70S 



IN RARIFIED GASES 



not, however, attempted to enter into the calculation of the state of 

 Ceady motion. 



[I have recently applied the method of spherical harmonics, as described 

 in the note to auctions (l) and (5), to carrying the approximations two 

 onkt* higher. I expected that this would have involved the calculation of 

 two new quantities, namely, the rates of decay of spherical harmonics of the 

 fourth and sixth orders, but I found that, to the order of approximation 

 required, all harmonics of the fourth and sixth orders may be neglected, so 

 that the rate of decay of harmonics of the second order, the time-modulus 

 of which is /* + /', determines the rate of decay of all functions of less than 

 6 dimensions. 



The equations of motion, as here given (equation 55) contain the second 

 tlwivatives of v, v, w, with respect to the coordinates, with the coefficient p.. 

 I find tliat in the more approximate expression there is a term containing 

 the fourth derivatives of u, v, w, with the coefficient p-' + pp. 



The equations of motion also contain the third derivatives of with the 

 coefficient p? + p9, Besides these terms, there is another set consisting of the 

 fifth derivatives of 6 with the coefficient p.* -=- 



It appears from the investigation that the condition of the successful use 

 of this method of approximation is that Z-j, should be small, where -^ denotes 



differentiation with respect to a line drawn in any direction. In other words, 

 the properties of the medium must not be sensibly different at points within 

 a distance of each other, comparable with the "mean free path" of a molecule. 

 Note added June, 1879.] 



