66 Statics. 



Geom. 113. With respect to solids of revolution, the general value 

 * ' of <J is xy a ; the general expression for the distance of the cen- 

 tre of gravity, will thus be-' n y ^ * . This formula may 



be applied to the sphere, the ellipsoid, &c., and by means of the 

 ellipsoid may be determined the centre of gravity of masts. 



114. What we have said with respect to centres of gravity, 

 will enable us to arrive at a solution in any case that may occur ; 

 we shall, notwithstanding, point out particularly the course to be 

 pursued in order to find the centre of gravity of the immersed 

 part of a ship's bottom, or rather of a homogeneous solid of this 

 form. 



We may suppose the centre of gravity to be in a vertical plane 

 passing through the axis of the keel, and we have only to deter- 

 mine its horizontal distance from a vertical line drawn through a 

 given point of the stern-post, and its vertical distance from the 

 keel. 



For each of these objects we must begin by determining the 

 centre of gravity of a surface ANDFPB. bounded by two par- 

 Fig. 50. allel lines AB, DF, and two equal curves, similar to A\"D, 

 BPF. 



If we had the equation of this curve, nothing would be more 

 easy than to determine its centre of gravity G, by the preceding 

 methods. But not having it, we must conceive the line CE to 

 pass through C and , the middle points of AB, DF, respective- 

 ly, and that this line is divided by the perpendiculars 77/, AW, 

 &c., into equal parts, so small that the arcs comprehended be- 

 tween any two adjacent perpendiculars, shall not differ sensibly 

 from straight lines. We must next take the moments of the trape- 

 zoids DTHF^ TKMH, c., with respect to the point , and 

 divide the sum of these moments by the sum of the trapezoids, 

 Geom. that is, by the surface ANDFPB. This surface, being compos- 

 l78 * ed of trapezoids, is readily determined. We have, therefore, 

 only to find a simple expression for the sum of the moments. 

 Now the distance of the centre of gravity of the trapezoid THFD, 

 from the point E. is 



