336 



Scientific Proceedings, Royal Dublin Society. 



and a sloping section, intersecting along mn, a line parallel to the- 

 axis round which the moment of inertia is to be taken. This- 



llgggja 



m. 



y 



latter, in nearly all the cases required by engineers, is a line 

 passing through the centre of gravity of the cross-section. 



Balance S and W on knife edges, and so determine y, the 

 distance of the centre of gravity of S, and Y, the distance of the 

 centre of gravity of W, from mn. Then will the moment of 

 inertia of the cross-section round mn be 



I f = AYy, 



where A is the area of the cross-section. From this it at once- 

 follows that the moment of inertia round an axis parallel to m n 

 through the centre of gravity will be 



I=Ay{Y-y). 



This last is the quantity required in engineering problems, and, to 

 determine it, it is only necessary to measure Y and y as above, and 

 to determine A, the area of the cross-section. 



To prove this — Let y be the horizontal [distance "of any point 

 from m n. Let z be the breadth of the section at that distance 

 from m n. Let A be the area of the cross- section. Let V be the 

 volume of the wedge W. Let a be the angle of the wedge W. 

 Let y be the distance of the centre of gravity of the cross-section S 

 from m n. Let Y be the distance from m n of the centre of gravity 



