considered geometrically. 408 



cularly to B is equal to the whole attraction of a on p, and 



•*• ~ YTTTTi X attraction of A on C, which, as already observed, 



QcAPP', which is the distance of P (now supposed within A) 

 from B. 



4. The preceding proposition shows that the attraction of an 

 ellipsoid on any point on its surface, or within it, can be got at 

 once from the attraction of the same ellipsoid on a point placed 

 at the extremity of an axis; and this latter attraction is found 

 and reduced to elliptic functions as follows. 



Let be the centre, and 0A = «, OB = i, OC=e the semiaxes, 

 and let the attracted point C be the vertex, and CO the axis of a 

 cone of revolution D whose semiangle is 6, and let d-\-d6he, the 

 semiangle of another such cone E very close to D, and having 

 the same vertex and axis, and let be the common vertex of two 

 other cones D' and E' parallel to D and E. Conceive the por- 

 tion of the ellipsoid between D and E to be divided into elemen- 

 tary pyramids by planes passing through CO; let /be the length 

 of a side of one of these little pyramids, which is a chord of the 

 ellipsoid and a side of D ; and let g be the parallel side of D', 

 which is a radius of the ellipsoid; and let/' and g' be the pro- 

 jections of/ and g upon c, and let co be the small angle between 

 two consecutive /® (or g^) ; then the attraction of the little 

 pyramid whose side is / on its vertex C is =/a) . dO (see Airy's 

 Tract on the Figure of the Earth, Prop. 4), and .-. its component 



f f q''^ 



along CO is =f(o.dd, .-. =2c x |-o) . dd ; but ^ =\, .-. said 



2 4 



component = — y^w . dd, .'. — — x |y^cos* dco . dd ; now ^g'^co 



is the area on D' included between the two consecutive g^, and 



the sum of all such elements is the entire surface of D' which we 



shall still name D' ; .'. the attraction of the slice of the ellipsoid 



T^ IT. n 1 no • 4D' cos2 0.d0 ,, 

 between D and E upon C along CO is = . Now 



the projection of D' on the plane of ab is obviously an ellipse 

 whose area D" is = D' sin 9 ; let r and r' be the sides of D' (radii 

 of the ellipsoid) Ipng in the planes of ca and cb, then the semi- 

 axes of D" are plainly the projections of r and ?•', or r sin 6 and 



r'sin^: and .*. D" = 7nr'8in*^, and soD'=-t— ^, .-, =7r/T'sin^; 

 ' sin^ 



and so the attraction of the slice on C along CO is 



= — rr* cos' Q sin Q . dd. 

 c 



Now 1 cos^^ . sin^^ ,1 cos^^ . sin*^ 



sin u J A cos u 



2E2 



