418 Mr. G. J, Stoney on Polarization Stress in Gases. 



a, #, y being the director angles of the element of solid angle 

 da. Introducing polar coordinates, we have 



d<r= tin 6 d6d<j>, 

 cosa = cos 6, 

 cos/3 = sin #cos <£, 

 cos 7= sin 0sin <£, 



by which the expressions for the stresses become 



P, = /- I \ IV 2 cos 2 6 sin 6 d6 d<f>, 



47r Jo Jo 



F=-f-\ I IV 2 sin 3 6 cos 2 'QdBdfa r. . (E) 



47r Jo Jo v f 



\ IY 2 sin 3 6 sin 2 cf> dd dcf>. \ 



These are the most general expressions for the stresses in 

 three rectangular directions within gas polarized in any way ; 

 and they will be the only stresses between portions of the gas 

 separated by planes parallel to the planes yz, zx, xy, if the axes 

 are so chosen that there are no moments round them arising 

 from the molecular encounters*. 



21. This condition is easily secured in the case which we are 

 investigating, viz. when heat is making its way between a 

 heater and cooler that are parallel to one another, and of large 

 extent compared with the interval between them, since the 

 polarization of the intervening gas will evidently be disposed 

 symmetrically round the direction in which the heat is travel- 

 ling. Hence, taking this direction as our axis of x, there can 

 be no moments round this axis, or round any axis at right 

 angles to it. The stresses (E), therefore, are the only ones to 

 be taken into account. Moreover we can integrate equations 

 (E) at once by <£, since IV 2 is, in this simple case, a function 



* Equations (E) cannot be integrated unless IV 2 is given as a function 

 of 6 and <p, i. e. unless the law of polarization in the gas is known. But 

 they show that in general the stresses in different directions are unequal, 

 which is here what is chiefly insisted on. 



When the gas is unpolarized, I becomes equal to unity, and V 2 is in- 

 dependent of the direction, and may therefore be put outside the integrals. 

 In this case all three equations concur in giving the well-known expres- 

 sion for the stress in unpolarized gas, viz. |pV 2 . 



