Stoney — On a Method of Determining Moments of Inertia. 337 



of the wedge W. Let T be the moment of inertia of the cross- 

 section round m n. Let I he the moment of inertia round a line 

 parallel to mn, through the centre of gravity of the cross-section. 

 Then will 



r = j zfdy. (1) 



Again, VY = tan a j zy 2 dy, 



which, by (1), = tan aT. (2) 



Also V = tan a jzydy = tan a Ay. (3) 



From (2) and (3) we get 



I' = AYy. 



And since, by a well-known theorem, 



l' = I+Ay\ 



it follows that the moment of inertia round a line parallel to m n 

 through the centre of gravity of the cross-section will be 



I = Ay(Y-y), 



which is the theorem to be proved. 



In order to find the positions of the centre of gravity of the 

 sections, I have found it convenient to use a table, levelled, and 

 with a row of points standing in a straight line, and projecting 

 -g- inch. The section is then balanced on these points, and when 

 the position is found at which it is exactly balanced a slight pres- 

 sure, if the section is not of hard material, drives the points into 

 the section, and then with a straight-edge a line over which the 

 centre of gravity lies can be drawn. A sharp knife-edge is in 

 some cases more convenient than a row of points. The edge of 

 the wedge is to be made parallel to the knife-edge, or row of 

 points, and to facilitate this adjustment it will be found convenient 

 to draw upon the table a number of lines parallel and perpendicu- 

 lar to the row of points. 



With wooden models, made with ordinary care, I found that 

 the moments of inertia by experiment and calculation did not in 

 any case differ more than 2 per cent., and were generally correct 

 to 1 per cent. This for almost all engineering purposes is of 

 quite sufficient accuracy, and in fact is, wherever the section is of 



SCTEN. PROC, R.D.S. — VOL. V. FT. V. 2 A 



