414 THE STRENGTH 



a to bf hence a : b :: \ : — =. compfession at 

 bottom ; the force to produce that compression 

 is therefore — , which is another vahie for r. 



a ' 



Substituting this for it gives ~ / Yycly = /\ 

 and "^fYfdy =/' x P/. 



7. If we substitute, in equations (1) and 

 (3), their values from the preceding article 

 we have : 



From equation (1), -^fXxdx = ^fYydy, 



whence fX xdx =; nf Yydy (4) 



From equation (3), 



WxGf=-^fXx'dx + '^Vfdy - - - (5) 



These two equations, when n and / are given, 

 will suffice to find the position of the neutral 

 line, and the strength of any body ; but we 

 may put them under a simpler form as below. 



8. Call the area ADB = s, the distance from 

 AB of the center of gravity of that area =: g-, 

 and the distance of its center of percussion 

 from AB =z p, call too the area ACB = s', the 

 distances from AB of its centers of gravity and 

 percussion g' and p' respectively. We have 



