66 MOMENT OF INERTIA. [CIIAP. VI. 



at once the moment of inertia of the given four forces with 

 reference to X X and Y Y as axes. If we were to suppose the 

 same forces with the same points of application to act parallel 

 to Y Y instead of X X, the distances q^ q 2 being measured par- 

 allel to X X instead of Y Y, we should have the force polygon 

 G! O li 2 t 3i i instead of C 1 2 3 4, and a precisely similar 

 construction would give us a? multiplied by pole distance for 

 the moment of inertia, and a' for the radius of gyration. We 

 recommend the reader to follow through the construction as 

 shown by Fig. 34-. 



56. Curve of Inertia Ellipse and Hyperbola of Inertia. 

 If having found the radius of gyration as above, we lay it off 

 from the axis on either side, in a direction parallel to the direc- 

 tions in which q q z , etc., are supposed measured, and through 

 the points thus determined draw two parallels to the axis M' 

 and M" on either side, and then suppose the axis to revolve in 

 the plane of the forces about any point as O situated in the 

 axis ; the lines M' and M" also revolve and enclose a curve of 

 the second degree, whose centre coincides with O. Thus, if in 

 PI. 10, Fig. 34, we lay off O b along Y Y on both sides of X X 

 equal to o V = k already found, and then let X X revolve 

 about O, K J and J K will also revolve, and enclose either an 

 ellipse or hyperbola. 



In order to prove this, take O as an origin of co-ordinates. 

 Let the co-ordinates of the points of application of the forces A, 

 A,, etc., be a;, y,, a?, j a , etc. From each of these points A draw 

 parallels to the axis of y, intersecting the axis of x in the points 

 C. Then O C = x, A C = y. Now pass through the point O 

 an axis of moments M in any direction, and project for each 

 point OCA parallel to this axis upon the line q, which meas- 

 ures the distance of each point from the axis of moment (not 

 necessarily perpendicular distance). This projection is evi- 

 dently equal to q. Denote by a and ft the ratios by which dis- 

 tances along X and Y must be multiplied, in order to obtain 

 their projections upon q, by lines parallel to M. Then 



q= ax + fty 

 for each point of application, and hence 



2 P <? = 2 P (a x + ft yf 



or since for one and the same axis M, and direction q, a and ft 

 are constant, 



