35, 36] PARAMETRIC REPRESENTATION OF SURFACE. 109 



energy in this case involves both the coordinates # and (p and the 

 velocities &' and <p'. Inasmuch as the particle in any given position 

 may have any given velocity, the variables #, qp, #', cp' are to be 

 considered in this sense as independent, although in any given 

 actual motion they will all be functions of a single variable t. 



The form of the function T is worthy of attention. It is a 

 homogeneous quadratic function of the velocities & and <p' ', the 

 coefficients of their squares being functions of the coordinates #, (p, 

 the product term in &' cp' being absent in this case. We may prove 

 that if a point moves on any surface the kinetic energy is always 

 of this form. 



In the geometry of surfaces it is convenient to express the 

 coordinates of a point in terms of two parameters q and q%. Suppose 



a = fi (& > &)> y = f* (ft > 0a) * = /3 (0i, a )> 



from these three equations we can eliminate the two parameters q l9 q 2 , 

 obtaining a single equation between x, y, s, the equation of the 

 surface. The parameters q and q 2 may be called the coordinates of 

 a point on the surface, for when they are given its position is 

 known. If q^ is constant and q 2 is allowed to vary, the point x, y y s 

 describes a certain curve on the surface. This curve changes as we 

 change the constant value q . In like manner putting q 2 constant 

 we obtain a family of curves. The two families of curves, 



q l = const, q 2 = const, 



may be called parametric or coordinate lines on the surface, any 

 point being determined by the intersection of two lines, for one of 

 which q^ has a given value, for the other, g 2 . 



We may obtain the length of the infinitesimal arc of any curve 

 in terms of q 1 and g 2 . We have 



dx i , dx -. 



25) 



Squaring and adding, 



26) ds* = dx* + df + dz* = Edq^ + ^Fdq, dq 2 





