380 



Lord Kelvin 



[April 27, 



any point J between U^ and U 4 , or between U 2 and TJ 3 , it will never 

 cut the principal section containing the umbilics, either between JJ 1 

 and U 2 or between U 3 and U 4 . This for the extreme case of C C =0 

 is illustrated in Fig. 3. 



§ 33. It seems not improbable that if the figure deviates by ever 

 so little from being exactly ellipsoidal, Maxwell's condition might be 

 fulfilled. It seems indeed quite probable that Maxwell's condition 

 (see §§ 13, 29, above) is fulfilled by a geodetic on a closed surface 

 of any shape in general, and that exceptional cases, in which the 

 question of § 29 is to be answered in the negative, are merely par- 

 ticular surfaces of definite shapes, infinitesimal deviations from which 

 will allow the question to be answered in the affirmative. 



§ 34. Now with an affirmative answer to the question — is Max- 

 well's condition fufilled ? — what does the Boltzmann-Maxwell doctrine 

 assert in respect to a geodetic on a closed surface ? The mere 

 wording of Maxwell's statement, quoted in § 13 above, is not applic- 

 able to this case, but the meaning of the doctrine as interpreted from 

 previous writings both of Boltzmann and Maxwell, and subsequent 

 writings of Boltzmann, and of Kayleigh,* the most recent supporter 

 of the doctrine, is that a single geodetic drawn long enough will not 

 only fulfil Maxwell's condition of passing infinitely near to every 

 point of the surface in all directions, but will pass with equal 

 frequencies in all directions ; and as many times within a certain 

 infinitesimal distance ± 8 of any one point P as of any other point P' 



Phil. Mag., January 1900. 



