CHAP. VI.] MOMENT OF INERTIA. 73 



curve, whose centre lies upon an axis through the centre of 

 gravity of the forces, at a distance i from this centre. 



Any two curves at equal distances either side of the centre of 

 gravity are therefore equal. If the semi-diameter of the central 

 curve a is real, and therefore a? positive, a 2 is also positive and 

 greater than a 2 . All the inertia curves are therefore of the 

 same kind as the central curve, and enclose the centre of grav- 

 ity. If, however, a 2 is negative, and the central curve there- 

 fore an hyperbola; all those inertia curves whose centres are 

 distant from the centre of gravity by a distance i less than a 

 are hyperbolas also. For a distance i equal to a, the curves re- 

 duce to straight lines equal and parallel to the conjugate diame- 

 ter of the central curve. For i greater than a, the curves be- 

 come ellipses. 



60. Centre of Action of the Statical Moments of the 

 Force.* We again suppose, through the centre of action of 

 the forces S [Fig. 35, PI. 11] a line N N drawn which cuts the 

 central curve at A and A'. Two such points we have in every 

 case, except when the curve is an hyperbola, andN N coincides 

 with an assymptote. 



Let (5 be the conjugate axis to NNin the central curve, E a 

 parallel to it through any point o distant i from S, and also 

 conjugate to N N in the inertia curve whose centre is o. Then 

 since the statical moments of the forces with reference to N N 

 is zero, the centre of action of the statical moments with re- 

 spect to E, considered as forces acting at the points of applica- 

 tion, will be somewhere upon N N. It is required to find 

 .where. 



We call q the distance of any point of application from E, 

 measured parallel to N N, and positive when upon the same 

 side of E as S, then i is essentially positive. 



As before, q is the distance of the points of application from 

 (5, also measured parallel to N N, and positive in the same 

 direction as q. 



Then we have always 



q = q+i. 



and for the moments of inertia of the forces with respect to E 

 and(* -ZP q a = .?(+)"= 2 Pj'-H'-SP 



or when a is the semi-diameter of the central curve, S A = S A' 

 and 



* See Supplement to Chap- VII., Art. 10, latter part. 



