794 PROFESSOR G. G. STOKES ON THE COEFFICIENT OF VISCOSITY OF AIR. 
When the modulus of ma is small, it is rather more convenient to expand (17) 
according to ascending powers of ma. This may be done by actual division, or more 
conveniently by assuming a series with indeterminate coefficients, and using the non¬ 
linear differential equation of the first order in z obtained from (9) by putting 
f\( r )—z/i(r). Carried as far as to a 12 , the development is 
m 2 a~ m i a i 
96" 
m 6 «. 6 
1536 
m 8 a 8 
23040+ 
13m 10 a 10 
4423680 
11 m 12 W 
55050240 ? 
and, denoting the modulus of ma by f and taking the real part, we have 
Nap. log. dec. 
2My f / 
MK 2 196 
P , ll / 12 
23040“*“55050240 
(19) 
This series must not be used when f is at all large, as the convergence is too slow, 
and, as appears by a theorem due to Cauchy, it becomes actually divergent when 
f— 3‘340' 1 ' nearly, whereas the series in (17) are always convergent, and when f has 
the above value converge rapidly. 
When /is decidedly large the series in (17), though ultimately convergent, begin 
by diverging, so that the calculation is troublesome, and moreover my Table giving k 
and k' is not carried beyond /= 8, as the calculation by a different method then 
becomes very easy. In this case we should employ the integral of (9), which is of the 
form e~ mr or e mr multiplied by a descending series. The former exponential only will 
come in when we are treating of the external air, and the latter only when of the 
internal. 
For the external air the integral is of the form 
/(r) = Be mr r 1 
1.3 
l.(8mr) 
1 2 .3.5 
1.2 (8m?*) 2 
1 2 .3 2 .5.7 
1.2.3(8?n?’) 3 
( 20 ) 
the signs being alternately + and —, and the new factors in the numerator being two 
less and two greater than the last factor in the term before. We get from (12), (20), 
and the expression for the logarithmic decrement in terms of T and the moment of 
inertia, 
* The square root of the smallest real root of the equation 
x + x 2 
274 2A46 
. . = 0 . 
The series would have become divergent still earlier if the equation just written bad had an imaginary 
root with a modulus smaller than 3’340 . . . 2 . 
