202 JOHN C. KOCH 



least stiffness, are drawn. In most cases these are at right 

 angles to each other and usually simple inspection will at once 

 locate these axes, which intersect at the center of gravity of the 

 cross section. Then a series ot small squares are drawn or pro- 

 jected over the cross section with their sides parallel to the 

 principal axes. In the study of the femur these lines forming 

 the squares were drawn at intervals of ^ inch, so that the 

 area of each square is 4^0 o^ ^ square inch. Then for each 

 axis, a tabulation is made successively of the area of the cross 

 section included between the adjacent lines parallel to the axis 

 considered. Then each separate area thus found is multiplied 

 by the square of the distance from the axis to the center of 

 gravity of that particular area. Each product thus found is the 

 moment of inertia of that area about the given axis. Each of 

 the separate areas is multiplied by the square of the distance 

 from the axis to the center of gravity of that particular area; 

 and the sum of all these products is the moment of inertia of the 

 cross section about that axis. 



48. Example. For example it is required to determine the 

 moment of inertia about the two principal axes of a rectangular 

 figure 1.0 inch by 1.2 inches. 



Let A-A and B-B denote the two principal axes as shown in 

 figure 9. If the area is expressed in units of ^ inch squares 

 and the distances in terms of ^ inch units, it will siniplify the 

 tabulations and the results may be accurately reduced to the 

 usual units. 



In the particular case assumed above, the exact integration 

 by the formula derived from calculus may be used, and the 

 graphical calculation of the moments of inertia about the two 

 axes checked. For a cross section of rectang\ilar shape the 



value of the moment of inertia is I = .^ , where b is the breadth 



and h is the height or dimension of the section at right angles 

 to the axis about which the moment of inertia is to be com- 

 puted. Substituting in this formula we find for axis A-A, 



/ = -T^ = 0.1440 inches*. By the graphical method / for 



