Ellipsoidal Shells and of Solid Ellipsoids. 399 



19. The result stated in (20) seems to have been first 



given by Plana in a " Note sur l'integrale I ~- = V" which 



appeared in Crelle's Journal, vol. xx. (1810). It was given 

 also by Lejeune Dirichlet in the same journal in 1816. His 

 process o£ demonstration is, however, indirect. The result 

 in (20) is first assumed, and then verified by showing that 

 this value of V satisfies Laplace's differential equation of the 

 potential 



fI + fI + P = (21) 



a.r d>/~ dc- 



at every point external to the ellipsoid, and that V vanishes 

 at infinity. This latter fact is important. It is fairly obvious 

 for both the homoeoid and the solid ellipsoid. The integral 

 (16) for the homoeoid, for example, may be compared with 

 the greater integral obtained by putting for each of a 2 , b 2 , <r 

 the value of the smallest. This integral is then evaluated, 

 and it is seen that it vanishes at infinity ; a fortiori so does 

 the integral for the homoeoid. 



20. If the point /, g, h be on the surface of the ellipsoid 

 the value of X is zero in (16) and (20). In the former case 

 the modified equation gives the potential at every internal 

 point for the homoeoid ; in the latter case 



Y= ^' d ' c ) VW^WTWT^ ■ ■ (22) 



for a surface point on the solid ellipsoid. The case of an 

 internal point requires examination in the latter case. 



It is easy to show that equation (22) is applicable without 

 change of form to the case in which the point /, g, h lies 

 within the surface of the solid ellipsoid. For let the point be 

 on the homoeoidal surface given by 



f 2 

 S W (?3) 



then the potential is made up of two parts, Y 1 due to the 

 ellipsoid internal to the surface (23), and V 2 due to the 

 homoeoid of finite thickness external to the point. By (22) 



/.. (1-X J~-)du 



T7 7 4 V a- fx + u J 



V x — irpabcu, 1 I . — 



Jo </{a 2 Li + u)\b 2 L* + u)(c 2 fjL + u) 



