358 Mr. Rankine on the Hypothesis of Molecular Vortices, 



]MV is the volume of the atom, 



/* 1 

 ■Tjr^ the mean density of the atmospheric part, measured by 



weight, the nucleus being supposed to be of insensible 

 magnitude, 

 and we have the following equations : — 



The suffix (,) denoting that the integration is to be extended to 

 all points within the surface 



(^-4)1 = 0). 



According to the hypothesis now under consideration, heat 

 consists in a revolving motion of the particles of the atomic 

 atmosphere communicated to them by the nuclei. Let v be the 

 common mean velocity possessed by the nucleus of an atom and 

 the atmospheric particles, when the distribution of this motion 

 has been equalized. I use the term mean velocity to denote that 

 the velocity of each particle may undergo small periodic changes, 

 which it is unnecessary to consider in this investigation. 



Then the quantity of heat in unity of weight is 



being equal to the mechanical power of unity of weight falling 



through the height pp. The quantity of heat in one atom is of 



coiu'se MQ, and in the atmospheric part of an atom /xQ. 



I shall leave the form of the paths described by the atmo- 

 spheric particles indeterminate, except that they must be closed 

 curves of permanent figure, and included within the surface 

 ($ — ^j = 0). Let the nucleus be taken as the origin of coordi- 

 nates, and let «, /3, 7 be the direction- cosines of the motion of 

 the particles at any point {x, y, z). Then the equations of a 

 permanent condition of motion at that point are 



1 dp' d^ „ / d ^ d d\ _ - 



p dx dx \ dx dy dz) 



1 dp' d<^ ^r^( d ^ d d\^ ^ 



—p'i-dH-^'^Wx'^^Ty^'^l^F-'' ^' (^) 



_1 ^'_^_2o('«i- B — -¥ — "i =0 

 p dz dz \ dx dy ' dz) 



Let r be the length, and «', y8', y' the direction -cosines of the 



