CHAP. III.] CENTRE OF GRAVITY. 29 



CHAPTER III. 



CENTRE OF GRAVITY. 



3O. General Method. One of the most obvious applica- 

 tions of our method, as thus far developed, is to the deter- 

 mination of the centre of gravity of areas and solids. We 

 shall confine ourselves to areas only, merely observing that all 

 the principles hitherto developed apply equally well to forces 

 in space. The forces being given by their orthographic pro- 

 jections upon two planes after the manner of descriptive geo- 

 metry, the projections upon each plane may be dealt with as 

 forces lying in that plane, and thus the projections of the force 

 and equilibrium polygons, the resultant, etc., determined. 



A body under the action of gravity may be considered as a 

 body acted upon by parallel forces. The resultant of these 

 forces being found for one position of the body [or the body 

 being considered as fixed, for one common direction of the 

 forces] may have its point of application anywhere in its line 

 of direction. 



For a new position of the body [or another direction of the 

 forces] there is another position for the resultant. Among all 

 the points which may be considered as points of application of 

 these two resultants there is one which remains unchanged in 

 position, whatever the change in direction of the parallel forces. 

 This point must evidently lie upon all the resultants, and is 

 therefore given by the intersection of any two. 



It is hardly necessary to give illustrations of the method of 

 procedure. 



Generally, we divide up the given area into triangles, trapez- 

 oids, rectangles, etc., and reduce the area of each of these fig- 

 ures to a rectangle of assumed base. The heights of these 

 reduced rectangles will then be proportional to the areas, and 

 hence to the force of gravity acting upon them ; i.e., to their 

 weights. Consider then these heights as forces acting at the 

 centres of gravity of the partial areas. Construct the force 



