28 



INTRODUCTION TO DYNAMICS. 



[52. 



52. As an illustration let us determine the centroid of the 



volume OABCD (Fig. 13), 

 bounded by the three co-ordi- 

 nate planes and the warped 

 quadrilateral (hyperbolic 

 paraboloid) ABCD. The latter 

 is generated by the line LM 

 gliding along AB and CD so 

 as to remain parallel to the 

 plane yz. The data are OA 

 CD=a, OB=b, OC=AD = c. 

 We take as element an 

 infinitesimal prism PQ off 



Fig. 13. 



base dxdz and height y. From similar triangles we have 

 = (c-z)/c, and RL/b=(a-x)/a\ hence 



_rdX C Z 



a c 



Thus we find, rejecting the constants which cancel in numerator 

 and denominator, 



***x(a-x)dx-(*-- 



*x(a-x)(c-z)dxdz 



. . 





a-x)(c-z}dxd 



I 



_ /o 



x(a-x)dx - g 



a* 



X(ax)dx a 2 

 2 2 



y=- 



,ax c z j j 

 b dxdz 



b 



2ac 



Finally, z = \c, by analogy with x. 



