FLUXIONS. 
Inverse’ limits may be found, between. which the curvilineal 
space oF the fluent is always contained. Besides, these 
may differ by as small a quanti 
dr diferente i manifestly tbe resangle PMT: 
for | I e 
which is containedby PM’=QQ’, andPR= 
the difference of the extreme ordinates; and QQ’ may 
Re sa soll 8. e pleas: 
If chords PP’, fasens Bs Gals, the gan tees 
lineal trapeziums PQQ’P’, &c. be a nearer approxi- 
pean iE ory aloe leccay pprpee pees p74 
or inscribed - . As an example 
Set eyelicanians eA tis: iecthodp We tube pero Oo 
approximate to the fluent gr? between the limits 
of 220 Gnd 2 231: FH Aser ok ts 
— the equation of the curve CD is y = 
Let 9a” equal parts, 
’ be divided into ten then, put- 
ting z= 0, x=.1, z=.2, &. to x=1, we oltain 
eleven equid ordinates ; the numeral values will 
be as WS: 
The Ist, 1.00000, . - The 7th, .73529. 
The 2d, .99010. | The 8th, .67114. 
_ The 3d, .96154. The 9th, .60975. 
The 4th, .91743. The 10th, .55249. 
The 5th, _.86207, The 11th, .50000. 
The 6th,. .80000. 
By the elements of etry, the area of the rectili- 
neal figure forured by the trapesiume, is found by add- 
ing together all the ordinates ex the first and last, 
and half the sum of the first and and multiplying 
the result by the common breadth of the iums, 
which is .1. peerala: ines 7.84981 for the area or 
value of fia veeween the proposed limits This 
fluent is the arc to the w, (art. 35). We have 
srpmanss hy.8. cutee) art. 148). If in that series 
we put «= 1 fluent, between ron ropa abe 
mits, will be 1 —+ 44 —44 &e. ; but is converges 
too slow to be of any use. The fluent ought to be 
glans 
153. If the ordinates PQ, P’Q’, &c. go on continual- 
increasing, the inscribed will be construct- 
ed on the Ist, 2d, 3d, &c. ordinates, and the circumscri- 
bed parallelograms on the 2d, $d, 4th, &c. Observing, 
now, that the ordinates are the values of the function i 
corresponding to AQ, AQ’, &c. values of x which differ 
from each other by the common interval QQ’, we have 
manifestly the following rule for approximating to a flu- 
ent pes between the limits of x=a and r=b. 
interval between a and & be divided into 
equal parts, each to h. J 
per BER . . Y®, be the values of y corre- 
©>= 4,@>=a+h, r=a+2h, &e. tox = 
+n h respectively, and let us suppose that Y, Y, &e. 
go on continually increasing ; then 
Syd UY4Y 4... 4¥P-), 
 Sydz ah (VEY 4... pO). 
The difference of these is h(Y—Y), which, by ta- 
be.as as we 
Sep Reelicentiy small, may small 
‘in ie oe 
which Y, Y’, ¥”, &c. curgitinibilivindtarmecet 
pip Y’, &c. first increase, but afterwards 
=a and «=é may be divided 
VOL. IX, PART I. 
449 
into two or more portions ; so that y may increase or 
decrease continually; from one extremity of each to the 
other. i cra 
Whatever be the values of y, provided they be al- 
ways finite from x=ato v= 6, if Y, Y’, &c. be de- 
termined as before, we have evidently, i 
Sydea=Vh+ Wap Yh... 4 YO h, neatly; 
and the smaller the increment / is, the more correct, 
will be the approximation to the fluent ;. so that the 
number 7 peng supposed to increase continually, and 
uently A to decrease, the expression will a 
- continually to the fluent, which will be its 
Leibnitz, and such as have taken his view of the 
subject, considered the fluent f‘y dx as the swm of the 
infinitely little elements Yh, Y’ kh, &c. Hence the 
origin of the terms integral, to integrate, integration, 
&c. (art. 112.) And as each was the difference between 
two infinitely near values of the’ integral, regarding hk 
as the differential of x, it followed that yd 2 (the general 
expression for each element) was the differential of the 
i This manner of considering a’ fluent is ex- 
tremely convenient, and on that ' account is ire 
employed in the application of'the integral or fluxional 
calculus, to physics and the higher geometry. 
154; We shall now make some general remarks re- 
‘ches 
Inverse 
1. If an area s is contained between two curves CD, Fig. 29. 
ed, or two branches of the same curve, (Fig. 29.), let 
PQ=y, and p Q= y’ be their ordinates corresponding 
to the common abscissa AQ = 2, then fy dz = area 
CEQP, and fy’ dx = area cEQ p; therefore {(y—y') 
dx=area Cep P, 
2. Or employing the calculus of infinitesimals, we 
may the area CP pc =<s as, made up of an in- 
finite number of trapezoids PP’p'p; each having dx 
for its breadth, and these again as made of an infinite 
number of rectangles m, of which the sides are dx and 
dy, 86 that dxdy will be an’ element of the second 
order of the areas: Then, to obtain s, the fluxion 
or differential dx dy must be integrated from y = PQ 
to y= pQ, and again the result between the limits 
2 AO and a= AE, so that we arrive at the same 
final result as before. 
3. The entire area of a curve that returns into itself, 
is found by taking the fluent of (y—y') da from the 
least to the greatest value of x. 
4, The ordinate y of the curve ought never to be- 
come infinite between the limits of the area. 
5. The fluxion yd changes its sign with y or 2; 
hence the area becomes negative if « and y have con- 
6. If a curve cuts the axis of the abscisse between 
the limits of the area, the parts on each sicle of the axis 
must be found separately, because the one is positive 
and the other is negative, and the fluent requires to be 
taken without any rd to the latter sign: 
For example, let KOACD be a curve of which the Fig, 30. 
equation is y=2— 2; (Fig. 30.) the origin of the co- 
ordinates bang at A, ae dave thee through A, and 
meets the axis in H and & so that are 
The al expression for the area 5 is 4: atc, 
If we su oD th bogie a6. dhe posta Rha a = 
AE = 4/4, then, at the origin of the fuent, O= 7— 1, 
bd hence c = — +, and s=> fat —fot— ys. If 
area is to end at F where «= AF = 4/3, we shall 
find.s = 0, which pao that the areas ECH, HDF’ 
L 
