482 
FLUXIONS. 
direction BC, which is to be a maximum; or 
-v x « 2 2 — zxxy s x 
— a maximum 
<z 2 +* 2 
or the nu- 
a 2 -\-x 2 
merator a 2 x-\-x 2 x —2v 2 .f—o ; hence, x—a. 
iix. i 3 . Given the folidity of a cone, to find the 
bafe and height, when the time of its vibration (hall 
be a maximum, fuppoftng the point of fufpenfion to 
be the vertex. Put x — radius of the bafe, 72= the 
altitude, p =rz 3,14159, &c. then $pxy 2 — s and 
"2 J _ r 2 
(Example 8. Prop. 28.) -———the diftance from 
5 X 
the point of fufpenfion to — the center of ofcillation = 
za 
4X 2 -1- 
_ „ s s 2 a , x 
minimum. But y 2 = -—= (if— —za) —; hence*- 
ipx V ip J X _ S x 
4.x 3 + 2 a . izx^xy'ix- —ioxxX 4 * 3 + 2a 
■zr- ■ „ —2= nun. and - 3 -=0; 
25X 4 
2 < 2>5 _ 
——er v 2 X « i > confe- 
$ x : 
tience, x— a-f, therefore 72: 
yjx' 
quently x : y :: 1 : yj 2. 
Ex. 19. To find when (A) x 3 —iSx 2 + 96X— 20 becomes 
a maximum or minimum. Affume the fluxion — o, and 
3>; 2 A'— 36xx-{-g6x=z$xX.x 2 —12x4-32=20, hence, ^=24 or 
S. Now to determine which value gives the maximum 
and which tlie minimum, find whether the value of the 
fluxion, juft before it becomes =0, be pofitive or negative ; 
if pofitive, the fucceeding root gives a maximum ; if nega¬ 
tive, a minimum-, for whilft a quantity increafes its fluxion 
is pofitive ; but when it decreafes its fluxion becomes 
negative, by Art. 16. Now as 3-vX*—4X-v—8e=3.vX 
x- —•i2.r + 32 ; when x is lefs than 4, each fa&or being 
negative, the value of the fluxion is pofitive, therefore 
the root 4 gives (A) x 3 —i 8 x 2 -f 9 6 ^'— 20 i a maximum; 
and as, when x increafes from 4 to 8, one factor is pofitive 
and the other negative, the fluxion is negative, therefore 
the root 8 gives (A) a minimum. When we fay that by 
making x—4 it gives (A) a maximum, we mean that (A) 
firft increafes till x becomes 4 and then it decreafes, and 
not that it is then the greateft pofiible ; for by increafing 
x after it exceeds 8, the value of-( A) increafes Jine limite. 
^nd in like manner, (A) decreafes whilft x increafes from 
4 to 8, and then it increafes, and therefore when x=:8, 
(A) is faid to be a minimum, not that it is then the leaf! 
poffible, for by decreafing x below 4, (A) will decreafe 
fine limite. 
We have here ftippofed x to increafe ; if we fuppofe x 
to decreafe, and firft affume it greater than S, then as x 
decreafes till it becomes 8, each fadtor x —4, x—8 being 
pofitive, the produdl is pofitive, and therefore it might 
appear that the root 8 ought to give a maximum ; but 
as x is a decreafing quantity, its fluxion ( x ) is negative 
by Art. 16 ; hence, 3.VX1'—4X -—8 is negative till x- be¬ 
comes 8, and therefore this root gives (A) a minimum; 
and whilft x decreafes from 8 to 4, 3-v'x*—4X*—- 8 is 
pofitive, and therefore 4 gives (A) a maximum, agree¬ 
able to what was before determined. Tiiis inftao.ee (hews 
the neceffity of attending to the figns of the fluxions of 
increafing and decreafing quantities, without which we 
might have determined (A) to have been a maximum 
when it is a minimum, and a minimum when it is a maxi¬ 
mum ; for it is merely arbitrary whether we fuppofe x to 
•increafe or decreafe. 
When all the roots of the fiuxional equation are im- 
poffible, as no poffible value of x can make the equation 
=0, it (hews that by increafing x, the given quantity in¬ 
creafes or decreafes fine limite, therefore it admits of no 
maximum or minimum. 
