184 Mr. A. Schuster on the Disruptive Discharge 



some other way ; but the above series will be the most con- 

 venient to use whenever the shortest distance between the 

 spheres is greater than the tenth part of the radius of either. 

 The Tables I have prepared will, however, save future observers 

 all trouble of calculation. The following remarks will suffi- 

 ciently explain the way in which they have been calculated. 

 If, in the general term of series (3), the value of q 2m has be- 

 come sufficiently small to be neglected compared to unity, the 

 remaining terms can be added up without further trouble ; for 



?n + 1 m— 1 



(l+ y — =mg 2 -0+ 2 )? 2 +(™ + % 2 -•• 



When the distance between the spheres is small, it is found 

 more convenient to change the form of series (3) so as to 

 render it more quickly convergent. We may expand each 

 term of the series in powers of q and add up all the first 

 terms, viz. : — 



l-3? + 5ff 2 -7?' + ... =(T^"r 



Treating the second and subsequent terms similarly, we 

 obtain 



J ™~(l+g)* (1 + g 5 ) 2 (1 + ? 9 ) 2 ' (1+Y m+1 ) 2 * 



Here, again, the terms may be added up as soon as q Am + 1 can 

 be neglected. Finally, we may transform the equation still 

 further by multiplying numerator and denominator with 

 (l-2 4m+1 ) 2 , finding thus 



f(a\ — (1 ~? )3 ' + ^ 2(1 -^ )3 4- tH=£l 

 JW— (i-2»)» + (l-g lu ) 2 + (l-? 18 ) 2 ' 



In this equation the denominators very quickly approach 

 the value unity. I have given the series in its various forms 

 as I find that individual computers differ much in their pre- 

 dilections for particular forms, and as the relative trouble of 

 calculating the different series differs according to the degree 

 of accuracy required. 



Table I., which I have calculated for the objects of this 

 paper, may be useful to others. The first column gives the 

 distance between the centres in terms of the radius of either 

 sphere, which is the value called c above. 



The second, third, and fourth columns give q, f(q), and y, 

 defined by equations (2), (3), and (4). Column 5 gives a 

 quantity, #, which will be found useful for purposes of inter- 



