86 Sir W. Thomson. On the Maxwell-Boltzmann [June 11, 



where V is the potential energy, which may be any f unction of x, y, 

 subject only to the condition (required for stability) that it is 

 essentially positive (its least value being, for brevity, taken as zero). 

 It is easily proved that, with any given value, E, for the sum of 

 kinetic and potential energies there are two determinate modes of 

 periodic motion ; that is to say, there are two finite closed curves 

 such that if m be projected from any point of either with velocity 

 equal to v/[2(E V)] in the direction, eitherwards, of the tangent to 

 the curve, its path will be exactly that curve. In a very special class 

 of cases there are only two such periodic motions, but it is obviour 

 that there are more than two in other cases. 

 13. Take, for example, 



1 d y = 1 



/ X = I COS (/-/) / 



For all values of E we have 

 x = a cos (a*-e 



y = / X = I COS (/-/) 



as two fundamental modes. When E is infinitely small we have 

 only these two; but for any finite value of E we have clearly an 

 infinite number of fundamental modes, and every mode differs 

 infinitely little from being a fundamental mode. To see this let m be 

 projected from any point N in OX, in a direction perpendicular to 

 OX, with a velocity equal to v / (2E-a 2 ON 2 ). After a sufficiently 

 great number of crossings and re-crossings across the line X'OX, the 

 particle will cross this line very nearly at right angles, at some point, 

 N'. Vary the position of N very slightly in one direction or other, 

 and re-project m from it perpendicularly and with proper velocity ; 

 till (by proper "trial and error" method) a path is found, which, 

 after still the same number of crossings and re-crossings, crosses 

 exactly at right angles at a point N", very near the point N'. Let m 

 continue its journey along this path and, after just as many more 

 crossings and re-crossings, it will return exactly to N, and cross OX 

 there, exactly at right angles. Thus the path from N to N" is exactly 

 half an orbit, and from N" to N the remaining half. 



14. When cE/(<* 2 /i 2 ) is a small numeric, the part of the kinetic 

 energy expressed by ^cx 9 y~ is very small in comparison with the total 

 energy, E. Hence the path is at every time very nearly the resultant 

 of the two primary fundamental modes formulated in 13 ; and an 

 interesting problem is presented, to find (by the method of the 

 " variation of parameters ") a, e, 6, /, slowly varying functions of t, 

 such that 



x = a sin ( e), y = b sin (fit /), 



" = a cos (at e), y = I fi cos (fitj), 



