1 84.] 



MOTION ON A FIXED SURFACE. 



99 



the following condition must be satisfied : 



f=f=f< " < 4) 



fV 9y 9* 



where </> x , <f> v , <f> e denote, as usual, the partial derivatives of 



x, y t z) with regard to x, y, z, respectively. 1 



If the acceleration of the particle be zero, the equations (i) 

 reduce to the conditions of equilibrium of a particle on a sur- 

 face, as given in Statics (Part II., Art. 222). 



184. A particle of mass m, subject to gravity, is constrained to 

 remain on the surface of a sphere of radius r. If the constraint 

 is produced by a weightless rod or string joining the particle 

 to the centre of the sphere, the rod or string describes a cone, 

 and the apparatus is called a conical or spherical pendulum. 



Taking the centre O of the sphere as origin (Fig. 26), and 

 the axis of z vertically down- 

 wards, we have for the equation 

 of the sphere 



whence <f>Jx= $ y /y <$> z /z. The 

 direction cosines of ./Vare x/r, 

 y/r, z/r. Hence, the equa- 

 tions of motion, as there is no 

 friction : 



Fig. 26. 



my 



mz mg 



-N*- 

 r 



(6) 



1 This abridged notation is readily extended to the second and higher derivatives : 



<t>xx = 2-2, <p xy = d ft f etc> it w ni a i so sometimes be convenient to use the fluxional 



dx dxdy 



notation for derivatives with respect to the time 



dx . dy . dz . d*x . . 



~dt~ X ' ~dt~ y ^ ~dt~ Z ' ~di*~*' 



dt* 



dr 



<tt. 

 dt 



In mechanics, this notation is of particular advantage, not only because the time 

 so often appears as the independent variable, but also because the initial values of 

 these derivatives (i.e. the components of the initial velocity and acceleration) can 

 then be indicated by zero subscripts. Thus, the components of the initial velocity 

 would be ^o> jo> 20- 



