THE EFFECTS OF MACHINES WHEN IN MOTIO%'. I45 



-^ — "!brceofthebiI^> ^'"^d /^ ~ "^^^vj^ X ^'•"'^'ce of the beam ; 



that is/)':z: -;;^ X Au. '-^•— v. ■£- = Triir^r^^T^ X by 

 the mafs. And by fubftituting the value ofj, we have 



/>' z: ■ .^i_^ . .,^. 2- X t)y the mafs, a general exprelTioii 



for the value of -p for any beam of equal thickneis, 

 and whofe weight is as the icr.f^th. 



Cor. I. Now when x vaniflies, p'~\ the mafs ; fa 

 that when the beam is fufpended at the extremity A, 

 then the weight which applied at the diftance AB to 

 an inflexible line vibrating with any given velocity, fo 

 as to have the fame force as the mafs of the beam col- 

 lected into its center of percullion, and moving with 

 the fame angular velocity, ihall be equal one-third 

 the weight of the beam. 



CoR. 2. If .V be taken to v in the ratio of i to » ; 

 then by fubftituting the value of x in the above e?c- 



prefllon,/r= -,^a_5^ . i, X by the weight, when that weight 

 is defined by i', its length ; and the weight of the 

 iliorter end, unity. Or fuppofe the whole weight to 



be W, then /=r"-^Fi^i^X W : and in this cafe the 

 weight of the iliorter end will be defined by — , 



and that of the longer by — ^, let W be what it wiiL 



CoR. 3. When «=2 then/=TW: but it mult be 

 remembered that />' is the power of the zvhok beam, 

 fmce it is compared with the whole mafs collected into 

 the center of percuffion ; and is therefore the/)' of both 

 ends reduced to B, and hence in cafes where the tv^'o 

 ends are equal, as in the prefent one, the p' of each 

 end is y of half the beam, which together are equal 

 ■ to Y W, the /)' of the whole beam. 



Pros. 



