298 
PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
have become identical with equation (1 1.), and will have supplied a verification of 
that equation. 
If the body to which the pressures ^15 ^2? P 3 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 P 3 in the preceding formula, so that, by that formula there is 
represented the relation between any two pressures P x and P 2 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 Pi and P 2 represent the moving and working pressures on the machine, and 
P 3 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 g sin (p, 
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 
P,=P 2 (^) +^{ P 2 2 L 2 + 2 p 2 P 3 M + P 3 2 <|}* • ■ • (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 by a. The value of — is evidently in this case independ- 
“ tty tty 
ent of the prejudicial resistances, so that a m = — , and — = 1. Assuming, there- 
fore, the direction of the moving pressure P L to be the same with that in which its 
point of application is made to move, representing by 6 the angle through which that 
(L S 
point has at any time revolved, and observing that -jj — a v we have by equation 9, 
U, = V 2 + P -^ ± f" W L 2 + 2 P 2 P 3 M + P 3 a, 2 } 1 d fl. (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 6 is described about the axis. 
If the directions of the pressures Pj and P 2 remain the same during the revolution 
of the body, and the working pressure P 2 be constant, then is the irrational quantity 
in the above expression constant, and the term involving the integral becomes by 
integration, 
| L , + 2 p 2 p 3 M + P/a 2 JU or | p 2 2 L 2 + 2 p 2 P, M + ty <*,*]• \ s 2 . 
tty ttg 
