144 O*^ THE MAXIMUM OF IvftcHANlC POWEllS> AND 



Proib. 7. Let S be the center of rotation, and let 

 the beam be made to revolve horizontally with any 

 given angular velocity : it is required to determine 

 the p' of the whole beam a6ling at B. 



The notation being the fame as in the laft problem 

 and V) being the diftance of the center of gyration 



- , , all the pd"^ p'd?- < ^ , 



from^S, then ^^^TK^b^ thTb^' therefore we have 



/=y!x the beam, ^^^lEll^+iJ^X the weight of the 

 beam the fame as in the laft problem. Hence in this 

 cafe, if :^ be to i; as i to n, then /=pii£^ X W, 

 and when the two ends become equal, fo that the 

 center of rotation coincides with the center of gravity, 

 then the beam may revolve either vertically or hori- 

 zontally, and the -p of both ends together will be ^ 

 the weight. 



Cor. I. Other forms may be derived for the value 

 of/, if the two arms be called d and b^ and their 

 weights c and d refpeftively. For by the general ex- 



preffionsj= f,,eeofi^ebeam ' ^"^ ^^'^IhfS^' ^OW by 

 the firft of thefe, if / be the power of the whole 



beam a6ting at B, we have / n-X the force of the 

 beam = 3^3,^.3.,^^ X ^+^; and by the fecond, f-'jr X 

 the beam =J^i^X^Td': in both cafes -j^^X 



the weight of the beam. Now when azzb, A'—"^ 

 or t the weight j and if ^=^, p'—l the weight alfo. 



Cor. 2. It further appears, that in all cafes of an 

 ofcillating motion of the beam, the p is defined by 

 multiplying the diftance of the center of percuifion 

 from the center of fufpeniion, by the mafs or 

 weight, and dividing by the fqudre of the diftance 

 at which p' is to aft : and that in all cafes of a 

 gyrating motion of the beam, the/;' is defined by -mul- 

 tiplying the fquare of the diftance ot the center of gyra- 

 tion from the center of rotation, % the mafs or weight, 

 and dividing by the fquare of the difiance at which/ is to 



