298 PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 



have become identical with equation (1 l.)j and will have supplied a verification of 

 that equation. 



If the body to which the pressures P^ Pg, P3 are applied have its centre of gravity 

 in the centre of the axis about which it revolves, as is commonly the case in 

 machines, then may its weight be supposed to act through the centre of its axis, and 

 to be represented by P3 in the preceding formula, so that, by that formula there is 

 represented the relation between any two pressures Pj and Pg applied to such a body 

 moveable about a fixed axis, the friction of that axis and the weight of the body being 

 taken into account. 



11. The modulus of a simple machine to which are applied one moving and one 

 working pressure, which is moveable about a fixed axis, and has its centre of gravity 

 in the centre of that axis, the weight of the machine being taken into account. 



Let Pj and Pg represent the moving and working pressures on the machine, and 

 P3 its weight, then is the relation between these pressures in the state bordering upon 

 motion determined by equation (12.), in which X represents the perpendicular upon 

 the direction of the resistance of the axis, and is therefore equal (section 8.) to f sin^, 

 if g" represents the radius of the axis, and (p the limiting angle of resistance. By the 

 substitution of this value of X, equation (12.) becomes 



Pi = P2(ff)+'-$^{P.^L^ + 2P,P3M + P3^«.^}* . . . (14.) 



Now it is evident that this equation is of the form assumed in equation 7, section 6, 

 the term involving the irrational quantity being represented by E (in equation 7-), 



and the coefficient of Pg, ^, by a. The value of — is evidently in this case independ- 

 ent of the prejudicial resistances, so that a^Q) = —, and — = 1. Assuming, there- 



fore, the direction of the moving pressure P^ to be the same with that in which its 

 point of application is made to move, representing by 6 the angle through which that 



d S 

 point has at any time revolved, and observing that ^^ = a^, we have by equation 9, 



V, = V, + ^-^f\p,^U + 2P,P,M + P,a,^)Ul), .... (15.) 



which is the modulus of the machine, and in which the term S, involving the integral, 

 represents the work lost by friction whilst the angle is described about the axis. 



If the directions of the pressures Pj and Pg remain the same during the revolution 

 of the body, and the working pressure Pg be constant, then is the irrational quantity 

 in the above expression constant, and the term involving the integral becomes by 

 integration, 



'-f^{P.^L^ + 2P.P3M + P3V}*.e,ore|^{p,3L^ + 2P,P3M + P3^«,^}*.S„ 



