FLUXIONS. 
its first term, provided that SY, the coeficient off in that 
term, is not aia its. hem. fhe, term itpel maul base 
now this must really be the case, that is, G/ must be =0; 
for were it otherwise, h might be taken-so small, that 
one of each pair of series (A), (B) would be a positive, 
and the other a negative quantity, and the same would 
also be true for every smaller value of 4. But this 
would not accord with the nature of a maximum or mi= 
nimum, which requires, that, .in the first case, the two 
series should be both less than 0; and that, in the se- 
cond, they should be greater than 0: therefore, when 
the function y is a maximum or minimum, we must hav 
dy_o ; and then it will follow, that in the former- case, 
dx 
te Gt 5c co. 
BER Bhd heen 
and in the latter, : 
qe et ae et ae utero 
is @ ae 6 tae oO 
In all these series, the sign of the first term is the 
same ; therefore, when h is a very small quantity, so 
that the amount of all the terms after the first is incom- 
ee ee tee a 
in pair will have the same sign as they ought, and 
this will also be true for every smaller value of h. More- 
over, as in the case of'a maximum, each series must be 
<<.0, therefore 77s, the coefficient of the first term, 
must be a negative quantity. In the case of a minimum, 
however, eachseries is 0, and therefore 4 must be 
positive. 
If, however, the substitution of a for x in the deve- 
lopements makes not only $Y — 0, but also $2 =0; 
then, to satisfy the condition of the maximum, we must 
have 
+52 M+ SY ™ 4 80.20, 
and in the case of the minimum, 
oy By AYE geo, 
Te ae 
By. ing the same train of reasoning as before, it 
ill abpear, that thediccatitionsieanalte, be entoted 
when we haye also “+2, = 0; and 20 on. 
63. Upon the whole, it appears that when y, an 
Retantigin Oa cinglé varidhle quailty 4 te’ itstinns 
or minimum, then 4 = 0; and that if the value of x, 
determined from this equation , make 7% a negative 
419 
quantity, the function y is a maximum ; but if it bea 
positive’ quantity, then y is a minimum. If, however, 
this value of x render Sa =0; then, unless at the 
5. 
same time it make zt} =0, the function can neither 
& 
be a maximum nora minimum. If we have a = 0, the 
dt 
function y will be a maximum when 9 is negative, 
and a minimum when this expression is positive ; and 
so on, the first fluxional coefficient which does not va~ 
nish, being always of an even order when y can have a 
maximum or minimum value, 
_ The correct theory of maxima and minima, was first 
given by Maclaurin in his Fluxions, Book.I. chap. 9. 
Examp.e 1. To divide a right line into two such 
le contained by its segments shall 
parte, that the Bt 
e ear possible. 
Let the whole line be a, and one of its segments 2 ; 
then the other will be a—x; and the rectangle, x«(a—w) 
=ax—z2*. Therefore, we must have y=ax— 2%, a 
ay hence, by the general rule, 
i= a—2x=0. 
This equation gives c—=}a, Moreover, since we have 
<i —2, a negative quantity, we infer that the va- 
lue x = }a corresponds: to a maximum ; which is also 
easily st upon other princip] 
“ane 2. To find the ee exceeds its cube by 
greatest quantity possi 
Let x be the fraction, then we must give such a va- 
lue to #, that y=x—z shall be a maximum. In this 
case, 
dy 
dz = 132° =05: 
Pt , 
therefore, v—=—t4/+. And.since cJ=—6e= = 6/5, 
it follows from the rule, that the maximum value of the 
function corresponds to a=+/5; but it has also a mi- 
ween salen Corse Ae ve 
Ex. 3. To determine greatest rectangle that can 
be inscribed in a given triangle. 
Let ABC (Fig. 2. Plate 
le, and E Gii the 
scribed in it. Draw 
EF inG. Put BC=a; 
of the le EFGH=y. By similar triangles, 
BC: AD::EF:AG; ba is, a:b:: ae: AG; hence, 
r 
=O) snd DG X EF= 
a a ‘ 
4 
ive x—x*)=y, therefore, 
dy_4& 
er oe z)=0; . 
hence a=2 2, and z=}<a. The altitude of the rectangle 
is therefore half that of the triangle. ; 
64. When the quantity which is to bea maximum or 
minimum is:multiplied or divided by any constant quan- 
tity, pre: rc may be rej : Thus we may re-~ 
ject —, and make y=a«—2*; which will lead to the 
same result.. In , when-a variable function is 
the greatest or least possible, any constant multiple or 
CLV.) be the given tri- Prater 
- SaypRamlab 4 that can be in. CCLV. 
perpendicular AD meeting F's: * 
—b ; EF=z; and the area 
