PLATE 
CcLy. 
Fig. 1. 
wi —tas* +7 a2—2a) —2 a's/ (20 2—a%) 
—2ar—a* 4 2ay/ ( 2ar—r*) 
and let its <minaeianes When @+h is 
substituted for x, the fraction becomes 
re er a 
+ a (a—h 
Stow Ieyhe Gnenilel yg ONTOS 
Me 2ah)aph— Ep To Be, 
i? “ie 
RIE LL 
When these series are substituted for the radical quan- 
les adh fo peodacpe), wage -abadeeemeae 
the fraction. 
60. If P and Q are funetions of x, such that when 
sa, then P=0, and Q= infinity, which is expressed 
by the character , it may be to find va- 
tha gfe peeing {Pt, e t R ae ae Put R= 
ae See OAS ae Rae Se 
make R=—0; and as PQ=4, the value. of x, which 
makes PQ=0x #, will make f= 2. The. problem 
may therefore-be resolved by the rules of ait. 58, and 
art. 59. 
Ex. Supposing " 1p stan. (dbo , and « any 
arc, 2 yg 1—x) x‘tan. (42) becomes 0 x 
when z= 1.  emrrad = tan.(422), R= 
ee (42), and— “y= = sates) when z=} this 
Q 
expression becomes Now dP =—dz, and dR 
d 
sm Ay (att 30), Hence in the case of «=1, 
2 
we have-Sp-=-— for the value of the expression 
(1—) tan. (} #2). 
Of the Greatest and Least Values of a Function. 
61. Ify be a function of a variable quantity x, of 
such a nature, that x being supposed to increase or de- 
crease continually, y increases to a certain value, but af- 
terwards : when y has‘that extreme value, it 
is said to be a maximum. , on the other hand, if y 
decrease to a certain value, and then increase ; when it 
has that particular value, it is said to be a minimwn. 
The co-ordinates of a curve par conveseaeiy tis 
relative changes of magnitude 
q , and any function of that quanti- 
Tm Plate cel, Fig. 1. ‘T let CQD be a curve, re | 
to an axis AB, such that, if any abscissa AP all 
© For, supposing the series to be tt. 
the same ; bi theenpreiong trate 
&e.) un any tatio of inequality whatever. 
5 
-term y becomes, by 
po abt mont oy t under this form 
and may be less than any thing mg 
FLUXIONS. 
between Q’ 
again between Q” and Q’", and returns to. 
wards the axis in the branch QD ; and so on. Suppose 
now the ordinate to move to itself from C 
along the axis, it increase from C to Q’: in the po- 
sition P°QY, it will be a maximum; then it will decrease, 
peared eweelfh Mae be a minimum ; after- 
wards it will increase, and be a mazimum in the 
position P'’’Q’”, and so on ; so that it may have vari- 
the values which or which immediate~ 
ly follow it, are smaller ; on the the minimum is 
exceeded by the values which precede and fol- 
62... It is an immediate consequence of this character- 
istic property, that if a be the value of x, which renders 
will both be less than a maximum va- 
Ios, oF tsa hans, endl both qrertes ter than a minimum 
value ; but when this substitution is made, the first 
the maximum or mini- 
alae Coa Therefore, in the case of a maximum, we 
must have 
* pay dy Ws 
y HP 
y—Z Hag FL 
sia Gai G Wed neat 
ae <—¥? 
and similarly ‘in the case of a minimum, 
“) 
+ ein Ze 4 ae astie a 
2 
— Hi. ties as Te ggtee oe 
thprit + &e) ——— remains 
$ therefore, ph may exceed hgh pr A? 
