378 Note 1 1 : electrification of two disks 



equal but of opposite signs, a term involving an harmonic of odd order would 

 give the surface-density everywhere zero. 



The potential arising from this distribution at any point whose ellipsoidal 

 co-ordinates are <a = ap and 7? = av 



is F=F +F 2 + &c. + V n , ...... (7) 



where F 2n = A r P* (M) <?'* (") ...... (8) 



In this expression Q' 2n (v) denotes a series, the terms of which are nu- 

 merically equal to those of Q Zn (v), the zonal harmonic of the second kind, 

 but with the second and all even terms negative. If we put i for V I, we 

 may write 



<?'* = (-)"*&(') ...... (9) 



&c 



_ 

 i . 3 5 4 + i 3 5 7 - (4 + 3) 



This expression is an infinite series, the terms of which increase without 

 limit when v is diminished without limit. 



It may, however, be expressed in the finite form * 



<?'*, M = P\ n (") tan-' - Z^ (v} , ...... (i i) 



where P\ n v = (-)- P^ (iv), ...... (12) 



that is to say P'^ (v) is a zonal harmonic of the first kind with all its terms 

 positive, and Z^ (v) is a rational and integral function of v of 2n - i degrees, 



which is such as to cancel all the terms of P'^ (v) tan -1 ( - j which do not vanish 

 when v becomes infinite. 



The expression (n) is applicable to small as well as great values of v. Thus 

 we find when v is o, as it is at the surface of the disk, 



Q'*> () = 2 2n M!n! 2 ' 



The potential at any point of the disk is therefore the sum of a series of 

 terms, the general form of which is 



On the axis, p = i and av = z, and the potential is the sum of a series of 

 terms, the general form of which is 



U* n = At n -_Q' 2n ( v ). ...(15) 



Since we have to determine the value of the potential arising from the first 

 disk at a point in the second disk for which z = c at a distance r from the axis, 

 and if we write r 2 = j ( x _ p*) t ...... (16) 



* See Heine, Handbuch der Kugelfunctionen, 28, 20. 



