

IS RAIUFIED OASES 



[Application f Spherical Harmonics to the Theory of Gases. 



If we suppose the direction of the velocity of M, relative to M t to be 

 d by the position of a point P on a sphere, which we may call the 

 of reference, then the direction of the relative velocity after the encounter 

 will be indicated by a point / x , the angular distance PP being 20, so that 

 the point I" lie* in a small circle, every position in which is equally probable. 



We hare to calculate the effect of an encounter upon certain functions of 

 the six velocity-components of the two molecules. These six quantities may 

 be expressed in terms of the three velocity-components of the centre of mass 

 of the two molecules (say v, v, w), the relative velocity of Jf, with respect 

 to J/, which we call V, and the two angular coordinates which indicate the 

 direction of V. During the encounter, the quantities u, v, w and V remain the 

 Mine, but the angular coordinates are altered from those of P to those of P on 

 the sphere of reference. 



Whatever be the form of the function of 17, , ,, &, rj,, ,, we may 

 consider it expressed in the form of a series of spherical harmonics of the 

 angular coordinates, their coefficients being functions of u, v, w, V, and we 

 have only to determine the effect of the encounter upon the value of the 

 spherical harmonics, for their coefficients are not changed. 



Let }' <-) be the value at P of the surface harmonic of order n in the 

 series considered. 



After the encounter, the corresponding term becomes what I^"' becomes 

 at the point P, and since all positions of P in a circle whose centre is P 

 are equally probable, the mean value of the function after the encounter must 

 depend on the mean value of the spherical harmonic in this circle. 



Now the mean value of a spherical harmonic of order n in a circle, the 

 cosine of whose radius is ft, is equal to the value of the harmonic at the pole of 

 the circle multiplied by P"> (p.), the zonal harmonic of order n, and amplitude p.. 



Hence, after the encounter, 7<" becomes P>P<"> (^), and if F n is the 

 corresponding part of the function to be considered, and SF n the increment of 

 F m arising from the encounter, oF u = F n (P M (/i)-l). 



This is the mean increment of F n arising from an encounter in which 

 cos 2$ = /t. The rate of increment is to be found from this by multiplying it 



