G. Jounstonse Sronzy On Polarization Stress in Gases. 55 
of spherical harmonics of the same simple type. Doing this— 
IV?=9,49, +g2+..- 
Tih th a 
the g’sand #’s representing spherical harmonics. Whence, and from the funda 
mental property of spherical harmonics— 
¢ Fl 
G—4Bp Wi SpOu 
Pr 
k=39 f  k,(3p?—1)dp 
Hence g,, is the only term of the first series that produces any conduction of heat, 
and k, 1s 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? in those directions. We thus obtain a solid of revolution 
which may also be arrived at by plotting down radii equal to k,, and successively 
correcting the solid so found by the addition of k,, k,, &c., to its radii. Now— 
kA 
k,—B.p 
k,=C.(3u?—1), 
&e., &e. 
Where A, B, C, &c., are independent of ». In the case we are considering B, C, 
&e., 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 radi. Next, &, +k, may be plotted down by shifting the centre of this sphere 
through the small distance B towards positive x, and by then very slightly distort- 
ing the form of the sphere. Again, to plot down k,+%,, we should elongate the 
sphere in the direction of the axis « by an amount equal to 4C and narrow it equatori- 
ally by an amount equal to 2C, without shifting its centre. Finally £,+k,+h, would 
be represented by radii drawn to the surface of this last solid, after it had heen 
slightly distorted and removed through the distance B towards the cooler. Through 
these mutations the mean value of all the radii drawn from the origin remains 
unaltered.* 
Comparing these figures with expansion (12), which is the value for LV furnished 
by Clausius’s hypothesis, we find that the form and position of the solid which 
* Since, by the fundamental property of spherical harmonics, 
+1 
fr tau=0 
“1 
+ 
if kdp=0 
=i 
&e, 
