CTTAP. VI.] MOMENT OF INEKTIA. 83 



the moment of each of these areas with respect to X X itself, 

 considered as a force. The method of procedure is then pre- 

 cisely as before. We draw a polygon the sides of which are 

 respectively perpendicular to those of the first polygon, and 

 thus find the statical moments 0'" 1'" V" 2'", etc., to basis H. 



Choosing then a pole C'" at distance H'" and drawing the 

 corresponding polygon, we have 8 1V for the moment of in* 



/ n TT W" O & IV 



ertia. The radius of gyration is then k = V 



a. 8 - 



We have taken H'" = -08, hence = ^ H ^ IV> Hence k 



1_ 



is a mean proportional between 2 8 IV and H. The construc- 



1_ 



tion is given in the Fig. by a semi-circle upon H+ 7j 8 IV . We 



thus find the semi-axis S b' = S b, and can now construct the 

 central ellipse. We have thus found graphically not only the 

 moments of inertia of the cross-section with respect to X X and 

 Y Y, but, by means of the central ellipse, for any other axis in 

 the plane of the Fig. passing through S. 



64. The above method of procedure holds good generally 

 for any cross-section, except that, when there is no axis of sym- 

 metry, the centre of gravity must be found by a second equili- 

 brium polygon whose sides are respectively perpendicular to 

 those of the first. When the moment of inertia with reference 

 to a single axis only is required, the above method becomes 

 quite short and simple, as well as accurate. In our Fig. the 

 scale used as also the number of divisions taken make the pro- 

 cess appear more complicated than it really is. 



With this we shall close our discussion of moment of inertia, 

 merely observing, that all the principles deduced in this chap- 

 ter for forces acting in a plane hold equally good for forces in 

 space. The central curve then becomes an area, we have a mo- 

 ment plane instead of moment axis M, and the ellipse and hyper- 

 bola of inertia become ellipsoid and hyperboloid respectively. 



For a much fuller discussion of the subject than is possible 

 here, we refer the reader to Culmanrts Graphische Statik, pp. 

 160-206 ; also Bauschinger's Elemente der Gra/phischen Statik, 

 pp. 116-168. To the latter we are largely indebted in the 

 preparation of the present chapter ; Plates 10 and 12 are, with 

 slight alteration, reproduced from that work. 



