CHAP. VI.] MOMENT OF INERTIA. 63 



ments of the forces, so the exterior areas. of the equilibrium 

 polygon represent the moments of the moments, or the moments 

 of inertia. Thus in PL 8, Fig. 30, the exterior parabolic area 

 o C h should be one-half the moment of inertia of the rectangle 

 or load area op r h, with reference to the resultant of the area 

 forces as an axis. 



Let us see if this is so. The area of the triangle o h C is -~ 



o k x the ordinate S C. This ordinate S C gives, as we have 

 seen, the moment, with respect to S, of the reaction. We can 

 therefore find its value. Thus if p is the load per unit of length, 



7 72 



and I is the length, *^ is the reaction, and r- this moment. 



7 72 ' " 



The area of the triangle o C h is therefore -^ x A = ~^-. 



The parabolic area odh is of the circumscribing rectan- 

 gle. This rectangle is I xSd, The ordinate S^ is equal to 

 S C dG. We have already found S C and d C is the sum of the 



moments of Pi and P 2 , or ^77- x - A = '*-zr. Hence S d = . 



248 4 



p 1? pi? 



*g = ^ . The area of the circumscribing rectangle is then 



v 1? 2 2> ? r> P 



^~. Two-thirds of this is - which subtracted from ^~ 



O .Ztb o 



gives for half the moment of inertia -^-. p Z 3 . Hence the 



Z't 



moment of inertia is ^ p P, as should be. 



54. We see therefore the significance of the area of the equi- 

 librium polygon. 



If, when a number of forces are given, we form t'he force 

 polygon, and then the equilibrium polygon, the ordinates to 

 this last give the moments to the assumed pole distance. If 

 now we take these moments themselves as forces applied at the 

 same points, form a new force polygon with new pole distance, 

 and new equilibrium polygon, the ordinates to this new polygon 

 to the new pole distance will give the moments of the moments 

 or the moments of inertia of the forces. The same method is 

 applicable to moments of a higher order, but in practice we 

 have only to do with those of the second order alone. 



55. Radiii of Gyration.* The moment of inertia of a 

 system of parallel forces P, P, etc., in a plane, with reference 



* In what follows we are indebted to Bauschinger. 



