I9i.] MOTION ON A FIXED SURFACE. IC >3 



(4) From the equations (5) and (6), Art. 184, derive the approxi- 

 mate path of the bob of a simple pendulum when the oscillations are 

 very small. 



191. Motion on Any Fixed Surface without Friction. The posi- 

 tion of a point on a surface can always be determined by two 

 variables, say q lt q^. Thus, on a sphere, the latitude and longi- 

 tude of a point determine its position ; and on any surface the 

 two systems of curves known as the curves of curvature of 

 the surface might serve as a system of co-ordinates. In other 

 words, the motion of a point on a surface is really a problem 

 of motion in two dimensions, just as the motion on a curve 

 takes place in one dimension (Art. 180). 



Let *=fi(ffi, 92)>y=A(4i> ft). *=/8(ft ft) ( l8 ) 



be the equations of the given surface, so that the elimination of 

 q-^y q<i from these equations would give the ordinary equation 

 < (x, y, z)=o of the surface. Then we have for the velocity 

 v the expression 



Kf 



If there be no friction, the equation of kinetic energy gives 



or say 



(20) 



If the forces depend only on the position of the particle, Q l 

 and <2 2 are functions of q lt q^ alone ; if, moreover, the expres- 

 sion Qidq^+Q^dq^ is an exact differential of a function of q^ and 

 # 2 , say dU(q^ q%), the equation (20) gives at once a first integral 



q^)+h. (21) 



