508 Mr. I. Todhunter on the Attraction of Spheroids. [June 20, 



those which make r greater than a. It is obvious that there may be 

 external particles for which r is less than a ; and for these the process 

 cannot be considered satisfactory, since it involves the use of a divergent 

 series. 



3. Still it has been usual with writers on the Attraction of Spheroids 

 and the Figure of the Earth to leave this point unexamined. They, in fact, 

 assume that formulae which are demonstrated on a certain condition are 

 true, even when that condition does not hold ; so that, for example, an 

 expression obtained strictly for the potential of an ellipsoid on an external 

 particle when r is greater than o, is assumed to be true for any external 

 particle. 



4. Poisson, however, has drawn attention to the difficulty; his discus- 

 sion of it is the main part of his elaborate memoir " Sur l'Attraction des 

 Spheroides," which was published in the 1 Connaissauce des Terns ' for 

 1829. He shows that the ordinary formulae, although obtained in an 

 inadequate manner, are really true as far as the terms of the order a 3 

 clusive, where a is the well-known standard small quantity of such inves- 

 tigations. I propose to extend his process so as to show that the result is 

 true for all powers of a. 



It will be necessary to give some preliminary transformations ; this I 

 shall do with brevity, referring to Poisson's memoir for detail. 



5. It is convenient to separate Vinto two parts, one being the potential 

 of a sphere of radius r, and the other the potential of the excess of the 

 spheroid above the sphere ; the word excess is here used in an algebraical 

 sense, for the surface of the spheroid is not necessarily all external to that 

 of the sphere. Thus we obtain 



where u denotes the radius vector of the surface of the spheroid corre- 

 sponding to the angles 0' and \p' ; so that the integration with respect to r 

 is to be taken between the limits r and u. The integration for // and i// 

 may be considered to be taken over the surface of a sphere of radius 

 unity ; and we may denote an element of this surface by dut', and use the 

 symbol J du instead of § § df* d\p'. 



6. Now, for those elements in the integral in (1) which have r' less than 



r' 



r, the radical must be expanded in powers of - ; and for those elements 



r 



which have r greater than r, the radical must be expanded in powers of 



-.. Thus we obtain 

 r 



Y ^^^^)^ +X -K^) V ^ (2) 

 where P» denotes Laplace's coefficient of the nth. order. In the second 



