11 s 
MAXIMUM. 
that of /7, its fluxion at that instant is evi- 
dently equal to nothing. T herefore, as the 
motion of the points m and n may be con- 
ceived such that their distance m n may ex- 
press the measure of any variable quantity 
whatever, it foHows, that the iluxion of any 
variable quantity whatever, when a maxi- 
mum or a minimum, is equal to nothing. 
The rule therefore to determine any flow- 
ing quantity in an equation proposed, to an 
extreme value, is : having put the equation 
into fluxions, let the fluxion of that quantity 
whose extreme value is sought be supposed 
equal to nothing; by which means all those 
members of the equation in which it is found 
will vanish, and the remaining ones will give 
the determination of the maximum or mini- 
mum required. 
Prob. I. To divide a given right line into 
two such parts, that their product, or rec- 
tangle, may be the greatest possible. This is 
the case when the line is bisected or divided 
into equal parts. See Fluxions. 
In any mechanical engine the proportion 
of the power to the weight, when they ba- 
lance each other, is found by supposing the 
engine to move, and reducing their velocities 
to the respective directions in which they act ; 
for the inverse ratio of those velocities is that 
of the power to the weight according to the 
general principle of mechanics. But it is of 
use to determine likewise the proportion they 
ought to bear to each other, that when the 
power prevails, and the engine is in motion, 
it may produce the greatest effect in a given 
time. When the power prevails, the weight 
moves at first with an accelerated motion ; 
and when the velocity of the power is inva- 
riable, its action upon the weight decreases, 
while the velocity of the weight increases. 
Thus the action of a stream of water or air 
upon a w heel, is to be estimated from the ex- 
cess of the velocity of the fluid above the ve- 
locity of the part of the engine which it strikes, 
or from their relative velocity only. The 
motion of the engine ceases to be accelerated 
when this relative velocity is so far diminish- 
ed, that the action of the power becomes 
equal to the resistance of the engine arising 
from the gravity of the matter that is elevat- 
ed by it, and from friction; for when these 
balance each other, the engine proceeds with 
the uniform motion it has acquired. 
Prob. II. Let a denote the velocity of the 
stream, u the velocity of the part of the engine 
which it strikes when the motion of the machine 
is uniform, and a — « will represent their rela- 
tive velocity. Let A represent the weight which 
would balance the force of the stream when its 
velocity is a, and p the weight which would ba- 
lance the force of the same stream if its velocity 
was oidy a — u; then p * A * * a — u l \ a 2 , or 
a = A x a — — - , and p shall -represent the ac- 
■ aa 
tion of the stream upon the wheel. If we ab- 
stract the friction, and have regard to the 
quantity of the weight only, let it be equal to 
qA, (or be to A Vs q to 1); and because the mo- 
tion of the machine is supposed uniform, p — q 
X A — 
X “ — T 
or q = 
The mo- 
a u x a — \ 
mentum of this weight is qAu 
which is a. maximum when the fluxion of 
v X « — k . , . 
vanishes, that is, when u X a — A 
aa 
— 2 uu X & — » — 0, or a — Su — O. Therefore, 
in this case, the machine will have the greatest 
rtf . • r « , .. . AX a — u 2 
effect ir u — — , or the wexgnt qA — 
3 aa 
4a 
= — ; that is, if the weight that is raised by 
the engine be less than the weight which would 
balance the power in the proportion of 4 to 9 : 
and the momentum of the weight is . 
Prob. III. Suppose that the given weight P 
(plate Miscel. fig. 156.) descending by its 
gravity in the vertical line, raises a given weight 
W by the cord PMW (that passes over the pul- 
ley M) along the inclined plane BD, the height 
of which BA is given ; and let the position of 
the plane BD be required, along which W will 
be raised in the least time from the horizontal 
line AD to B. 
Let AB r=: a, BD — x, t — time in which W 
describes DB ; then the force which accelerates 
aw . xx 
the motion of W is P , tt is as ' 
x P v • — aw 
and if we suppose the fluxion of this quantity to 
vanish, we shall find * = or P — — ^ ; 
p J 
consequently the plane BD required is that 
upon which a weight equal to 2W would be 
sustained by P ; or if BC be the plane upon 
which W would sustain P, then BD — 2BC. 
But if the position of the plane BD be given, and 
W being supposed variable, it be required to 
find the ratio of W to P, when the greatest 
momentum is produced in W along the given 
plane BD ; in this case, W ought to be to P as 
BD to BA -f A /BD + BA + yTEA 
Questions of this kind may be likewise de- 
monstrated from the common elementary geo- 
metry, of which the following may serve as an 
example. 
