ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 601 



Inserting these values in Equation I. we get 



being an equation of the ellipse, where a, and 6, are the radii of gyration 

 relatively to the axis of Xj and ?/,, and 



2_SP«,2/, 

 ' 2P 



If two other axes of coordinates be taken, an equation of the form 



is obtained, a and h being the radii of gyration relatively to the new axis, 

 and 



'-"sp • 



If the axes of x and y correspond to the direction of the major and 

 minor axes of the ellipse, for which case 



c=0 or SPa;2/=0, 

 then 



or when 



In the foregoing case the moment of inertia has the same sign as the 

 segment. If all the segments have the same sign (say + ) in this case, a* 

 and b^ being positive, the curve is an ellipse. If, on the contrary, the 

 segments have not the same sense, the moments of inertia relatively to 

 the principal axis may be of contrary sign ; that is, a^ and b^ may be re- 

 placed by — a- and — b^ ; the curve may then be an hyperbola. The curve 

 relatively to the point O in a plane is called the curve of inertia relative 

 to this point, and will be a curve of the second degree, the axis of the 

 curve being the principal axis of inertia. When the centroids correspond 

 with the centre of the conic it is called the ' central conic,' and is usually 

 an ellipse, that is, the central ellipse. Instead of taking the rectangular 

 axes Ox and Oy, take the new axes Ox^ and Oyi, the latter making an 

 angle 6 with Ox. 



Then y=yi sin 



x=Xi+yi cos 8. 



The above equation referred to the new axes is 



{a-Xi^ + 2a^coBd-2c'^smd) x^yy + ia- cos^ O + b^ sm^d-2c^amd coB6)yi^ 



Then since y{- is the square of the radius of gyration to the axis O j/i, 

 let by be this radius. Then, taking 



