Centre of Gravity. 47 



The lines FA", FO, are called the axes of the moments. 



79. If now we suppose the point F, which we at first took 

 arbitrarily, to be in G, G'G and G''G become each equal to zero. 

 Therefore the sum of the moments with respect to FA", and the 

 sum of the moments with respect to FC, must in this case be each 

 equal to zero. 



80. We now say, that if the sum of the moments of several 

 bodies with respect to the straight line TS, passing through the Fig. 28. 

 point G, is equal to zero ; and the sum of the moments with res- 



pect to the straight line DE, perpendicular to TS, and passing 

 also through Gt, is in like manner equal to zero ; the sum of the 

 moments with respect to any other straight line LH, passing 

 through the same point G, will also be equal to zero. 



Indeed, having let fall upon the lines DE, TS, LH, the per- 

 pendiculars AA', AA", AA"' ; if we suppose that the point / is 

 that in which AA meets LH, from the right-angled triangle GA'I, 

 we have 



sin GIA' : GA' : : sin DGL : A' I, 

 or 



TN^ r //i// r\/^ir /itr j4A"s\n DGL 

 cos DGL : AA" : : sin DGL : A' I = _-.. 



cos DGL 



whence 



cos DGL 



Now from the right-angled triangle IAA"', we have, radius being 

 supposed equal to 1, 



1 : AI :: sin AIA"> : A A'" 



: : cos DGL : A A" = AI x cos DGL; 

 that is, substituting for AI its value above found, 



AA" = AA cos DGL AA' x sin DGL ; 

 hence, if we multiply by the mass m to obtain the moment, we 

 shall have 

 m X AJf" = m X AA' x cos DGL m x AA" X sin DGL; 



in other words, the moment of the body m with respect to the 

 axis LH, is equal to the cosine of the angle DGL, multiplied by 

 the sum of the moments with respect to the axis DE, minus the 



