CHAP. VI.] MOMENT OF INERTIA. 67 



In this expression a and ft will vary with the position of M 

 and the direction of q, but 2 P a?, 2 P y* remain unchanged. 

 These last expressions are, however, nothing more than the mo- 

 ments of the second order (moments of inertia) of the given 

 force system with reference to the co-ordinate axis, the distances 

 of the points of application being measured in the direction of 

 the axis. They are known if the force system is given and the 

 co-ordinate system assumed. 



If we put ip a? = d? ZP, 2P f = tf S P, 2 P x y =f*2P, 

 b and a, etc., are the radii of gyration of the moments of inertia 

 with reference to os and y, and the above equation becomes 

 2P f = 2 P [a? a? + /3 2 b z + 2 a /3/ 2 ] 



If we conceive for the assumed position of M, the radius 06 

 gyration k to be found, and M' and M" drawn on either side 

 at a distance &, measured parallel to ^, and indicate the dis- 

 tances cut oft by these lines from the co-ordinate axes by # e 

 y w and then project these distances parallel to M upon the 

 direction of q or &, we have J& = a x e = /3 2/ e , whence 



K Q K 



a p = 



e ^ ' 



and these values substituted in the above equation give 



y<> e y* 



where 1& is essentially positive in the second term. 

 Hence, 



If we suppose the axis M to change its position revolving about 

 O, the segments x e y e cut off from the axes of x and y by 

 M' and M" alone will change in this equation. It is therefore 

 the equation of the curve enclosed by M 7 M". If this curve is 

 known for a given force system, then the moment of inertia for 

 any axis passing through its centre is easily found. We have 

 only to draw parallel to the axis two tangents to this curve, one 

 on either side, and measure their distance from M, in the direc- 

 tion in which the. distances q of the points of application from 

 the axis are taken. This distance is the radius of gyration, 

 and the moment of inertia is simply the product of its square 

 by the algebraic sum of the forces. 



