of Electricity through Gases. 187 



2/(**+ <*) = 2U = 1- ^ + ^|^ 



_ 61 . 2 6 , g 6 1385. 2 8 . £ 8 _ 5021. 2 10 .g 1Q 

 61 8! 10! 



This series is differentiated with respect to f, and the 

 differential coefficients are expressed in terras of log q by (8). 

 It will not be necessary to give all the steps of the numerical 

 transformations. The result is the series : — 



where 



a = i + A 2 (log q y + A 4 (log qY + A 6 (log qf + . . . (9) 

 A 2 =l= -02083, 

 A.^-003038, 



A 8 =-000494, 



A 10 = -000523. 



It will be seen that A 10 is greater than A 8 , and it is impossible 

 to say how large the A coefficients may ultimately become. 

 If we compare together the values of y obtained by means of 

 (6) and the series (7) with those obtained by the series (3), 

 we find that if log q is so small that all terms higher than 

 the second in the series may be neglected, the agreement is 

 perfect ; for larger values of q, although the series seems to 

 converge, it only yields approximate values of y. The con- 

 vergence of Kirchhoff's series does not necessarily imply the 

 convergence of the series (7), and we must confine ourselves 

 therefore to the use of the latter for such values of log q only, 

 for which the series can be shown to give correct results with 

 a moderate number of terms. If the explosive distance is small, 

 we may obtain a very simple formula by a further trans- 

 formation. We call e the explosive distance, and express 

 the normal force in the form Ya/e as in (5). Let x now be 

 expanded in a series proceeding by powers of k where k 2 = e/a; 

 a being as before the radius of the sphere. The function of 

 q occurring in the equation (6) is to be expressed in terms 

 of k. Remembering that ca is the distance between the 



