CHAP. VI.] MOMENT OF INERTIA. 69 



one of the common assymptotes, the radius of gyration and 

 moment of inertia with respect to it of the given force system 

 is zero. 



57. Contraction of the Curve of Inertia. The curve of 

 inertia for a given system of parallel forces and given centre 

 O, is determined by the direction of any two conjugate diame- 

 ters, since as we have seen in Art. 55, PI. 10, Fig. 34, these 

 directions being assumed we can find the radii of gyration with 

 respect to X X and Y Y, and can thus determine O a and O 5, 

 the semi-diameters. We have then to develop a principle by 

 means of which these directions may be determined. 



If we denote the distances of the points of application of the 

 forces from the axis of M measured in any direction by y, then 

 the statical moments of the forces, P y, are indeed dependent 

 upon the direction in which y is measured, but their relative 

 values remain the same. If then being found for any direction 

 of y, these statical forces are considered as being themselves 

 parallel forces acting at the points of application, and their 

 centre of action is found (for gravity centre of gravity) for 

 some other value of ?/, this centre of action remains unchanged. 

 For any axis passing through this centre of action the sum of 

 the moments of the forces is zero. If therefore we take a point 

 O in the axis M as origin of a system of co-ordinates, whose 

 axis O X may lie at will in the plane of the forces, while O Y 

 passes through the centre of action ; the sum of the moments 

 of the statical moments P y, considered as forces acting at the 

 points of application, with reference to O Y, will be zero. 

 These moments however, provided that the distances of the 

 points of application are measured along the co-ordinate axes, 

 are the moments of inertia, viz., 2 P y x. If these are zero we 

 see that the general equation of the curve of inertia (1) Art. 56, 

 becomes that of a hyperbola referred to its conjugate diameters 

 as axes. With the centre O therefore, the line joining O with 

 the centre of action, gives the direction of the conjugate di- 

 ameter of the curve. 



This is the principle required. By means of it we can find 

 the conjugate diameters of the inertia curve, for a given centre 

 O, and thus construct it. 



58. Construction of the Curve of Inertia for four paral- 

 lel forces in a Plane. Example. As an example let us 

 take the four parallel forces in PI. 10, Fig. 34, supposed 



