1872.] Mr. I. Todhunter on the Attraction of Spheroids. 507 



times, and with every conceivable modification and check. Some few of 

 them have already been published in the ' Chemical News/ but are here 

 referred to again for the sake of comprehensiveness. 



At present the writer does not venture to put forth any definite theory 

 respecting the presence and nature of the nuclei which are so universally 

 diffused throughout the atmosphere ; but when it is considered how much 

 sodic chloride is constantly present in the air, and what quantities of sul- 

 phurous acid are evolved daily, which becomes partly converted into sul- 

 phuric acid, the presence of particles of sodic sulphate in the air would not 

 be surprising ; and that it does exist is proved by drawing air through, water 

 and finding comparatively large quantities in the solid matter arrested by 

 water. 



Sodic sulphate solutions, too, crystallize on exposure much more readily 

 than those of any other salt. The other salts which form supersaturated 

 solutions are certainly less diffused than sodic sulphate. 



XXI. " Note relating to the Attraction of Spheroids." 

 By I. Todhunter, M.A., F.E.S. Received May 16, 1872. 



In a memoir on the Attraction of Spheroids, published in the ' Con- 

 naissance des Terns' for 1829, Poisson showed that certain important for- 

 mulae were true up to the third order inclusive of the standard small 

 quantity. The object of this note is to establish the truth of the formulae 

 for alt orders of the small quantity. 



1. Suppose we require the value of the potential of a homogeneous 

 body at any assigned point. Take a fixed origin inside the body ; let r\ 

 6', \p' denote the polar coordinates of any point of the body ; and let r, 0, 

 be the polar coordinates of the assigned point ; and, as usual, put // for 

 cos 0', and fi for cos 0. The density may be denoted by unity. 



Then the potential V is given by the equation 



r' 2 dr d\x d-d/' 

 t J{r 2 + r' 2 -2rr A)' 



\ = + V ( i -f) V ( i -V 2 ) cos 0// - . 



The integration must extend over the whole body. 



2. Suppose that r is greater than the greatest value of r ; then 

 (r 3 4-r' 2 —2rr'X)-* can be expanded in a convergent series of powers of 



-. Thus, for example, let the body be an ellipsoid, and take the centre 

 r 



as the origin ; let a, 5, c denote the semiaxes in descending order of mag- 

 nitude. Then, if r is greater than a y the expansion may be effected in the 

 manner just stated ; and so a convenient expression may be obtained for 

 the potential of an ellipsoid on an external particle. This expression, how- 

 ever, is not demonstrated to hold for every external particle, but only for 



