433 



of the internal axis of the cone whose vertex is S, and which 

 circumscribes the shell. 



To find the whole attraction of the shell, the expression 



mno cos Q (0 



must be integrated. Let ^ be the angle which a plane, pass- 

 ing through SP and the axis of 5, makes with the plane ^?j ; 

 then 



u) — sin ddOd^, 



1 _ /fcos'^e ?,\nWeos^(t> sin^flsin^^N 1 



r\- y \~ir + A' + k" J VfiTi' 



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

 be readily integrated, first with respect to 6, and then with 

 respect to <j>. 



It is evident that, by the same substitutions, the expres- 

 sion (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 {Memoires des Savants Etran- 

 gers, tom. ix.) He uses a theorem equivalent to the formula 

 (/), but deduces it in a diiFerent 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 ellipsoid confocal with the surface A of the shell. 



Let S' be another ellipsoid confocal with A, and indefi- 

 nitely near the surface S. The normal interval between the 

 two surfaces S and S', at any point S on the former, will be 

 inversely as the perpendicular dropped from the common cen- 

 tre of the ellipsoids on the plane which touches S at S. Hence, 

 supposing the point S to move over the surface 2, that per- 

 pendicular will vary as the attraction exerted by the shell on 

 the point S, when the force is inversely as the square of the 



