196 On the Maxima and Minima of Quantities. [Marcu, 
Therefore / a? + 2 + / (C— 2°) + B= y, and it is 
réquired to find x, when y = m (minimum). 
Substituting p and (p + e) for z, as in the last example, we 
have Ja? +p? + VW (—prp +e ay= Va + pre 
doiahKGaap oF Bas ity. « Now Whaipulgrek tapi aE pt 
* st ke. and VEG TO TRH 
Seca a (c — pe 
a Kciot BY Sk Die Fee 
Therefore fa +p? + /V (c— pyr +b=Va +p? + 
pe. ee (c—p)e 
Fi gr Sith (oirrup) ick Dimi eee ike: 
Taking away the quantities that are common to each side and 
Sie ilicc 1 h te SEO re 
sen et ar danni ie a FS PREC ET Ri: 
+ &e. 
0, which 
. ac 
reduced gives p= ——j = 2. 
Emerson, in his Algebra, has given a rule very analogous to 
this, for resolving problems relating to the maxima and minima, 
but it appears to have been suggested to him by the differential 
calculus. At any rate his notion of the symbol e is very different 
from ours; for he supposes it to represent an infinitely small 
quantity, and he rejects the terms containing its powers, with- 
out assigning a very satisfactory reason. e conceive e to be 
the difference of two affirmative roots of the equation fr = y, 
which is always a real finite quantity, except when y atrives at 
a limit, and then it actually vanishes. 
We shall conclude this article with a concise demonstration 
of the well-known theorem, that the fluxion or differential of a 
function = 0, when it is a maximum or minimum. For deve- 
loping the second term of the equation marked (A), by 
Taylor’s theorem,* we have 
afp fp 
fP=fP+O=fP rh Getape t+ ke. 
Taking away f p from each side of the equation, and dividing 
by e, we have ae + Pe+dz=o. But when the function 
attains a limit e = 0, and, therefore, all the terms, except the 
first, disappear from this equation; and we have in this case 
Bf prise pe GF cee bi 
+ mntant But p= 2 -. 7 =0,o0rdfr=o. 
* Tora demonstration of Taylor's theorem, see Calcal. differential et intégral 
de Lacroix, : 
