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infinitely small.

To find the yalue of the latter expression ; let - >: 2; then

(i + tt)" becomes f i + - j , in which z is regarded as infinitely

great. Suppose the limiting value of this expreamn to be re-
presented by the letter e, according to the usual notation. We
'can then find the value of ^ as follows by the Binomial
Theorem: —



(■n) -



2 I zlz-i) I



^ 1.2 1.2.3



+ &C.



The limiting* value of which, when ^ = 00, is eyidflnily



I

I + - +



I 1.2 1.2.3 1.2.3.4

By taking a sufficient number of terms of this series, we
can approximate to the value of « as nearly as we please.
The ultimate value can be shown to be an incommensurable
quantity, and is the base of the natural or Napierian system
of logarithms. When taken to nine decimal places, its value
is 2.718281828.

Again, sinoe (i +«)" = « when w = o, we get

dx X

Also, since the calculation of logarithms to any other
base starts from the logarithms of some numbers to the base e,

* It win 1w shown in Chapter 3, without aasnming the Binomial expaation,
that is the limit of the sum of the series

I + - + h +, &c., ad infinitum.

I 1.2 1.2.3'



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24 Firet Principlea — Differentiation.

as, moreover, the logarithms of all numbers are expressed hj
their logarithms to the base e multiplied by the modulus of
transformation, the system whose base is ^ is fundamental in
analysis, and we shaU. denote it by the symbol log without a
suffix. In this case, since log e= i, we have

^aog*)=^. (21)

Again

where logioC or -= is the modulus of Briggs* or the ordi-
nary tabulated system of logarithms. The value of this mo-
dulus when calculated to ten decimal places is

0.4342944819.

On the method of its determination see " Qalbraith's Algebra,'*

p. 379-

30. Differentiation of a^.

Let y -(^j then log y = « log a,

but rf<bgy)^^_Gogj()^^ i^

dx dy dx yda?

"i? = :l = yiog« = «'iog«- (23)

ve

= ^. (24)



dx dx
Also, since log e = i, we have
d .e



dx

Examples.

I. |f S3 log ^sinx).

Let vmx'^if then y slogs.



And since



dx~"di' dx'



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Logarithfnic Differentiation. 25



dy coax
^e get — = -: — = cot*.

° dx Bmx



2. y=iogv/tf«-«»a}iogc«*»-«a); £=-^rr^**



, /i - COS a;

Vaco8»- *

, . « dy I

.% y = log tan -, — = -; — .

2 ax fsmx

3 1 . Iiogarithmic Differentiation. — ^When the function to be
differentiated consists of products and quotients of functions,
it is in general useful to take the logarithm of the function,
and to differentiate it. This process is called logarithmic
differentiation.

Examples.

1. y =yi . y2 . ya . . . yn, log y = logyi+ logy2 + . . . + logy*.

ydx yi dx y2 dso ' ' ' yn dx'
This fdrnishes another proof of formula (4), p. 13.

an^x „ . , . ,

2. y = . Here, log y = m log sino; - « log cob x ;

t dy cos iT ' Bin dy. Bin»»"»« . . ...

.«. - -2- s m -, — + n .'. ~ = — - I- (m COS** + H sin'a?).

y dx mix cos 1; dx cose^^x

3. y = r^



(«-2)J(«-3)J'

Here logy = ^log(»- i) -3log(«- 2) -2iog(-r-3);

* 4 3



hence



idy _s ' 3 I 7^j 7a;« -t- 30a? - 97

y d^ "" a J? - I * 4 «- 2 " 3 a? - 3 12 , (« - i) (* - 2) (« - 3)*



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26 Mrst Principles — Differentiation.

d^ (a; - i)t (73^ + 30X - 97)

"dg"" ,2.(a;>-2)*(»-3)^

5. y 5= ic«. Here log y = a? log a?.

6. y = tf«^. Here log y = «•,

idy d,xF

7. y = «*'> where ti and v are both functions of af.
Here log y = r log «,

I dy . dv V du
.-. - -r- = log tt ~ + - — ;
y«te ® dx udx

dy /, <2r rd'tfX , do du

...^«t^(^logt,^ + - )-t^log..-+..i-i^.

