The Variation in Attraction Due to the Attracting Bodies. 205 



scribed. The construction of the remaining portions of the diagram where 

 not evident is explained. 



Draw B, B, parallel to P a, and draw A, A, through point of tangeucy 

 a and center C. A, A, and B, B, are, then, conjugate diameters. 



Ordinate a d equals ordinate a g, 1 k equals 1 h. Per known demonstra- 

 tion of Conie Sections; 



(a) g d : h k :: B, : B. 



When the angle e P g becomes infinitesimal. A, A, passes through the 

 points of tangency a and b. From the similarity of the ellipses having 

 C 1, C n, and C A for semi major axes; b a : n 1 :: A, : A. Draw s s 

 through point a, and peri^endicular to P a. Then s a = b a cos a 



(&) s a : n 1 :: A, cos. ex : A. 



Diagram 4. 



Draw t a perpendicular to P C. Per Art. 9, t a = C 1 cos. 5^. Then from 

 the similarity of the ellipses; 



(c) ta : CI :: A cos. S- : A. 



The product of the proportions (a) (b) (c) gives; 



gdXsaXta:hkXnlXCl :: A A, B, cos 3 cos a: A' B :: cos3: 1^ 



The last ratio is true because A^ B, cos nr ^ A B. 



In accordance with a demonstration in Art. 3, gdXsaX^a represents 

 a mass at point a, that attracts particle P in direction P a the same as the 

 mass cut from the ellipsoid by rotating chords e c and g d on axis P C; 

 and hkXo-lXCl represents a mass (m) from likewise rotating chords 

 o p and h k. 



