600 REPORT— 1893. 



Central Ellipse and Kern. 



Suppose the foregoing operation performed for a certain point O, 

 assuming tliat the segments which have their extremities on the boundary 

 of the area have a certain direction. Suppose, further, the operation 

 repeated with the segments in various directions and the respective radii 

 of gyration found. If these be plotted from the point O in the directions 

 perpendicular to their respective direction, and parallels to the directions 

 of the segments in the corresponding positions are drawn, these parallels 

 will be found to envelop an ellipse. This ellipse is known as the ' ellipse 

 of inertia,' and in the case when the point is the centroid of the figure 

 under consideration the ellipse is called the ' central ' ellipse. The term 

 ' momental ' ellipse is often used for the ellipse of inertia, the central 

 ellipse being called the ' eqnimomental ' ellipse. The foregoing is stated 

 as follows by M. Levy : — ' If upon different lines issuing from any point 

 whatever, in a plane, we plot their lengths, inversely proportional to the 

 radius of gyration relatively to these lines, the locus of the extremity of 

 their lengths forms an ellipse.' ' 



The proof of this and other properties of the central ellipse by that 

 writer are, with slight alterations, given in the following paragraphs, 

 where by using the idea of segments instead of that of forces the treat- 

 ment is made perfectly general. 



Let O X and y he axes of coordinates. Let any line u making any 

 angle a with the axis of x be that about which the moment of inertia 

 of a segment P whose point of intersection with the plane p is 

 required. 



Then 



Ip=P (y cos a — a; sin a)^, 



and the moment of inertia of a number of such segments becomes 



Slp=2P {y cos a — x sin a)-, 

 or 



r2SP=:SP|/2 cos2 a + SPa;2 sin^ a—2l,Fxy sin a cos a ; 

 or if 



ai22P=SP»2 

 &,2:SP=SP2/2 



Ci22;P = 2PiBM 



(I) 



r^=ai^ sin^ a + h^^ cos'^ a — 2ci' sin a cos a. 

 Take upon w a length 



{Jm= — , 

 r 



where d is some constant. 



Then x^ and t/i being the ordinates of M 



ajj = - cos o, 

 r 



cP . 

 y\= sm a, 



' La Statique graphique, vol. i. p. 400. 



