236 On the Theory of the Wedge, 



to keep either half in equilibrio will be the same, and conse- 

 quently the whole power must now be twice as great in propor- 

 tion to the resistance as before. Or the power will be to the re- 

 sistance as twice the sine of the semi-angle of the wedge to the 

 sine of the angle which the direction of the resistance makes 

 with the face of the wedge, and, consequently, when the direc- 

 tion of the resistance is perpendicular to ac, as 6 h to a c. 



Or thus : — let the resisting body be urged against the wedge, in 

 the direction ke; and let it be kept from sliding along the face of 

 the wedge by a force acting in the direction de, perpendicular to 

 k e. There are now three forces acting on the body, namely, the 

 resisting force k e, the sustaining force in the direction d e, and 

 the re-action of the wedge in the direction e o, perpendicular to 

 the surface b c. Draw k o perpendicular to ^ e ; and, since the 

 three forces are in equilibrio, they will be to each other as the 

 three sides of the triangle koe. Now, as the wedge is kept in 

 equilibrio by the fore e, the power na, acting perpendicularly 

 to b /i, and the re-action of the plane m r, these three forces are 

 to each other as the sides cb, bh, and he of the triangle cbh^ 

 and therefore na ', oe ;i sin. bch i sin. b h c ; and because o e l 

 ke : rad. : sin. koe, or sin. keb, therefore, by compounding, na 

 ; ke i\ rad. x sin. bch i sin. b he X. sin. keb ; or since rad. x 

 sin. 6 c ^ = 2 sin. bca X. sin. bhc^ na I ke ii 2 sin bca l sin. 

 keb ; SiS before. 



Mr. Nicholson does not take the proposition generally ; but 

 after showing that, when the resistance is pei*pendicular to a e, 

 the power required to keep one half, ab c, of the wedge in equi- 

 librio, is to the resistance as a 6 to a c, he (rightly I think) in- 

 fers, that " as the pressure on the other half of the wedge acts 

 with equal effect, a double force will be required to preserve the 

 equilibrium ; that is, a force as b h to h c [the same as above]. 

 Or, in general terms ; in any wedge, as the line b h, joining the 

 two equal sides cb and c A, is to the distance between the vertex 

 c, and the middle point a of bh, so is the force [or power] im*- 

 pressed to the resistance." I am, Sir, 



Your obedient humble servant, 



Evesham, May llth, 1825. I. T. 



