76 MOMENT OF INEKTIA. [OHAP. VI. 



straight line gives the direction for the central curve and every 

 inertia curve whose centre lies upon it, of the diameter conju- 

 gate to an axis passing through the centre and parallel to the 

 lines joining the points of application. 



For if we take any such axis, the points of application of the 

 forces in each group are equally distant. The statical moments 

 for each group are then proportional to these distances. If, 

 therefore, they are considered as forces, their centre of gravity 

 coincides with that of the forces themselves, and lies therefore 

 in the line joining the centres of gravity of the groups. The 

 centre of gravity of the whole force system lies then in this 

 line, whicji is therefore the direction of the axis conjugate to 

 the line parallel to the lines joining the points of application, 

 in the central curve, and also all curves whose centres lie upon 

 this line. 



(3.) When the forces can be so grouped that the centres of the 

 central curves of each group lie in the same straight line, and 

 the diameters in each curve conjugate to this line, are parallel. 

 Then in the central curve of the entire system, the diameter 

 conjugate to this line is also parallel to these diameters. For, 

 for any axis parallel to these diameters, the centres of gravity 

 of the moments of the forces in each group lie upon the line 

 joining the centres of the curves. The centre of gravity of the 

 moments for the entire system lies then also upon this line, 

 which is therefore the direction of the axis conjugate to an axis 

 parallel to the diameters of the curves, for any inertia curve 

 whose centre lies upon this line. 



In all these cases, if the directions thus found are perpendicu- 

 lar, we have to do with the principal axes. 



62. Practical Applications We can now apply the above 

 principles to practical cases, and as in the determination of the 

 moment of inertia of irregular figures, we have to deal with 

 triangles, parallelograms and trapezoids, we have first to con- 

 sider these three cases. 



1st. The Parallelogram. PI. 11, Fig. 36. 



The moment of inertia of a parallelogram is, as is well known, 



M = a J*,* a being the breadth and b the depth. 



