PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
301 
the same side of the axis. It is then a condition necessary to the most economical 
working of any machine (whatever may be its weight) which is moveable about a 
cylindrical axis under two given pressures, that the moving pressure should be ap- 
plied on that side of the axis of the machine on which the resistance is overcome, or 
the work done. It is a further condition of the greatest economy of power in such a 
machine, that the direction in which the moving pressure is applied should be in- 
clined to the vertical at an angle q 3 determined by the formula 
h.3 — X — ‘23 + tan 1 { p 2 + p 3 cos <2 3 } ( 20 ‘) 
When i 23 — 0, or when the work is done in a vertical direction, < 13 = cr, whence it 
follows that the moving power also must in this case be applied in a vertical direc- 
7 r 
tion, and on the same side of the axis as the work. When < 23 = 7 y, or when the work 
P 
is done horizontally, tan y = W- 3 ; 
*2 
h.2 
= % + tan - 
The moving power must therefore in this case be applied on the same side of the 
axis as the work, and at an inclination to the horizon whose tangent equals the frac- 
tion obtained by dividing the weight of the machine by the working pressure. 
3 7T 
Since the angle i X2 is greater than t and less than — , therefore cos i X2 is nega- 
tive ; and, for a like reason, cos < L3 is also in certain cases negative. Whence it is ap- 
parent that the function (18.) admits of a minimum value under certain conditions, 
not only in respect to the inclination of the moving pressure, but in respect to the 
distance a x of its direction from the centre of the axis. If we suppose the space Sj 
through which the power acts whilst the given amount of work IJ 2 is done, to be given, 
and substitute in that function for the product S 2 a x its value Sj a 2 , and then assume 
the differential coefficient of the function in respect to a x to vanish, we shall obtain 
by reduction, 
U2 2 + 2U2P 3 S 1 COS, 1 .3 + P3 2 Si* 
«1 — «2 U 2 2 COS q 2 + Uo P 3 S t COS < 2-3 * ^ ’’ 
If we proceed in like manner, assuming the space S 2 instead of S : to be constant, and 
substituting in the function (18.) for S x a 2 its value S 2 a x , we shall obtain by reduction, 
.. _ ^2 a 2 _ ( 22 .) 
1 P 2 COS< 12 + P 3 COS < 2,3 
It is easily seen that, if, when the values of i 12 and q 3 determined by equations 
19 and 20. are substituted in these equations, the resulting values of a x are positive, 
they correspond, in the two cases, to minimum values of the function (18.), and de- 
termine completely the conditions of the greatest economy of power in the machine, 
in respect to the direction of the moving pressure applied to it. 
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