20 Mr. G. F. Fitzgerald on the Mechanical 



for it is evident that the pressures in the three directions can- 

 not be equal unless b± and b 2 both vanish,. which will not in 

 general be the case. Without a knowledge of the nature of 

 the distribution of the velocities and numbers of molecules 

 moving in the different directions, it would be impossible 

 to calculate the values of b l7 b 2 , b 3 , 6 4 , and b 5 ; but I think 

 we can see that they will in part at least vary as the square 

 of the quantity of heat passing. This can be seen from 

 the following considerations. ISTo matter what the distri- 

 bution of the velocities and numbers of molecules moving in 

 the different directions may be, it is plain that terms occur- 

 ring in the coefficients of V 1 — jjl 1 sin . <£\/ 1— /a 2 cos <p (i. e. 

 in the spherical harmonics of the first order in u and v) will 

 occur in the terms of the same order in nv nv 2 and nv*, and 

 that linearly ; while these same terms will occur squared 

 in the spherical harmonics of the second order in nv nv 2 and 

 nv z . Hence we see that terms occurring linearly in the sphe- 

 rical harmonics of the first order in nv s will occur as squares 

 in the spherical harmonics of the second order in nv 2 ; so that 

 bi, b 2 will contain c l7 c 2 , and c 3 in the second degree, i. e. will 

 contain terms varying as the squares of the quantities of heat 

 passing. It is also to be observed that terms occurring in the 

 spherical harmonics of the second order can never come into 

 those of the first, except as products with terms belonging to 

 spherical harmonics of the third order ; so that a hypothetical 

 distribution which gave correct values for the quantities of 

 heat passing might very well be quite inadequate as a means 

 of calculating the difference of pressure in different directions. 

 This remark is of importance when we come to consider the 

 results of Clausius' hypothesis, and was suggested to me by 

 Mr. Stoney in conversation. 



As an example of what I am insisting upon, we may take 

 two opposite extreme cases : — first, the case of B 2 vanishing, 

 and, secondly, the case of C 1 doing so. In the first case there 

 would be a distribution of velocities and numbers such that, 

 though heat would be conducted across the layer, nevertheless 

 there would be no resultant inequality of stress ; while in the 

 second case, though no heat would be conducted, yet there 

 would be inequality of stresses. It seems, however, certain 

 that neither of these extreme cases can exist as a permanent 

 distribution in gases. Before calculating the values of these 

 quantities upon particular hypothetical distributions, it may 

 be well to see what they are in the simple case of two parallel 

 planes, each at a uniform temperature. 



In this case it is evident from symmetry that, if we take X 

 normal to the planes, we must have all our equations indepen- 



