Mr. G. J. Stoney on Polarization Stress in Gases. 421 



The general formulae for the cond action of heat and for the 

 polarization stress are the following — 



=yp 



IY 2 .(Sfi 2 -l)dn 



(see Clausius' memoir, p. 514, and equation (Gr) above). 

 Now fM and 3yW 2 — 1, which occur as factors in these integrals, 

 are the first and second terms of a series of spherical harmonics 

 (Laplace's coefficients) of the simple kind that are functions 

 of [m only, and which therefore represent the radii of solids of 

 revolution from points on their axes. It is moreover obvious 

 that we can expand IV 2 and IV 3 in series of spherical harmo- 

 nics of the same simple type. Doing this, 



lY 2 = k Q + k 1 + k 2 + ... y 



the #'s and &'s representing spherical harmonics. Whence, 

 and from the fundamental property of spherical harmonics, 



fc=lp\ k 2 (3{ju 2 — l)d{i. 



Hence g x is the only term of the first series that produces any 

 conduction of heat, and k 2 is the only term of the second series 

 that produces any polarization stress. 



Let us suppose radii drawn from a point in all directions, of 

 lengths proportional to the values of IV 2 in those directions. 

 We thus obtain a solid of revolution which may also be arrived 

 at by plotting down radii equal to k Q , and successively correct- 

 ing the solid so found by the addition of k ly k 2 , &c. to its radii. 

 Now 



k =A, 



* 2 = C.(3/* 8 -l), 



&c. &c. 



where A, B, C, &c. are independent of //,. In the case we are 

 considering, B, 0, &c. are small compared with A. From the 

 foregoing values it follows that if k is plotted down by itself, 

 it will produce a sphere with its centre at the origin of radii. 



