70 



Sir W. Thomson on an Apparatus 



[June 20, 



kept permanently charged from year to year by very slow water-droppmg 

 arrangements, a drop from each nozzle once every two or three minutes 

 being quite sufficient. 



Pig. 3. 



The mathematical theory of the action appended below * is particu- 

 larly simple, but nevertheless curiously interesting.^ 



The reciprocal electrostatic arrangement now described, presents an in- 

 teresting analogy to the self-sustaining electromagnetic system recently 

 brought before the Eoyal Society by Mr. C. W. Siemens and Pro- 

 fessor Wheatstone, and mathematically investigated by Professor Clerk 

 Maxwell. Indeed it was from the fundamental principle of this electro- 

 magnetic system that the reciprocal part of the electrostatic arrange- 

 ment occurred to me recently. The particular form of self-acting electro- 

 phorus condenser now described, I first constructed many years ago. 



* Let c, & be the capacities of the two jars, I, I' their rates of loss per unit potential 

 of charge, per unit of time, and D, D' the values of the water-droppers influenced by 

 them. Let +-2; and -v' be their potentials at time t; and v' being both of one sign, 

 in the ordinary use of the apparatus described in the text. The action is expressed by 

 the following equations, ^ 



dt dt 

 lie D I c', D', V were all constant, the solution of these equations would be, for the 

 case of commencing with the first jar charged to potential 1, and the second zero, 



c'{p — (j) ' c'(p — (t)' 



with the corresponding symmetrical expression for the case in which the second jar is 

 charged, and the second at zero, in the beginning ; the roots of the quadratic 



{cx-^l){c'cc^l')-T)T>'=0 

 being denoted by p and <t. When W > DD', both roots are negative ; and the electrifi- 

 cation comes to zero in time, wliatever may be the initial charges. But when W < DD', 

 one root is positive and the other negative ; and ultimately the charges augment in pro- 

 portion to if p be the positive root. 



