



o 7 \ v idv v i^y 7 i^y 



f? ?-*?;? r i = ^' z i = A ar 



When A has been determined as above we have for the magnitude 

 of the reaction, 



As an example let us consider the motion of a particle acted 

 upon by gravity and constrained to move on the surface of a fixed 

 sphere of radius I. If the constraint is caused by attaching the 

 particle to a fixed point by means of an inextensible string whose 

 mass is negligible, we have the so-called ideal pendulum. The 

 equation of constraint is 



and does not contain t, so that ~ = 0. If the ^-axis be taken 

 vertically downward the equations of motion are 



