112 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



33) ds 2 = Ei/dq^ -f E^dql -\ h E^ m dg 2 m 



where 



., F (r)_ 0*r fy dyrdyr d*r<)*r 



~ 2q s $$* dq. dq t ^ 3q s 3 q t ' 



Thus the square of each infinitesimal arc is a quadratic form in the 

 differentials of all the coordinates q. Dividing by dt 2 , denoting the 



time derivatives by accents as before, multiplying by y#V, and 

 taking the sum for all the particles, we obtain 



35) T = Q n q? + fcrfj 1 + + ft--*.* 



r = m s= 



where 



36) 



(In the double sum the factor is introduced because there occur 



both a term in Q rs and one in Q sr , both being equal.) Thus the 

 kinetic energy possesses the characteristic property mentioned above 

 of being a quadratic form in the generalized velocities q', the 

 coefficients Q rs being functions of only the generalized coordinates q. 

 They must satisfy the conditions necessary, in order that for all 

 assignable values of the #"s T shall be positive. Of the form of 

 these functions no general statement can be made. They are linear 

 functions of the masses of the particles of the system and depend 

 upon the choice of the parameters q used to denote the configuration. 

 We may call them coefficients of inertia. It is evident from 36) that 

 every Q ss is positive, for E$f is a sum of squares. If no product 

 terms occur we may by analogy with 28) call the coordinates 

 orthogonal. 



It is sometimes convenient to employ the language of multi- 

 dimensional geometry. This signifies nothing more than that when 

 we speak of a point as being in m dimensional space we mean that 

 it requires m parameters to determine its position. Inasmuch as in 

 motion along a curve, that is in a space of one dimension we have 

 for the length of arc 



on a surface, that is in a space of two dimensions, 