Pros. IV. Let a fluid, moving with the velo- 
city and direction A.C (plate Miscel. fig. 157), 
strike the plane CE ; and suppose that this plane 
moves parallel to itself in tlue direction CB, per- 
pendicular to CA, or that it cannot move in any 
other direction ; then let it be required to find 
the most advantageous position of the plane CE, 
that it may receive the greatest impulse from 
the action of the fluid. Let AP be perpendi- 
cular to CE in P, draw AK parallel to CB, and 
let PK be perpendicular upon it in K ; and 
AK will measure the force with which any par- 
ticle of the fluid impels the plane EC in the di- 
rection CB. For the force of any such particle 
being represented by AC, let this force be re- 
solved into AQ parallel to EC and AP -per- 
pendicular tp it ; and it is manifest, that the 
latter AP only has any effect upon the plane 
CE. Let this force AP be resolved into the 
force AL perpendicular to CB, and the force 
AK parallel to it; then it is manifest, that the 
former, AL, has no effect in promoting the 
motion of the plane in the direction CB ; so that 
the latter, AK, only, measures the effort by 
which the particle promotes the motion of the 
plane CE, in the direction CB. Let EM and EN 
be perpendicular to CA and CB, in M and N ; 
and the number of particles moving with di- 
rections parallel to AC, incident upon the plane 
CE, will be as E M. Therefore the effort 
of the fluid upon CE, being as the force of 
each particle, and the number of particles to- 
gether, it will be as AK x EM ; or, because 
AK is to AP (=: EM) as EN to CE, as 
EM 2 EM X EN . , 
— — — ; so that CE being given, the pro- 
C E 
blem is reduced to this, to find when EM 2 x EN 
is the greatest possible, or a maximum. But 
3 
because the sum of EM* and of EN S (— CM 2 ) 
is given, being always equal to C’E 2 , it follows 
that EN 2 X EM 1 is greatest when EN 2 = I-CE 2 ; 
for when the sum of two quantities AC and CB 
(fig. 158.) was given, AC X BC 2 is greatest 
when AC — as will be very evident if a 
semicircle is described upon AD. But when 
EN J x EM 1 is greatest, its square root EN X 
EM 2 is of necessity at the same time greatest, 
Therefore the action of the fluid upon the plane 
CE, in the direction CB, is greatest when Eb{ 2 
= l-CE 2 , and consequently EM 2 = |CE 2 ; that 
is, when EM, the sine of the angle ACE, in 
which the stream strikes the plane, is to the ra- 
dius, as \P2 to ^/3; in which case it easily ap- 
pears from the trigonometrical tables, that thi* 
angle is of 54° 44'. 
Several useful problems in mechanics may be 
resolved by wliat we have just now shewn. If 
we represent the velocity of the wind by 
AC, a section of the sail of a windmill per- 
pendicular to its length by CE, as it follows 
from the nature of the engine, that its axis ought 
to be turned directly to the wind, and the sail 
can only move in a direction perpendicular to 
the axis, it appears, that, when the motion be- 
gins, the wind will have the greatest effect to 
produce this motion, when the angle ACE, in • 
which the wind strikes the sail, is of 54° 44*. 
In the same manner, if CB represent the direc- 
tion of the motion of a ship, or the position of 
her keel, abstracting from her lee-way, and AC 
be the direction of the wind perpendicular to 
her way, then the most advantageous position 
of the sail CE, to promote her motion in the 
direction CB, is when the angle ACE, in which 
the wind strikes the sail, is of 54° 44 y . The best 
position of the rudder, where it may have the 
greatest effect in turning round the ship, is de- 
termined in like manner. 
Prob. V. To find the internal dimensions of 
a cylindrical cup, whose capacity is equal to a , 
when the cup is made with the least possible 
quantity of silver of a given thickness. 
Put the diameter =: ; and .7854 (the area of 
a circle whose diameter is 1) — c: then, by 
El. xii. 2, cx~ = the area of the bottom, and 
therefore — — the altitude ; but 4cx — the 
cx 
circumference of the bottom, and therefore 4 cx 
X — 2 — — A le inside curve superficies. 
CX X 
^tci 
Hence cx 2 -j- - — — the whole inside superfi- 
cies, which is a minimum ; and therefore its 
fluxion is zn 0 : that is, 2cxx — = 0, or 
x i 
2cx 3 x — 4 ax — 0, or cx 3 — 2 A— o, therefore 
, ’ 3 /2a 
cx — 2a-, and x — \^/ — — diameter. By 
substituting this quantity for x in — we have 
C A 
2T,F 
2 ac 
the diameter is 
altitude. Since then 
and the altitude is half that 
quantity, they will be to one another as 2 to 1, 
to answer the conditions of the problem. 
Prob. VI. To find the -greatest cone that can 
be inscribed in a given sphere. 
Let AD (plate Miscel. fig. 159) the diameter of 
the sphere = «; .7854 (the area of a circle whose 
