1887.] Electric Time-constant of a Circular Disk. 289 



conditions of the formation and composition of the hydrate of 

 phosphine. 



I must express my best thanks to Professor Dewar, F.R.S., for his 

 many kind and helpful suggestions, and also to the Master and 

 Fellows of Christ's College, who have permitted me to retain my 

 scholarship during the continuance of this work. 



II. " On the Principal Electric Time-constant of a Circular 

 Disk." By HORACE Lamb, M.A., F.R.S., Professor of Pure 

 Mathematics in the Owens College, Victoria University. 

 Received March 29, 1887. 



The time-constant for currents of any normal type in a given con- 

 ductor is the time in which free currents of that type fall to 1/e of 

 their original strength. In strictness there are for any conductor an 

 infinite series of time-constants, corresponding to the various normal 

 types, but in such a case as that of a coil of wire one of these is 

 very great in comparison with the rest, which belong to types 

 in which the current is in opposite directions in different parts of 

 a section of the wire. And in all cases the time-constant correspond- 

 ing to the most persistent type which can be present under given 

 circumstances is, of course, the one which is most important from an 

 experimental point of view. 



A determination of the time-constants of a uniform circular disk 

 would be of interest for two reasons : first, in relation to Arago's 

 rotations, which are entirely due to the greater or less persistence of 

 currents once started in the disk ; and, secondly, in connexion with 

 Professor Hughes's experiments with the induction balance, in which 

 the disturbance produced in the field by the currents induced in metal 

 disks (such as coins) was studied. Unfortunately, the mathematical 

 problem thus suggested would seem to be difficult. Restricting our- 

 selves, for simplicity, to cases where the currents flow in circles 

 concentric with the disk, so that the problem is not complicated by the 

 existence of an electric potential, then if be the current-function, the 

 electric momentum at a distance r from the centre of the disk will be 

 — dP/dr, where P is the potential of an imaginary distribution of 

 matter of density over the disk. Hence, if p' be the resistance per 

 unit area, we have — 



P dr dtdr 

 In any normal type, and P will vary as e -x *, and, therefore — 



