Equilibrium of Vis Viva, 

 determinant ® is equal to unity, that is 



c n c 12 . . 



155 



^21 



= 1 



and (2) the doubled kinetic energy takes the form 



1=1 



In this investigation the coefficients c and /uu are functions 

 of the generalized coordinates b l . . . b ny bat the a's cannot in 

 general be regarded as momenta which belong to any system 

 of generalized coordinates whatever. I will therefore call 

 them "momentoids" in order to prevent confusion. 



The angular velocities about the three principal axes of 

 inertia of a solid in the most general case of rotation furnish 

 an example. 



I shall denote by ^/^a* 2 that portion of the whole vis viva 

 which belongs to the momentoid tx ( . Since = 1, we have 

 da ± , da 2 . . . da n =dot l , du 2 . . . da n . Let us insert the variable T 

 in place of a 1 on the left and of ol x on the right, then 



dT 

 da x 



c/T da 2 da 3 . . . da = -^ dT da 2 da z 

 n ell 



doi. v 



. . da,. 



Dividing by dT and observing that 





da-. 



= ^1*U 



we obtain 



— da 2 da z . . . da n = - — da 2 du 3 . . . da n . 



From Maxwell's equation (28) (I c. p. 721) we find for 

 the number of systems whose generalized coordinates lie 

 between the limits b ± and b 1 + dbi,,\b n and b n -\-db n , and 

 whose momentoids lie between a 2 and a 2 + doi 2 • • • a « anc ^ 

 a n + da n (a 2 being determined from the equation of w viva) 

 the value 



NC Al 11 1 



ao L . . . ab n atxo . . . du n . 



If we now proceed with the integration exactly as Maxwell 

 does, we arrive at his equation (-15), which is thus perfectly 

 correct. 



By calculating the probability that the vis viva hfin»n 



M2 



