324 Note on Surfaces of the Second Order. 



the same method to the case where the law of force is that of 

 nature, probably from not perceiving that, in this case, the ellip- 

 soid ought to be divided (as Poisson has divided it) into concen- 

 tric and similar shells. This application requires the following 

 theorem, which is easily proved : 



Supposing A' to be another ellipsoid, concentric, similar, and 

 similarly placed with A\ let the right line SPP' intersect 

 it in the points p and jt/, respectively, adjacent to P and P'; 

 then if the direction of that right line be conceived to vary, the 

 rectangle under Pp and P'p (or under Pp' and P'p') will be to 

 the rectangle under SP and 8P f in a constant ratio. 



Denoting the constant ratio by m, and combining this theo- 

 rem with the formula (/), we have 



Pp x P'p mr 



'-~- 



Now let the two surfaces A and A be supposed to approach in- 

 definitely near each other, so as to form a very thin shell, then 

 ultimately P'p will be equal to PP', and we shall have 



D ry ' mr 

 Pp = P'p = _, 



where m is indefinitely small. Therefore if the point S, external 

 to the shell, be the vertex of a pyramid whose side is the right 

 line SP, and whose section, at the unit of distance from the 

 vertex, is o>, the attraction of the two portions Pp and P'p' of 

 this pyramid, which form part of the shell, will be equal to mru. 

 Hence it appears, as before, on account of the symmetry of the 

 surface B round the axis of , that the whole attraction of the 

 shell on the point 8 is in the direction of that axis, and conse- 

 quently (as was found by Poisson) in the direction of the inter- 

 nal axis of the cone whose vertex is S, and which circumscribes 

 the shell. 



To find the whole attraction of the shell, the expression 



cos (i) 



