W HEEL-CARRIAGE. 
or, in words, the velocity varies inversely as the 
square root of the length of the day estimated in 
hours of working. It has already been shown, that 
the maximum yelocity of a strong cart - horse is 
about 7’5 miles an hour, and that of ordinary coach- 
horses about 15 miles an hour, and that they cannot 
keep up that speed-without injury for more than one 
hour. On these principles, the following table 
shows the duration of labour, and greatest velocity 
of a horse of each kind, in miles per hour, which, 
by interpolation, may serve for any other velocity 
required. 
TABLE IL. 
Duration of H. | 4H. 
Labour, 1 
See ee SS ee ee ee 
M.} M.| M.| M.| M.| M.} M.| M. 
M. 
5'3 | 4:3] 3°7| 3°4| 3-0} 2°8) 26) 2°5| 2-4 
Greatest velocity. 
Coach- uM. | mM. | mM] mM. | M.] M.| M.| M.| M.] M. 
Horse, 15:0 |10°6 | 8:6} 7:5] 6°7| 6:1) 5:7] 5:2} 5:0) 4°8 
This table will represent very nearly the rate of de- 
crease of velocity by an increase of the duration of 
the time of labour on level road. But if the road 
be inclined the velocity of ascent will be diminished 
in the ratio of the sine of the angle of elevation, or 
it will be increased in descent in the same ratio; that 
is, if the road rise or fall one foot in a hundred, the 
velocity will be diminished or increased by about 
one-hundredth of that stated in the table. ‘The re- 
| sults in the table are those corresponding to a horse 
loaded very slightly, so as to continue to do so day 
after day without being injured, and must therefore 
be the utmost limits of velocity, though the work 
done must be little or nothing. It is also obvious, 
that if the load be so great that the horse can just 
move it, the work done in this case will also be no- 
thing; but between these limits of velocity and 
power there must be a ratio at a certain point, which 
will produce the greatest quantity of useful effect, 
and must therefore be the most advantageous for the 
application of horse-power. To obtain this quantity, 
let V be the maximum velocity when unloaded, v 
the actual velocity, as in formula (B), or E= Fv 
v 
¢ = 5) . Let V—v=y, then F= (V—v)?= 
v 2 
Bie x uz2=—F a Qe): 
stant quantities, and making uw? (V — uw) a maximum, 
by putting its fluxion equal to nothing, that is, 2 V 
udu — 82 udu= oo. Whence u = ;V, and 
Omitting the con- 
2 
therefore v = V — u = V — 5 V, or 
vor Vv (L). 
Hence the velocity required to produce the maximum 
effect is one-third of the greatest velocity at which 
the animal is capable of moving. This gives about 
two and a-half miles an hour for cart-horses, and five 
miles an hour for coach-horses, when working eight 
or nine hours a-day. The time thus obtained seems 
to answer very well for draught-horses, but it is too 
long for mail-horses, where less effect but greater 
speed is required. It is easy to regulate this by the 
table. Thus, if a set of coach-horses have to run 
two stages of an hour each, or two hours on the 
whole in a day, then under 2 h., and opposite coach- 
horses, will be found 10°6 m., or about ten and a-half 
717 
miles, the rate at wnich they may run without much 
injury, when slightly loaded, during one day of 
twenty-four hours, when well fed, and receive a 
good rest between the times of their running their 
prescribed stage. Since, by formula (A) f = F 
V — v\2 U2 2 
ce) =F ;,, and u = 5 V, we have 
4 
VA 4 
fee” Tao F (d), 
as already remarked, or the power at the maximum 
is four-ninths of the force exerted by the muscles at 
their greatest capability. This conclusion ought 
therefore to be attended to, when the greatest quan- 
tity of work is required from strong sound horses. 
It will now be useful to show the addition of ex- 
pense for a variation of velocity from the maximum 
of useful effect. Since the expense must be inversely 
as the effect produced, if the maximum, Table I., be 
taken for unity, then will be obtained 
TABLE IIL 
VELOCITY. PROPORTIONAL EXPENSE. 
Miles per hour. Cart-Horses. Coach-Horses. 
0 Infinite. Infinite. 
1 1°48 12°55 
2 1:03 1:48 
21 1:06 1:28 
3 1:03 1:16 
4 1:27 1:03 
5 1:99 1:00 
6 4°55 1:03 
zl Sei 1:12 
“4 Infinite. 1:19 
8 as 1:28 
9 1:53 
10 2-00 
11 2°80 
12 4-76 
13 10°26 
14 ASS?/ 
1d Infinite. 
From this table it will appear that, if a cart-horse 
walk at the rate of two and a-half miles an hour, the 
expense will be the least; but if slower or faster 
than this, the expense will be increased. Thus, at 
5 miles an hour the expense will be doubled for a 
heavy cart-horse, while it will be the least for lighter 
coach-horses. The latter would be doubled at ten 
miles an hour, the usual rate of mail-coaches, conse- 
quently our mails are accelerated at great expense. 
By comparing the three tables just given, a fair 
estimate of useful effect may be obtained, involving 
the velocity, duration of labour per day, and relative 
expense; and the proper value of each may be so 
chosen as to suit the object most particularly in view. 
Thus, a draught horse moving on a railway will pro- 
duce the greatest effect at least expense, when mov- 
ing at the rate of 24 miles an hour, exerting a force 
of traction of nearly 175 lbs. (Table I.), and will 
work eight or nine hours a day (Table II,), on an 
expense denoted by 1 (Table III,), or the least pos- 
sible expense. If the velocity be increased to 4 
miles an hour, the animal will be able to exert a 
force of 86 lbs. only, work about three and a-half 
hours, at an expense of 1°27, or an increase of 27 per 
cent. In like manner, similar consequences may be 
drawn for other velocities and for other kinds of 
horses. Thus a coach-horse running at the rate of 
10 miles an hour, can exert safely a force of traction 
of about 20 lbs. (Table I.), can work two hours a- 
