72 MOMENT OF INEBTIA. [CHAP. VI. 



is parallel to the line joining the two centres of gravity of the 

 statical moments considered as forces, then the moments of 

 inertia 2P y x are zero, and hence as in the preceding Art. 

 this axis ofYis conjugate to X X. 



This holds good not only for the central curve, but also for 

 every inertia curve, whose centre O instead of coinciding with 

 the centre of gravity of the forces, lies in the axis passing 

 through that centre. In this case also the axis through the 

 centre O parallel to the line of union above, is a conjugate to 

 X X. Still more, the half length of this conjugate diameter is 

 in both cases the radius of gyration of the force system for the 

 axis X X and the direction of Y. 



Hence in every inertia curve of a system of parallel forces, 

 whose centre lies in an axis passing through the centre of grav- 

 ity of the forces, the diameters conjugate to this axis are paral- 

 lel and equal. All these inertia curves are therefore touched 

 by two lines parallel to this axis and' equally distant on either 

 side. This distance is the radius of gyration for this axis. 



For any such inertia curve, whose centre O is distant i from 

 the centre of gravity S of the forces, we call E and @ the par- 

 allel conjugate axes to S O for this curve, and the central curve 

 respectively ; q and q the distances from them of any point of 

 application, these distances measured parallel to S O, and con- 

 sidered positive when the point of application lies on the same 

 side of E or (5 respectively as the centre of gravity S from E. 

 Then i, the distance apart of E and G is essentially positive, 

 and if we indicate by a and a the lengths of the semi-conjugate 

 diameters for the inertia and central curve respectively, we 

 have 



a z 2 P = S Pq 2 and a 2 2P = 2 Pg* 

 where q and q stand in the simple relation 



q = q + i. 

 Hence 



S P q 2 = IP (g + i) 2 = 2P /+ 2 i 2P q+? 2 P. 



Since Q. passes through the centre of gravity %P q = o, and 

 therefore 



2 P q 2 = 2 P / -t- ^ 2 P = a 8 2 P . 

 Herice 



tt?=s<#f, 



an equation which gives the relation between the lengths of 

 the semi-conjugate diameters of the central and any inertia 



