CHAP. VI.] MOMENT OF INERTIA. 71 



we find k I, the mean proportional of O k and & <z, and lay it off 

 twice from O to D along the assymptote O K, O D is the diag- 

 onal of a rectangle whose sides are the principal axes. We 

 thus find the vertices A, A, B, B. 



We can thus construct the curves. Then for any position 

 of the axis X X as it revolves about O, we can find the cor- 

 responding radius of gyration and consequently the moment of 

 inertia, by simply drawing tangents to the curve above and 

 below the new position of X X and parallel to it. The radius 

 of gyration thus obtained measured to the scale of length and 

 multiplied by the algebraic sum of the forces, or 4 to the 

 scale of force, will give the moment of inertia required for the 

 assumed position of the axis. 



59. Central Curve. Central Ellipse. If the point O about 

 which the axis turns coincides with the centre of action (or 

 gravity) of the forces, we call the curve enclosed by the paral- 

 lels M' M" at the distance 7s on either side, the central curve. 

 When the parallel forces all act in the same direction this curve 

 is always an ellipse. 



For the central curve the principle proved in Art. 57 and 

 the method of construction given in Art. 58, are no longer 

 applicable, for the algebraic sum of the statical moments of 

 the given forces is zero for every axis through the centre of 

 gravity. We cannot therefore find the centre of gravity of the 

 moments of the forces, when considered as forces themselves 

 and applied at the given points of application. 



If we divide, however, these moments considered as forces 

 into two portions or groups, and find the centre of gravity of 

 each group, the line joining these two points has an important 

 property, viz., that for every moment axis parallel to it, the 

 algebraic sum of the moments of the statical moments consid- 

 ered as forces, that is, the algebraic sum of the moments of 

 inertia of the forces, is zero. In other words, 2 P e e' is zero, 

 e being the distances of the points of application from the first 

 axis, which passes through the centre of gravity of the forces, 

 and e' the distances from the axis parallel to the line joining 

 the two centres of gravity of the two groups of statical moments 

 considered as forces. If we draw then through the centre of 

 gravity of the forces themselves the moment axis X X, and 

 take it as the axis of abscissas of a co-ordinate system whose Y 

 axis passes also through the centre of gravity of the forces and 



