Criterion of Potential Energy* 697 



similar to that followed in Kelvin and Tait^may be reduced 

 to the form in which they are obtained in that treatise, 

 namely. 



+ «'{(f-|f)*H^-i)»-} + ••• + i >n!, 



Here M, N, 0, . . . M\ N', 0', . . . are best defined for our 

 purpose by the relatious 



X=|5-(MVr+N0 +00 +...) = |q +*, (say)'] 

 x'=|§-(M'f+N'^ + 0'5+...)=|§-+*' 1 . I 



16. It is clear that (12) are precisely the equations of 

 of motion of a system in which *C is the kinetic and K the 

 potential energy, ^provided that for all values of yfr, <£, Q, . . . 



*{(f-l->($-!>"}-°.] 



^{(i-f)H(i-i)^...}-», . ( H) 



Let m be the number of the working coordinates t|t, <£, 0, . . . : 

 then since in each of the m equations (14) the coefficients of 

 ty, <p, f), ... will have to vanish separately, and since -v^ is 

 absent from the first equation, <£ from the second, and so on, 

 we obtain m(m— -1) equations which must be satisfied if K is 

 to be treated as the potential energy of the system. These 

 conditional equations, however, are not all independent, but 

 consist of hn(m — 1) independent equations, each occurrino- 



