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XII. On a Theorem relating to the Attraction of the Ellipse. 

 By G. R. Dahlander*. 



CERTAIN investigations relative to the attraction of the 

 ellipse have led me to a theorem analogous to Ivory's well- 

 known theorem concerning the attraction of the ellipsoid, which 

 I will here communicate, as, so far as I am aware, it has never 

 before been remarked. 



It is supposed that from every element of the surface of the 

 ellipse emanates an attracting force proportional to the area of 

 the element, and varying according to a function P(r) of the 

 distance of the element from the attracted point, which is sup- 

 posed to be in the plane of the ellipse. 



Let the equation of the ellipse be -^ + 4* — 1, and let A* and 



K y denote the components of attraction parallel to the x and 

 y axes, and let ct and ft be the coordinates of the attracted 

 point. Then 



| \Zb'2- 7J 2 



A*=/aI dy\ dx^—-~F(r), 



in which expression jjl is a constant quantity, and 



The result of the integration according to x can be expressed in 

 the form 



a*=/u,I Ar(#ri)*~$CrJ)> 



ct — x 

 where <f>(r) = 1 dx F(r), an d f\ and r 2 denote the values of r 



corresponding to the values ^ x/tf—y' 2 and — ^ \/b*—y 2 of x. 



If these limiting values of x be denoted by x x and — x v it fol- 

 lows that 



r,= i /(i-. a?| )«+(0_y)a and r 2 = • (a + *!>•+ ^-.y)*. 



x Ji y 12 

 If we now consider another ellipse, -^ -f ju ~1 } confocal with 



the former, and passing through the point (a, (3), we find in the 

 same way that the attraction which it exercises on the point 

 (a', /3') has a component parallel to the x axis, 



A^jV^R,)-^)), 



* Communicated by the Author. 



