fne •ate of six feet per second, while our an- ! 
thor has seen one of 33 feet high move very 
steadily and well with a velocity of little more 
than two feet. The reason of this superior j 
velocity in the 24-feet wheel, may probably 
be owing to the small proportion that the 
head requisite to give die proper velocity to 
the wheel bears to the whole height. 
4. The maximum load for an overshot- 
' wheel is that which reduces the circumfer- 
[ encc of the wheel to its proper velocity ; 
which is known by dividing the effect it 
J ought to produce in a given time, by the 
.space intended to be described by the cir- 
! cumference of the wheel in the same time: 
| the quotient will be the resistance overcome 
j at the circumference of the wheel, and is 
j equal to the load required, including the fric- 
j tio'n and resistance of the machinery, 
5. The greatest velocity that an overshot- 
! wheel is capable of, depends jointly upon the 
! diameter, or height of the wheel, and the ve- 
j io city of falling bodies ; for it is plain that the 
i velocity of the circumference can never be 
i greater than to describe a semicircumference, 
! while a body let fall from the top describes 
; the diameter, nor even quite so great ; as the 
difference in point of time must always be in 
j -favour of that which falls through (he diame- 
ter. Thus, supposing the diameter of the 
I wheel to be 16 feet and an inch in diameter, 
a heavy body would fell through this space 
in one second ; but such a wheel could never 
arrive at this velocity, or make one turn in 
two seconds, nor could an overshot-wheel 
ever come near it: because, after it lias ac- 
quired a certain velocity, great part of the 
[water is prevented from entering the buckets, 
and part is thrown out again by the centrifu- 
gal force ; and as these circumstances have a 
considerable dependance upon the form of the 
buckets, it is impossible to lay down any gene- 
ral rule for the velocity ot this kind of wheels. 
6. Though in theory we may suppose a 
wheel to be made capable of overcoming any 
resistance whatever, yet as, in practice, it is 
necessary to make the wheel and buckets of 
some certain and determinate size, We al- 
ways find that the wheel will be stopped by 
'Such a weight as is equal to the effort of the 
water in all the buckets of a semicircunifer- 
ence put together. This may be determined 
from the structure of the buckets themselves ; 
but, in practice, an overshot-wheel becomes 
unserviceable long before this time : for when 
it meets with such an obstacle as diminishes 
■ its velocity to a certain degree its motion be- 
comes irregular ; but this never happens till 
the velocity of the circumference is less than 
the two feet per second, when the resistance 
ie equable. 
7. From the above observations we may 
easily deduce the force of water upon breast- 
wheels, &c. But, in general, all kinds of 
wheels where the water cannot descend 
through a given space unless the wheel moves 
with it, are to be considered as overshot- 
wheels ; and those which receive the impulse 
or shock of the water, whether in an hori- 
zontal, oblique, or perpendicular direction, 
are to be considered as undershots. Hence in 
a wheel in which the wate- strikes at a certain 
point below the surface of the head, and after 
tii at descends in the arch of a circle, press- 
ing by its gravity upon the wheel, the effect 
of such a wheel will be equal to that of an 
undershot whose head is equal to the differ- 
VOL. II. 
MILL. 
! once of level between the surface of the wa- 
ter in the reservoir and the point where it 
: strikes' the wheel, added to that of an over- 
! shot whose height is equal to the difference 
of level between the point where it strikes 
the wheel and the level of the tail-water. 
In the 66th volume of the Transactions 
our author considers some of the causes 
which have produced disagreements and dis- 
putes among mathematicians upon this sub- 
ject. He observes, that soon after sir Isaac 
Newton had given his definition, “ that the 
quantity of motion is the measure of the 
same, arising from the velocity and quantity 
of matter conjointly,” it was controverted by 
his contemporary philosophers. They main- 
tained, that the measure of the quantity of 
motion should be estimated by taking the 
quantity of matter and the square of the ve- 
locity conjointly. On this subject he re- 
marks, that from equal impelling powers 
acting for equal intervals of time, equal aug- 
mentations of velocity are acquired by given 
bodies when they are not resisted by a me- 
dium. Thus- a body descending one second 
by the force of gravity, passes through a 
space of 16 feet and an inch ; but at the end 
of that time it has acquired a velocity of 32 ft. 
