PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
299 
(observing that 9 a 2 = S 2 ), or bringing S 2 under the radical sign, 
~ { p 2 2 S 2 2 L 2 + 2 P, S 2 2 P 3 M + P 3 2 S 2 2 «, 2 ] 4 , 
or 
{ U * 2 U + 2 U 2 S 2 p 3 M + P 3 2 S 2 2 a, 2 
so that in this case of a constant direction of the moving pressure, and a constant 
amount and direction of the working pressure, the modulus becomes, 
U, = U, + ^ { U 2 2 L 2 + 2 U 2 S 2 P 3 M + P 3 2 ^ j *; . . . (1G.) 
and the work lost by friction whilst the space S 2 is described by the working point, 
is represented by the term involving the irrational quantity in this equation. 
12. A machine working about an axis of given dimensions under two pressures, 
P x and P 2 , the direction and amount of one of which P 2 are given, it is required to 
determine that constant direction in which the other pressure Pj must be applied, so 
that the machine may be worked with the greatest economy of power. 
It has been shown in the last section that the work lost by friction is represented, 
in the case here supposed, by the formula 
p sin 
a 
j^(u 2 2 L 2 + 2 U 2 S 2 P 3 M + P 3 2 S 2 2 a,, 2 } 4 (17.) 
The machine is evidently worked then with the greatest economy of power to yield 
a given amount of work, U 2 , when this function is a minimum. Substituting for L 2 
its value 
<q 2 + 2 a 1 a 2 cos q 2 + a 2 2 > 
and for M its value 
it becomes 
a x { a 2 cos q 3 -f- cos / 23 } (section 10.), 
p sin <p 
Cl-I dc) 
U 2 2 («! 2 +2 a x a 2 cos q 2 + a 2 2 ) + 2 U 2 P 3 (a 2 cos q 3 + a x cos / 2 3 ) -f P 3 2 S 2 2 a x 2 j 2 ( 1 8.) 
Now let us suppose that the perpendicular distance a 2 from the centre of the axis 
at which the work is done, and the inclination / 23 of its direction to the vertical, are 
both given, as also the space S 2 through which it is done, so that the work is given 
in every respect ; let also the perpendicular distance a x at which the power is applied, 
be given ; it is required to determine that inclination q 2 of the power to the work 
which will under these circumstances give to the above function its minimum value, 
and which is, therefore, consistent with the most economical working of the machine. 
Collecting all the terms in the function (18.) which contain (on the above suppo- 
sitions) only constant quantities, and representing their sum 
U 2 2 ( a l 2 + «2 2 ) + 2 P 3 S 2 a \ ( U 2 C0S '2.3 + P 3 S 2 ) 
by C 2 , it becomes 
{ 2 a, a 2 U, (U, cos , h2 + P 3 S 2 cos ,, 3 ) + C 2 } 
2 R 
MDCCCXLI. 
