1894.] MaxiuelVs Law of Partition of Energy. 251 



and the kinetic energy takes the form of a sum of squares, viz. 



The angular velocities co^, w^, w^ and the corresponding angular 

 momenta Aca^, Bw^, Gco^ are proportional to what Boltzmann in 

 his important paper "On the Equilibrium of Vis Viva"* calls 

 " momentoids " of the body, and they cannot be regarded as true 

 generalised velocities or momenta to which Lagrange's equations 

 are directly applicable. 



Now let Ilj, Ilg' ^s denote the initial angular velocities of the 

 body about its principal axes, w^, w^, (o^ being its angular velocities 

 at any time t. Then w^, co^, eog are functions of 12^, H^j ^s '^^^ ^' 

 and we may readily show that the Jacobian 



8(0)^, 6),, ft?3) ^ 



the differentiations being made on the supposition t = constant. 

 3. To prove this relation we have 



dt \d (a^, o„ Hg)! 8(fi„ n„ n,) "^ d (a,, n„ nj "^ d (ii„ n„ n^) • 



But by the equations of motion (1), 



8 (d)^, ft),, 6)3) ^„ . V + (^ —2 ^ 



-(5-c)2,±(^«.8n^ + «3afi;' an/ 8O3 



If this determinant be written in full, with the operators 



8n/ ana' aOg 



arranged in columns, the first row is equal to co^ times the second 

 row plus ft), times the third, therefore the determinant vanishes, or 



8(^,a„03) ^• 



Therefore - | ^K>^2 .3l I = q 



ineretore dt\d{Q.„a^,a^)] "' 



a^£,'a;>^lr ''"'''^' 



= 1, 



since its initial value is unity. 



* Translated in the Phil. Mag. for March, 1893. 

 VOL. VIII. PT. IV. 19 



