THE EFFECTS OF MACHINES WHEN IN MOTION. 1 39 



gravity, the power of ofcillation will ceafe, and the 

 motion, if there be any, will be rotatory. 



CoR. I. If it be required to determine the diftance 

 of the center of fupenfion when the vibrations are 



the quickeft poflible, thenj^', or its equal ^—"^JH^x^ 

 becomes a maximum, and ther efore its flu x ion, by 

 rhakinn^ .v variable, is z:<7. Hence i2.v.^; — 6wx 3'z/ — ^x 

 — yvy^^rj^ — 6^^v+6A:^zro, and Arz:-J±-^Vt or -z— -iV-j 

 when S is taken towards A, or on that Tide of the 

 center of gravity. 



Cor. 2. Other forms may be obtained if AS=Ar 

 and SB=z, both variable quantities : — for then^-fi 

 will exprefs the fluxion of both ends, and x'^Ar'^-^ the 

 fluxion of all the /)^^ in AB. And fince the force of any 

 ofcillating body may be exprefled by multiply ing the dil- 

 tance of the center of gravity from the center ot fufpen- 

 fion, into the body itfelf, the force of AB is therefore 



defined by — ^ Hence j- — -zr—i -^jrz^^K^. 



1. 

 Then when x—o, y becomes equal ^x ; and if x^\Zy 

 y—z ; — and when x and z are equal, jy vanilhes. 



Prob. 2. The notation remaining, as in the laft pro- 

 blem : let the center of gyration be required, while the 

 beam A B is made to revolve round a center S at any 

 variable diftance from A: and let w exprefs the diftance 

 of the center of g yrati on from S . Then we Ihall have 



IV —V = \^ . Hence ir xzzo^ 



w=:Vx/-\. and when .v=:x, fo that S may be in the cen- 

 ter of gravity of A B, then w— 1'/\, When xzn yoy *o 

 becomes equal 4**^ alfo, and in this cafe the center of 

 gyration will be at the fame diftance from B, that the 

 center of rotation is from A. 



Cor. 



