290 



Prof. H. Lamb. On the Principal [Apr. 21 r 



or, in the case of uniform resistance — 



p'<j> = XP+0 



over the disk. 



In the absence of a rigorous solution of this problem (which seems; 

 well worthy the attention of mathematicians), a good approximation 

 to the principal time-constant may be obtained on the following- 

 principles : — * 



1. An increase of resistance in any part of the disk will diminish 

 the time- constant ; and 



2. If the time-constant be calculated on any arbitrary assumption 

 as to the distribution of current, the result will be an under- estimate, 

 and will, moreover, be a close approximation to the true value if the 

 assumed law be not very wide of the mark, on account of the 

 " stationary " property of the normal types. 



Some distributions of density 0, and corresponding potentials P, 

 convenient for our purpose, are obtained by considering the disk as a 

 limiting form of heterogeneous ellipsoid, in which the surfaces of 

 equal density are similar and coaxial ellipsoids. f If the density at any 

 point Q in the interior of the ellipsoid — 



^il 2 ,^ _ -. 



be c(i-e*y, 



where 6a, 6b, 6c are the semi-axes of the similar ellipsoid through Q,, 

 the corresponding potential at internal points will be — 



"» + lJ \ a*+k P+k c* + Jc) 



2 V +1 die 



V{(a2-rk)ib* + k)(cZ + k)} 



Putting a = b, and passing to the case of a disk, by putting c = 0, 

 2Cc = 1, we find that to the surface- density — 



/, r2V +i f i7r + i 7 ' »* r(»+l)/, r*Y+* , Q \+ 



* = ( 1 -^) ].°^ lx * c=s T-ifeFi)( 1 -p) ' (30t 



corresponds, for points of the disk, the potential — 



I 



F - 2(n + l) J A a?+k) 



* See Kayleigh's ' Sound,' §§ 88, 305, &c. 



+ See Ferrers, ' Quart. Journ. Math.,' vol. 14, p. 1. 



X Iu the electrical application we must suppose n + 2 > 0. 



