Note on Surfaces of the Second Order. 325 



must be integrated. Let be the angle which a plane, passing 

 through SP and the axis of , makes with the plane rj, then 



o> = sin6dOd<j> 



1 / 



~ = 

 r \ 



/cos 2 sin 2 cos 2 sin 2 sin ! 0\ 1 



(~T ~iT ~ll' JSf^i 



When these values are substituted in (i), that expression may be 

 readily integrated, first with respect to 0, and then with respect 

 to 0. 



It is evident that, by the same substitutions, the expression 

 (g] may be twice integrated. 



An investigation similar to the preceding has been given by 

 M. Chasles, for the case in which the force varies inversely as 

 the square of the distance.* He uses a theorem equivalent to 

 the formula (/), but deduces it in a different way. 



From what has been proved it follows that, if V be the sum 

 of the quotients found by dividing every element of the shell by 

 its distance from an external point S, the value of V will be the 

 same wherever that point is taken on the surface S of an ellip- 

 soid confocal with the surface A of the shell. 



Let S' be another ellipsoid confocal with A, and indefinitely 

 near the surface S. The normal interval between the two sur- 

 faces S and S', at any point 8 on the former, will be inversely 

 as the perpendicular dropped from the common centre of the 

 ellipsoids on the plane which touches S at 8. Hence, supposing 

 the point 8 to move over the surface S, that perpendicular will 

 vary as the attraction exerted by the shell on the point 8, when 

 the force is inversely as the square of the distance, or as the at- 

 traction exerted by the whole ellipsoid A on the point 8, when 

 the force is inversely as the fourth power of the distance. 



When the point 8 is on the focal hyperbola, the integrations, 

 by which the actual attraction is found in either case, are sim- 

 plified, for the surface B is then one of revolution round the 

 axis of , and its semidiameter r is independent of the angle 0. 



* Memoires des Savants Etrangers, torn. ix. 



