On Printing Presses and their Theory. 323 



being fixed, and A being moveable along CA ; the power 

 applied at B" and acting perpendicularly to CA is to the 

 resistance overcome at A, as twice the tangent of BAC is to 

 n — 1. 



Suppose in the first place that the power is applied atB : 

 by Prop. 1. Cor. 3. it will be to the resistance : : 2 tan 

 BAC: 1. But the nature ofthe combination requires that the 

 rods should in all states be parallel to each other ; hence 

 velocity of B' = 3 xvel. B ; vel. B" = 5x vel. B, he, and 

 power at B"= } power at B ; so that power at B' : resist- 

 ance at A : : I x 2 tan BAC : 1 : : 2 tan BAC : 5. In the 

 the same manner it may be shewn, whatever be the num- 

 ber of rods combined, that power : resistance : : 2 tan 

 BAC:«— 1. 



Prop. VI, (Fig. 5.) Let the two levers of Prop. I. instead 

 of being united at B, act on each other by means of circu- 

 lar cheeks BD, B'D, having equal radii BO, B'O', less than 

 BA : it is required to determine the power which, applied 

 to B at right angles to BA, shall overcome a given resist- 

 ance acting at A in the line AC. 



To simplify the investigation of the relative velocities of 

 B and A, let it be supposed that when B suffers an indefi- 

 nitely small change of position, A and C move equally in 

 opposite directions. Then the centres O, O', ofthe arcs B 

 D, B'D, will describe lines perpendicular to AC ; and if 

 the motion of B be continued, the point of contact D, the 

 centres 0,0', and the points B, B' ivill all fall upon AC to- 

 gether. Let ab (Fig. 6.) be the position which AB assumes 

 when it has moved an indefinitely small distance : the point 

 o will be in the perpendicular OP, and ao will be equal to A 

 O. Drawing the perpendiculars, hg, oh, Ae, and placing / 

 at the intersection of AO and ao, it may be shewn as hi the 

 demonstration of Prop. 11. that Bg = Oh — ea. Hence Ae : 

 ho : : tan OAP : cot OAP : : sia'^ OAP : cos^ OAP : : O 

 P- : AP2. By sim. tri. ae : ho : : Af:fo. But/O may be 

 taken as=/o ; therefore A/:/0 : : OP^ : AP^ By compo- 

 sition, /O : AO : : PA^ : AO^; hence, putting A to denote 



the ande BAP, fO = AO cos ^ A. Likewise oh = Ae. —r—-r, 



o ^ J sins A 



cos 2A_ 



sin A 



oh : : B/: Of; that is, (by substituting the values already found,) 



and Ae ~ A«. sin A ; so that oh = Aa. — • By sim. tri. srb i 



' sin A •' ° 



