458 Colonel A. R. Clarke on the Potential of 



ence is collected on C2 when Ci is charged up to a certain 

 potential-level, and that which in the charging of C2 up to the 

 same potential is through influence collected on Ci, are equal 

 the one to the other. 



Beside these two conclusions, here cited as examples, refer- 

 able to two very simple special cases, from equation (I.) many 

 other similar inferences can of course be drawn. 



LX. On the Potential of an Ellipsoid at an External Point. 

 By Colonel A. R. Clarke, C.B,, F.R.S* 



IN connexion with the subject of the attraction of the earth 

 upon external objects, there is a series which is usually 

 brought forward by writers on astronomy, viz. that which 

 results from expanding into a series in descending powers of r 

 the expression for V, 



y= {{{ dxdydz 



r being the distance = (/^ +^^ + ^^)^ of the attracted external 

 particle from the origin, x^ y, z the coordinates of any ele- 

 ment of the mass, the density being unity throughout. Put 

 ;c^+y^ + 2^ = /3^ oLX-k-^y-^^z — d^ where a, /5, 7 are the direc- 

 tion-cosines of r ; then the radical in the expression for V 

 may be expanded in the form 



1 . Qi . Q2 . Q3 . 



" T" ~o r '~q" "T ~4" "r • • • 5 



m (T** *»" rp*^ ' 



where Q. is a homogeneous function of x^ ?/, z of the degree i. 

 When the body is an ellipsoid referred to its principal axes, 

 the terms in which % is odd will disappear in integrating over 

 the volume of the ellipsoid ; so that in this case, M being the 

 mass of the ellipsoid, 



^ 1 ^^dm + ^ j Q4^m+ ^ j QecZmH- . . . , (1) 



which is the series alluded to. Lagrange's investigation of 

 the terms (as far as just written down) of this series is referred 

 to in Todhunter's ^ History of the Theory of Attraction and 

 the Figure of the Earth,' vol. ii. p. 161. See also Thomson 

 and Tait, ' Natural Philosophy,' pp. 401, 402 ; Pontecoulant, 

 Theorie Analytique du Systeme du Monde, ii. p. 233, where 

 the above expression for the potential is given as one suitable 



* Communicated by the Author. 



r r 



