CHAP. VI.] MOMENT OF INERTIA. 75 



bola, we must observe whether the diameter conjugate to the 

 moment axis is real or imaginary. In either case the centre of 

 gravity is the pole of the line -symmetrical to the moment axis 

 in that hyperbola for which that diameter is real or imaginary. 



The construction is given in PI. 11, Fig. 35. 



Upon S o' = S o we describe a semi-circle. With S as cen- 

 tre, and S A' a = semi-diameter of the central curve, describe 

 an arc, and from the intersection with the semi-circle drop a 

 perpendicular upon So'. The point M thus found is the centre 

 of gravity of the moments. For : a 2 = a M? + m 2 and a M 3 

 = m (i m) hence a 2 = m 2 + m im* = m i. The central curve 

 being known as also the distance i, the point M can be readily 

 found. 



61. Case where the Direction of tbe Conjugate Axis of 

 the Inertia Curve can be at once Determined. There are 

 certain special and practical cases in which the conjugate direc- 

 tions or axis of the inertia curve can be at sight determined, so 

 that only the length of the semi-diameters remains to be found. 

 The most important of such cases are as follows : 



(1.) When in a system of parallel forces, these forces can be 

 so grouped in pairs, that the lines joining the points of appli- 

 cation of each pair are all parallel, and the centres of gravity 

 of each pair all lie in the same straight line. Then for the 

 central curve and all inertia curves whose centres lie upon this 

 straight line, the direction of the axis conjugate to this line is 

 the same as that of the lines joining the points of application 

 of each pair. 



This is easy to prove. For, for each pair, the sum of the 

 moments with respect to the line joining their centres of gravity, 

 is zero. These moments regarded as forces and applied at the 

 points of application, give therefore for each pair two parallel 

 opposite and equal forces, the sum of the moments of which 

 for any line parallel to the line joining the points of applica- 

 tion, is zero. This is the case for all the pairs, and therefore 

 the direction of the lines joining the points of application is 

 that of the axis conjugate to the line joining the centres of 

 gravity, for the central curve as also all inertia curves whose 

 centres lie upon this last line. 



(2.) When the forces can be so grouped that the points of ap- 

 plication of each group lie in parallel lines, and the centres of 

 gravity of the groups lie in the same straight line. Then this 



