110 Statics. 



The essential parts of the machine are represented in figure 

 101, where AMN is the plane of the wheel, FF the axis of the 

 cylinder, and BDL a section of the cylinder, parallel to AMN, 

 and passing through the cord Dp. 



Having drawn the radius EA to the point A, where the pow- 

 er q acts upon the wheel, suppose a plane FEA passing through 

 FF, and EA^ and meeting BDL in IB ; IB will be parallel to 

 EA. Join AB, and through this line and the direction A q of 

 the power, imagine a plane q AG to pass meeting the axis FF in 

 some point G. Lastly through B and G draw B & and G r par- 

 allel each to A q. 



This being supposed, the force q may be decomposed into 

 5,^ two other forces w, r, directed according to B , G r ; and as this 

 last passes through the axis of the cylinder, it can have no effect 

 in turning the machine about this axis, and consequently can 

 contribute nothing toward the support of the weight p. It will 

 therefore be expended against the supports F, F. There will 

 accordingly be only the force & by which an equilibrium with 

 the weighty is to be effected. Now (1.) This force is directed 

 in the same plane BDL in which the action of the weight is ex- 

 erted. (2.) The two lines B w, 727, being parallel respectively to 

 the two A 9, AE, which are at right angles to each other, B & is 

 Geom. perpendicular to 7?7, and consequently a tangent to the circum- 

 70 ' ference BDL. We may therefore consider BID as an angular 

 lever, of which the fulcrum is at 7; and since the distances of 

 the directions of the two powers w, p, from the fulcrum are equal, 

 these two powers must be equal ; we have accordingly v = p. 

 Let us now see what is the ratio of & to q. 



According to what has been laid down, we have 



q i TUT 1 1 BG i AG j 

 but the similar triangles G7?7, GAE, give 



BG : AG :: BI : AE; 

 whence 



q : w : : BI : AE ; 



or, since = />, 



q : 'p : : BI : AE 5 



