480 
6 x—a, hence, x— -a 
6 
FLUXIONS. 
In like man- 
i i 
,'.y—-a, z—~a 
3 2 
ner, whatever be the number of unknown quantities, 
make any one of them variable with each of the red, 
and the values of each in terms of that one quantity will 
be obtained ; and by fubftituting the values of each in 
terms of that one, in the given equation, you will get 
the value of that quantity, and thence the values of the 
others. 
Ex. 3. To find when y is a max. in x 3 -\-y 3 \ a —a^x 7 -. 
cr 
9 x '' 
Take the fluxions of both fides, and 2 x s* 2 •*•-{-3 yy 
X x 3 -\-y 3 —2a i xx ; but when y is a maximum, j'—O ; 
- — — . ft ^ ■ —■ 2 
hence, 6 x*x x x 3 -j-y’ > —2a i xx, -y 3 ——•, and x 3 -\-y 3 = 
3 X 
” 3 , r . a 3 a* a 
therefore avc“=-, and x =—, or x— —hence, 
■ 9 V 3 
3 , 3 . « 3 a 3 1 12 
y\=za'x— * 3 )=———— a 3 y~ ~ -r=« 3 X-=; y 
__ V 3 3 * V3 3 l 3v / 3 ’ 
~ a ^ 3V3 
Otherwife. As y 3 2=.a‘‘x — x 3 , • •. 3 yy — a*x — ^x 2 x~o, 
a 
becaufe y—o, .\x— —— • 
V3 
Ex. 4. To infcribe the greatefl parallelogram DFGT in 
a given triangle ABC. Draw BH 1 AC; put AC=«, 
BH=^, BE—x,thenEH= 
b —x ; and by fun. s, 
ax 
b : a \ : x : DF=:— \ hence, 
b 
dyC -- 
thearea DFGI =—xb — x 
b 
— max. or *x b—x — lx 
LL j 
c hence, — = 3x2, or a == 
x £ 
1 2 
— confequently EH— ~BII. 
3 3 
Ex. 7. To cut the greatefl: parabola DEF from a given 
cone ABC. Let AGC be that diameter of the bafe which 
is J_ to DGF now EG is 
parallel toAB; put A C~a, 
AB=i,CG=r, thenAG— 
a — x; andby the property of 
the circle, DG— y/ax — x-, 
.-. DF—2 ^ ax — x‘ ; alfo, 
by fim. £\s, a : b : : x : GE 
=—; hence, we have the 
a 
area of the parabola =- 
3 
l) X _ 
X — X 2 i/ ax—x- — max. 
hence, x ax — x- = max. or x 
max. .•. 3 ax-x —4-\ 3 .v=o, 
X o.x — x 2 — ax 3 — a 4 — 
2a—xx, and x~-a. 
Ex. 8. To divide a given arc A into two parts, fuch that 
the 7» th power of the fine of one part, into the w th power of 
the fine of the other, may be a maximum. Let P and QU>e 
the two parts, x and y Hieir tines, radius being unity ; then 
x"‘ X y" ~ maximum ; hence, my"x n — x x-\-nx m f — 'J—O, 
and my x——nxy 
y 
V 1 —j 2 
y 
Now (Art. 46.) P— 
and as P-|-Clj= A,P-|-(L=:o, 
—, Qj= 
V 1 — X 2 
. P =— Q, or 
V 1 —} 2 V 1 —’ 
tion myx ■=. — nxy, and my. 
multiply this equation by the equa- 
y 
2«X- 
>°r?«X 
2xx—o ; hence, x—-b, therefore 
2 
EH—-BH. 
2 
Ex. 5. Let ABC reprefent a cone, AC the diameter of 
thebafej to infcribe in it the greatefl cylinder DFGI. Put 
/>=.78539, &c. then (the 
fame notation remaining) 
it will appear when we 
come to treat on the me¬ 
thod of finding the areas 
of curves, that ^ a * =2 
the area of the end DEF 
of the cylinder; hence, 
the content of the cylinder 
pa*x* -- 
=—7 — X b—x — max. or 
■J 1—j 2 V 1— * 2 
tan. P —n y tan. Q, . •. m : n : : tan. Q : ; tan. P, and m-\-n 
; m—n : : tan. Q + tan. P : tan. Q — tan.P: : tin. (Q-)-P): fin. 
(Q—P) : : fin. A : fin. (Q—P) = fin. A x -—- ; hence 
m-\-n 
w r e know the fine of the difference of the two parts of 
the arc, therefore we know the difference Q—P of the 
arcs themfelves; and knowing the fum Q+P, or A, 
we know the two parts Q and P. 
Ex. 9. To determine at what angle the wind muft Al ike 
againft the fails of a mill, fo that the effect to put it in motion 
may be thegreateft pollible. Put a— the cofine of theangle, 
then 1— x 2 — the fquare of the fine, radius being unity; 
hence (by the Principles of Ilydroftatics) the effedt is as 
x* x b — x~bx* —x 3 —max. .2bxx-\-^x—o ; hence, x— 
-b; therefore EHzz-BH. 
3 . 3 
Ex. 6. To infcribe the greatefl parallelogram DFGI in 
a given parabola ABC. Put BH=a, p— the parameter, 
a=BE, then by the property of the parabola, DE -—px 
3 _ _i 11 
• DE =zp 2 x 2 , and DF=i/ 5 j( 2 ; hence, the area DFGI= 
2 p 2 x~y a —max. or x 2 
X a .— x—ax~ —a 2 — max. 
1 A 3 i 
-ax 2 -v- x 2 x—o, 
2 2 
a 
*Xi— x 2 =zx — x 3 , which is to be maximum ; ,-.x — ix 2 x 
17 
=0 ; hence the cofine of 54 0 44'. 
Ex. 10. Given two elaftic bodies A and C, to find an in¬ 
termediate body x, fo that the motion communicated from 
A to C through x, may be a maximum. Put a— the given 
velocity of A, w— the velocity communicated to C, and 
z the velocity communicated to x ; then (by Mechanics) 
A-(-* : 2 A :: a : zu 
* 4 -C : 2X :: w : z 
.‘.comp. Ax-|-A 2 -(-AC-i-Cx : 4A* a : z, or 
AC 
A +a- 1-fC ; A :: a : z ; now as the two middle terms 
x 
are conftant, the laft term varies inverfely as the firft ; 
and as the laft is to be a maximum, the firft muft be a 
„ . „ . . ACi- 
minimum; therefore its fluxion x - -j-=o ; hence, 
x 2 =AC, and A : x :: x : C. 
Ex. 11. Given the altitude BC of an inclined plane A B, to 
find its length, fo that a weight Padtingupon another Win a 
line parallel to the plane, may draw it up through AB in the 
leaft time. Put a=zBC, a=AB j then (by Mechanics) the 
accelerating 
