78 MOMENT OF INERTIA. [CHAP. VI. 



The conjugate axes of the central curve are by principle 1 or 

 2 of the preceding Art. E F and A D. 



The above value of a is then the length of the semi- diameter 



along A D, or a = \ -z h x -~ h. That is, a is a mean propor- 

 tional between -% h and -n A. This is found by the semi-circle 

 o o 



O D Fig. 37. 



The moment of inertia of the triangle with respect to A D is 



g M~o a ) 8 ' The radius of gyration then is V ~- \->ay = 



~2 \H a / X ^\ 1 > a ] or a mean P ro P or tional between -^ and 

 iof|orDC. 



This is given by the semi-circle on D G = s D C, and we 



a 



thus have the four points 1 2 3 4 of the central ellipse, and the 

 semi-diameters 1 and 3, and can therefore construct it. 

 From the central ellipse as before, we can find the centre of 

 gravity of the moments considered as forces for any axis par- 

 allel to B C or A D, as also in either case, the radius of gyration 

 and therefore moment of inertia, for any axis passing through O. 



3d. Trapezoid. PI. 11, Fig. 38. 



Here the lines E F joining the centres of the parallel sides, 

 and G H parallel to these sides, and passing through the centre 

 of gravity 0, are the conjugate axes of the central ellipse. 



For the axis A B and direction E F, the moment of inertia is 



1 (0+3 5) A 8 ,* 



a and b being A B and C D, and h = E F. The square of 

 radius of gyration is then 



* T Va (a-b) 



