1872.] Mr. T. Todhunter on the Attraction of Spheroids. 509 



term on the right-hand side of (2) the integration with respect to w' is to 

 extend over so much of the surface of a sphere of radius unity as corre- 

 sponds to negative values of u — r ; and in the third term the integration 

 with respect to u/ is to extend over so much of the surface of the sphere as 

 corresponds to positive values of u—r. By 2 is denoted a summation with 

 respect to the integer n for all values from zero to infinity. 



7. By adding a certain quantity to the second term on the right-hand 

 side of (2), and subtracting the same quantity from the third term, we 

 obtain, finally, 



V=l^ + 2 JL J V+W )P' n M,-Z jU P' B dJ t . . (3) 

 where U stands for 



In the second term on the right-hand side of (3) the integration for &/ 

 extends over the whole surface of the sphere of radius unity ; and this I 

 denote by explicitly putting the limits and 4tt. But in the third term 

 the integration for J extends only over that portion of the surface which 

 corresponds to positive values of u—r ; and this I denote by leaving the 

 limits unspecified. 



8. For the rest of this paper the notation just explained will be strictly 

 preserved. If the integration with respect to w' extends over the whole 

 surface of the sphere, the limits and Air will be expressed ; if the inte- 

 gration extends only over that portion of the surface which corresponds to 

 positive values of u—r, the limits will not be expressed. 



9. The value of V obtained in (3) is quite general, but it is specially 

 convenient for the case of an external particle. Poisson gives also another 

 form which is specially convenient for the case of an internal particle. 



It will be sufficient for us to confine ourselves to the case of an external 

 particle, as the same process is readily applicable to the case of an internal 

 particle. 



10. For an external particle which is sufficiently remote, the third term 

 on the right-hand side of (3) vanishes, because in this case w— r is never 

 positive ; so that we have then simply 



Now what we have to show is that this formula will also hold for every 

 external particle. In other words, it must be shown that for any external 

 particle 



SjUF.ik/ ss (4) 



