FUNCTION. 
dy ve a 49) ,anddV = *! y dy ./(a” + 49°), con- 
aay Vv a oP ce <= UT = 2) rd ae t 4X )i + 
e, (x being a eoant arbitrary quanti , 
‘The manner in which Mr. Woodhoufe applies thefe prin- 
eferv t 
10d to that moft Melty fal followed by renee se in 
The laws of motion are here beautifully illuftrated from 
‘principles cad by on poner from the oS - 
common algebra, where iofe writers ufually pur 
method entirely the ens "ad inftitute algebraical oe 
from the theory of motion 
o determine the pieumiaces under which a function 
$x admits of amaximum or minimu 
Let x be increafed by Aw; then 2 G +A es = ie é 
Dex. Ax + Deez. (Ax)? + &e. ory + Diy. Ax 
+ D?.y (Az)? + &c.; now in cafe of a maximum 9 is 
>y + 4, or >yt Dey, Ax eens 
&c.; and in oo san <y+D.y.Av+D?. 
y+ (Ax) or Diy. Oa + Diy. (Ax) + &e. 
is < 0 in‘cafe eo maximum, one > o in cafe of a mini- 
may be taken fuch, that 
o wit, D.y. 4a is greater a a wis 
of. all the ee os conte tly 
9+ (Se r <0; sear as SD. 
-Swis pofitive or negative : Ag —o its 
figns with Aw, con eee taking 4 Bay +O x. 
o +b x)'.+ &e. cannot i al 
: <0 cafe of ae haar nor > 0 in cafe : 
sy ia, except D. ; hence, when y or 9 x 
d 
at a ftate of maximum or minimum, D. y" ‘or on 
a “7 
=0, 
ee et ee a aed - dy 
Again, fince the co-efficient D . y or Te =O then D ; 
ye (A we)? + Ds.y. (Ax) + &e. ‘muft be’ < o incafe 
. of amaximum, ‘and > o incafe of a minimum: take A x fuch 
that the firft term, namely, De -y + (A x)*, is greater than 
the fum of all the fucceeding terms; then fince (A x)? is 
ets whether A x be pofitive or negative, DP.y.(Ax)? 
e 
D3.y. (Ax)? + &c. is < oor > 0, accordingly as 
@? 
J ig ‘negative or - pofitive ; hence, for a maxi- 
- Ly or da 
“mum D° ) is negative, for a minimum, pofitive. 
; ‘Next, dae the co-efficients D.y, D*. y, both to 
equal o, then D (Axvy + Dt.y (Ax) + &c. is 
r a maximum, es o for a minimum } 
fuch that the firft t D’y wee w)? is ae an me 
fum of all the lieeedne terms: then finc : = 14 
changes its fign as Aw does, Ds.y , mn ay? mf 74 < 
(A «)* + &e. cannot always <o fora maximum, ae 
> o for a minimum, except D3, 
And generally, if D. y, D>. y, D?. yy, &e. D” .y 
; then that ce enon -y (@ #) may ad- 
mit a maximum “or © miviniu mutt -alfo be 
0; and for amaximum Dt, y mult on negative, and 
for a minimum pofitive. 
= Oy or ax’ = Ce. 
‘Such are the methods for determining the maxima _ 
minima of state 3 the principles on which they a 
ounded are ciently evident, and merely meat 
fuf 
‘examples cannot illuftrat e them. 
co ae of thefe oe 
It was in attemptin 
imi. inimis, that 
@ maximis et minintis, e 
faid to have originate ermat is the firft writer in 
as given a methed purely analytical. His copa con- 
fits in maki the quantity whofe maximu Ie 
m min 
mum is fought equal to the expreffion of the fame quantity, 
in which the unknown quantity is augmented 
de oe bebe ity: 
dicals ing ie 
1 of the aa Ny 
he divides the others by the soieene ta a tity by 
which aes are multiplied; then he makes i 
Zero, an tains an equation which eee 
the aeooae quantity. The following ee ex ample 
will ferve to illuftrate the method of Ferm 
Let it be required to divide a right line i ne two parts, 
in fuch a manner we ae rectangle contained by the two 
parts may ma 
Let a be ‘the ce Tne # one pat, and a — «x the: 
other. The expreffion en to be a maxi- 
mum. Add tie arbitrary ne eto the unknown 
quantity «, and we obtain this new expreffion, 
w+e)— (e+ +e 
Thefe creo being io equal 
= 2 (x ras : is 
ee 
es 
_ = ax 
fabieeaee a— x = cach ‘fide, a ad divide i e 
a 
now fuppofe e= o, thena — 2m=0,anda= -. 
> 
This is the fame refult as. obtained by the fluxional 
r differential method, and the bafis on which they are 
al founded is ver y fim il “The t which are’ re~ 
jected as infinitely fmall by fome writers on the infinitefi 
"Ss 
os 
"QO 
5 
3 
—— 
Thus x being the abiciffa. and y the ordinate, if # is the 
iibeapedt at the point of the curve which anfwers to # 
and y, then by fimilar triangles aoe for the ordi. 
- te to the tangent — to the bli xe; and’ 
this a fhould ap ual to’ tha ‘the curve. for the 
fame abfciffaw + es ther refore have the fe wale 
equation, Pe in the Gets for the curve we fubititu 
x + ein the place of » and y + de ‘in the place of. ye. 
La —. = = divifible by ¢; all its terms then 
o be divi e, and afterwards thofe fupp 
e zero, 