It may happen that the fluxion may be 2220, and yet 
the quantity (A) may not be a maximum or minimum, 
which takes place when two of the roots of the Suxional 
equation are equal, becaufe in that cafe, the fign of the 
fluxion is the lame both before and after the equation 
becomes =0 from 'the fuhftitution of one of the equal 
roots. For let the given quantity be x 4 —i6x 3 4-()ox 2 —- 
2i6x, whofe fluxion is 4.x 3 a - — e[.Zx 2 x-{-iSoxx- —2i6x—4* 
Xx 3 —1 2X 2 +45X—54—4xX-r—3X x —3 X x —6. Now juft 
before X2223, this fluxion is negative, and juft after x=3, 
it is alfo negative ; therefore as the fluxion continues 
negative whilft x pafles through 3, that root does not 
give (A) a minimum; but as the fluxion pafles from 
negative to pofitive whilft x pafles from lefs than 6 to 
more than 6, the root 6 gives (A) a minimum, its fluxion 
after that time being pofitive (hews that (A) then begins 
to increafe. * 
Let the flifxional equation have three equal roots, as 
in xXx —«>(x— aX x —^X*— b, and let a be lefs than b. 
Then it is manifeft, that when x is lefs than a, this fluxion 
is pofitive, and when x pafles through a and lies between 
a and b, the fluxion is negative ; therefore x—tz-gives (A) 
a maximum. Hence it is manifeft, that, in general, when 
the fiuxional eqviation lias an even number of equal roots,, 
one of thofe roots gives (A) neither a maximum nor mi¬ 
nimum ; but when it has an odd number, that root gives 
(A) either a maximum or minimum. If the reader wi(h 
to fee any thing further on this point, he may confult 
Lyons’s Fluxions, p. 91. ' 
Ex. 20. To find the value and pofition of the greateft 
and leaft ordinates of a curve, w hofe equation is 7222 
x 3 — px- -j- qx — r, x being the abfeifla and y the or¬ 
dinate. Take the fluxion, and 7=22 jx 2 .*-—2 pxxf-qx; 
but when 7 becomes a maximum, 7=0; lienee, 3x 2 x— 
, p fp 2 q 
zpxx-\-qx—0 ; confequently x—-±.f— -, the values 
of the abfeifla correfponding to the required ordinates; 
and if tliefe values of x be refpedtively fubftituted into 
the given equation, the values of the ordiiyites themfelves 
will be known. Which of the values of x gives the or¬ 
dinate a maximum and which a minimum, may be found 
by Ex. 19. If p— 18, q— 60, r— jo, then x—z. and 10, 
the two abfciffae ; which fubftituted for x in the given 
equation, give 46 and —210 for the two ordinates, the 
latter of which being negative, (hews that the curve at 
that point lies below the abfeifla. 
22. In qneftions of a geometrical and philofophical na¬ 
ture, there are frequently reftrictions, which enter not 
into the analytical expreflion. For inftance, if a body 
revolve in an ellipfe and the force tend to the center, it 
varies as the diftance (a) from the center. Now this 
force in the ellipfe is a maximum at the extremity of the 
axis major, and a mininum at the extremity of the axis 
minor; but it is manifeft, that the quantity x is not re¬ 
dacted within either of tliefe limits, as it may be in- 
creafed or diminiftied fine limite ; therefore if we put its 
fluxion (x) =0, we can determine nothing from that 
equation refpedfting the points where the force becomes 
a maximum or minimum. The analytical expreflion can 
never be applied, but where its value is neceffarily re- 
ftrained by, and buffers all the changes which, the quan¬ 
tity it expreffes is fubjebt to. 
To draw TANGENTS to CURVES. 
Prop. X .—Let the curve ACZ be deferibed by the extremity 
of the ordinate BC, which moves parallel to itfelf and varies in 
its length ; to draw a tangent to the curve at any point C, 
23. Let TCV be the required tangent; draw any other 
ordinate Dr and produce it to s; draw alfo CE parallel 
to BD ; join Cr, and produce it to t and W ; produce 
alfo CE to any point G, and draw Gmn parallel to.Er. ; 
Now let Dr* move up to BC, then by the motion of r, 
the line W rCt will revolve about C, and when r coincides 
with C, it ceafes to cut the curve between C and Z, and 
it does not cut it between C and A, for to cut CA, Ct 
muft fall below CT, and confequently CW mu ft lie 
above CV, or r muft have puffed s , Which it cannot have 
done. 