2 inc. in a second : at the'end of 2 sec. it has 
acquired one that would carry it through 64 
feet 4 inches in a second. If, therefore, in 
consequence of this equal increase of velo- 
city, we define this to be a double quantity of 
motion generated in a given time in a certain 
quantity of matter, we come near to sir 
Isaac’s definition : but in trying experiments 
upon the effects of bodies, it appears, that 
when a body is put in motion, by whatever 
cause, the impression it will make upon an 
uniformly resisting medium, or upon uni- 
formly yielding substances, will be as the 
mass of matter of the moving body multi- 
plied by the square of its velocity. The 
question therefore properly is, whether those 
terms, the quantity of motion, the momenta, 
or forces of bodies in motion, are to be 
esteemed equal, double, or triple, when they 
have been generated by an equable impulse 
acting for an equal, double, or triple time? 
or that it should be measured by the effects 
being equal, double, or triple, in overcoming 
resistances before a body in motion can be 
stopped? For, according to the meaning we 
put upon these words, the momenta of equal 
bodies will be as the velocities or squares of 
the velocities of the moving bodies. 
Though by a proper attention to the terms 
employed, however, we shall find both these 
doctrines to be true ; it is certain that some 
of the most celebrated writers upon mecha- 
nics have fallen into errors by neglecting to 
attend to the meaning of the terms they make 
use of. Desaguliers, for instance, after hav- 
ing been at pains to show that the dispute, 
which in his time had subsisted for 50 years, 
was a dispute merely about words, tells us, 
that both opinions may be easily reconciled 
in the following case, viz. that the wheel of 
an undershot water-mill is capable of doing 
quadruple work when the velocity of the wa- 
ter is doubled, instead of double work only : 
“ For,” says he, “ the adjutage being the 
same, we find, that as the water’s velocityjs 
double, there are twice the number of parti- 
cles that issue out, and therefore the iadle- 
board is struck by twice the matter; which 
matter moving with twice the velocitv that it 
B b 
lf>3 
had in the first case, the whole effect must b<“ 
quadruple, though the instantaneous strok e 
ot each particle is increased only in a simpl e 
proportion ot the velocity.” In another place 
the same author tells Us, that though “ the 
knowledge of the foregoing particulars is ab- 
solutely necessary for setting an undershot- 
wheel to work, yet the advantage to be reap- 
ed from it would still be guess-work, and we 
should be at a loss to find out the utmost that 
it could perform, had it not been for an inge- 
nious proposition of that excellent mechanic, 
M. Parent, of the royal academy of sciences, 
who has shewed, that an undershot-wheel can 
do the most work when its velocity is equal 
to the third part of that of the water ; because 
then two-thirds of the water are employed in 
driving the wheel, with a force proportion- 
able to the square of the velocity. By multi- 
plying the surface of the adjutage or opening 
by the height of the water, we shall have the 
column of water that moves the wheel. The 
wheel thus moved will sustain on the opposite 
side only four-ninths of that weight which, 
will keep it in equilibrio; but what it can 
move with the velocity it has, is only one- 
third ot the equilibrium.” This conclusion 
is likewise adopted by Mr. Maclaurin. 
Mr. Smeaton, in the year 1759, instituted 
another set of experiments; the immediate 
object of which was, to determine what pro- 
portion or quantity of mechanical power is 
expended in giving the same body different 
degrees of velocity. Having constructed a 
proper apparatus for the purpose, and with it 
made a number of experiments, he concludes, 
“ that time, properly speaking, has nothing 
to do with the production of mechanical ef- 
fects, otherwise than as by equally flowing it 
becomes 4 common measure ; so that, what- 
ever mechanical effect is found to be pro- 
duced in a given time, the uniform continu- 
ance of the action of the same mechanical 
power will, in a double time, produce twice 
that effect. A mechanical power, therefore, 
•properly speaking, is measured by the whole 
of its mechanical effect produced, whether 
that effect be produced in a greater or less 
time: thus, having treasured up 1000 tuns of 
water, which I can let out upon the overshot- 
wheel of a mill, and descending through a 
perpendicular of 20 feet ; this power,' ap- 
plied in a proper manner, will grind a certain- 
quantity of corn in an hour: but, supposing 
Ihe mill to be capable of receiving a greater 
impulse with as great advantage as a less ; 
then, if the corn is let out twice as fast, -the 
same quantity of corn will be ground in half 
an hour, the whole of the water being like- 
wise expended in that time. What time has 
therefore to do in the case is this: Let the 
rate of doing the business, or producing the 
effect, be what it will ; if this rate is uniform, 
when l have found by experiment what is 
done in a given time, then, proceeding at the 
same rate, twice the effect will be produced in 
twice the time; 011 supposition that I have a 
supply of mechanic power to go on with. 
Thus, 1000 tuns of water descending through 
20 feet perpendicular, being, as has been 
shewn, a given mechanic power, let it be ex- 
pended at what rate it will ; if, w hen this is 
expended, we arc to wait another hour till an 
equal quantity can be procured, then we can 
only expend 12 such quantities in 24 hours. 
But if, while the thousand tuns of water are 
expending in one hour, the same quantity is 
