14^ ON THE MAXIWUXf OF MECHANIC POWERS, AN1> 



Cor. It appears from Cor. i of the laft problem, 

 that when the vibrations of a beam are the quickeft 

 polTible, X is equal to^ — Wit when the point of fufpen- 

 iion is taken on that fide the center of gravity towards 

 A. Now fince \ is the diftance of the center of gravity 

 of the beam from A or B, it follows that IVf exprefles 

 the diflance of the centerof percuffion from the center 

 of gravity when the vibrations are the quickeft poffible. 

 But it appears from this problem, that i-^i exprefles 

 the diftance of the center of gyration from the center 

 of gravity, when the beam is made to revolve on that 

 center. Therefore if the beam be fufpended, by what 

 in this cafe is the center of gyration, the vibrations will 

 i)e the quickeft poffible. 



C o R . 2. I F the parts AS, SB be denoted by ^ and 2: as in 



Cor.ljofthe laft prob. then wr: v -^^^ jS+i^. 



Thcnifx=o,and z become equal A B, zvzzz>/~^=v»/~: 

 and when x and 2; are equal, wtzx>/^ or 2^/"|*=^ ^"J*, 

 and laftly, if x—-\Zy then wzz^z -, all which are pre* 

 cifely the fame as in the laft problem. 



Prob. 3. Let ABD be a folid beam of uniform 

 thicknefs, having an angle at D, and let ADrrDB, 

 and AE = EB=:;c, and if the line ED be continued to 

 the center of rotation S, ^^^ 



then S E will be perpe'ii- /y,^ 



dicular to AB, and there- / ^\ 



fore A S=B S, and the / p \ 



beam will be in the fame ,^ ^^^^4^ \ 



plane with the triangle /y^^ I ^^^^^*^ 



A SB, and being made to Z^:^ :^ ^^^^^-^ 



revolve round the center _^^^.«. 1 --^B 



S, retaining its polition E 



with refpe£t to the line S E: it is required to determine 



the diftance of the center of gyration from S. 



Put DS=<^,andAD=BD=''^,andaIfoED=^. Theia 



AS 



