1891.J Doctrine regarding Distribution of Energy. 87 



shall be the rigorous solution, or a practical approximation to it. 

 Careful consideration of possibilities in respect to this case [cE/( 2 /3 2 ) 

 very small] seem thoroughly to confirm Maxwell's fundamental 

 assumption quoted in 11 ; and that it is correct whether cE/(<* 2 /3 2 ) 

 be small or large seems exceedingly probable, or quite certain. 



15. But it seems also probable that Maxwell's conclusion, which 

 for the case of a material point moving in a plane is 



Time-av. x 2 = Time-av. if .............. (1) 



is not true when 2 differs from /3 2 . It is certainly not proved. No 

 dynamical principle except the equation of energy, 



i(M-y') = E-V ................... (2), 



is brought into the mathematical work of pp. 722 725, which is 

 given by Maxwell as proof for it. Hence any arbitrarily drawn 

 curve might be assumed for the path without violating the dynamics 

 which enters into Maxwell's investigation ; and we may draw curves 

 for the path such as to satisfy (1), and curves not satisfying (1), 

 but all traversing the whole space within the bounding curve 



=E ................ (3), 



and all satisfying Maxwell's fundamental assumption ( 11). 



16. The meaning of the question is illustrated by reducing it to a 

 purely geometrical question regarding the path, thus : calling 9 the 

 inclination to x of the tangent to the path at any point xy, and q the 

 velocity in the path, we have 



x = q cos 0, y=q8in& ............ (4), 



and therefore, by (2) q = v/{2(E-V)} ................ (5). 



Hence, if we call s the total length of curve travelled, 



J &dt = \ q cos 2 qdt = J ^{2(E- V)} cos 2 Q ds ..... (6) ; 



and the question of 15 becomes, Is or is not 



n 2 0? .. (7), 



J_ 



S Jo 



where S denotes so great a length of path that it has passed a great 

 number of times very near to every point within the boundary (3), 

 very nearly in every direction. 



17. Consider now separately the parts of the two members of (7) 

 derived from portions of the path which cross an infinitesimal area 

 ia having its centre at (a;, y). They are respectively 



