74 MOMENT OF INERTIA. [dlAP. VI. 



Let now m be the distance of the centre of gravity or action, 

 of the moments of the forces with respect to E, from E, and m 

 its distance from Cr, positive the same as q and q. Then 



m = m + i 



and since the sum of the moments is equal to the moment of 

 the resultant : 



But the sum of the moments P q of the forces with reference 

 to E, is equal to the product of the sum of the forces into the 

 distance i of the centre of gravity of the forces from E. Hence 



and therefore 



iS P = 2P q 2 = (a 2 



or, m i a? + &. 



Introducing the value for m 



(m + i) i = a* -f & 

 or m i = a 3 . 



If now a 2 is positive, which is always the case for an ellipse 

 as central curve, in is also positive, and is therefore to be laid 

 off from S along N N on the opposite side of from o. If 

 then we conceive an axis E' drawn parallel to E, and symmet- 

 rical with reference to S, which axis we shall call for conven- 

 ience the symmetrical axis to E, we see from the above relation 

 that M is the pole of this axis in the central curve. 



If, however, a 3 is negative, therefore a imaginary, in is nega- 

 tive, and must be laid off from S towards o, and the point M 

 thus found is therefore the pole of the axis E itself, or in the 

 case of an hyperbola is the pole of E' in that hyperbola which 

 is not cut by N N, and for which therefore A A' is imaginary. 



Hence we have the principle 



If we consider the statical moments of the forces with refer- 

 ence to any axis as E as themselves forces acting at the given 

 points of application, the centre of gravity of these moment 

 forces does not coincide with the centre of gravity of the origi- 

 nal forces, but is the pole * in the central curve of an axis E' 

 parallel and symmetrical to E. 



In those cases where the central curve becomes an hyper- 



* POLAR LINE OF A POINT, in the plane of a conic section, is a line such, 

 that if from any point of it two straight lines be drawn tangent to the conic 

 section, the straight line joining the points of contact will pass through the 

 piven point, which is called a pole. 