32. The expression to be differentiated frequently admits
of being transformed to a simpler shape. In such oases the
student will find it an advantage to reduce the expression to
its simplest form before proceeding to its differentiation.

Examples.



X at*

Here , a sin y, or :3 = sin* y, hence a; a tan y,

andweget g = oos«y=j^.

a. y = tan-i^5^^zz= . «

VI + «»-'v/ I -«•

„ V^I + ar* + \/i -a:»

Here tan y



V^l H- a:»- v^l - a«

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Hence



Hence



27



V I + «■ tany + i

._ (i+tany)»-(i~tany)» 2t«ny
• • *^ "■ (I + tany)2 + (I - tany)» " r+tiS^ " ■''' *^-






COB 2y = a;.



" ' <fo "" 008 ay '



/T-a*



=>



- log f A/l-^-g-f y/l-d? ^ I \/l+dg + \/l-g

* l+V^i-ar^l I y I

= -log (i + ^ I -a?2) - - log a?.



4r



aa;Vi -«*

4 ., a/ iT^ - I . ,
^ = tan*^ ^ + tan.">-



Let dT 3= tan s^ and the student can eaaily prove that

if = -B; hence ^a- z.



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^8 First Principles — Differmtiation.



I. y = sec-^ X, Am.



Examples.
dy I



3. y = « logaf. T^ = > + log*.

a9



3. y*=logtanar.

4. y = log tan-' a;.

5. y = a\/x.



dy^ 2
dx sin 2x'

dy I



dx (I + «2) tan-i «*



^ • ^ X <^ COS (log ar)

'6. y = BinGog*). g = — i^'.

. 1 « dy I



8. y = tan-i ^^ ^^ — ~



Here y = tan** v^« + tan*' v^a.



^' ^"/TXaZ^i*'



• log Ia /i + g» + g <^y^ I



11. y

12. y asm-' - 7=1. -pg y

^ (i + «> * dx (i + «) (f + X*)

i-x dy (i + a?)

14. y » r -i B - -i -.

^/i + *2 <*»(!+ »»)*

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ExamplcB. 2^



15- y



(i - x^)^ sin-i a? rfy i - «* i +



2«t^



^ I — tana? dy

19. y = lo6-— + Jbg— -— -+V'3tan-»-^^ ^-



i-x ■ *i-» + «a ^ •* !-«»' dlv i-«»*^



20. y = log {(2a; - i) + 2v^af* - a: - I } .



ifo ■ (:b» - a? - I)i"



\l-;r^/2 + a:* I-** ^ ! + «*'
22. y = tf*'' tan-'a?. ^f "•"'(737;^ + «'**^"^* (^+lo«*)y
23• Being giyen that ysx^li-aj^J |i J ;if

dy ex^ + ^a^ + g"a:*

detennine the values of <?, c', c". -4««. « = 3, c' = — 6, <j" = f .



24. y = log (log a?).



<f:i; «log«



, 3 + 5 COB j: <fy 4

25. y = cos-^ - — . -r •

5 + 3 cos a: dx 5 + 3 cosa:



26. y ss sin-' r.



- 2



I +«* <£r I + «•



27. y = ««* 8m"» rr. 1^ ~ **" 8in«»-V» (a sinrx + mr cosrv).

28. y = tf«*8mr«. j^ = *'»*V^a«+ r» sin (ra? + ^).



<£b



where tan ^ a -.



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30 First Princ^lea — Differentiation.

,9. y = log (v/^::7 + ^/r^*). /«». ;! = —^======.

30. y = 2tan-i(i^]*.

__ I — a? y <fy I

Here = tan» - .*. ar = cos y .% ~ = - -; r:?.

I + a? 2 ^ dx (I - ««)!

31. y = af***. ~ = af*'*+'*-^(nlog«+i).

32. y = (i+«2)igin(mtan-i«). ;^ = m(i +««)"i~cos {(m- Otan-'dr}.

__. /a COS 09 - 6 sin a; <^__ — «J

33' y- °*^\flfoo8» + i8iiia;' dx "" a^cos'o: - i^sin^a?*

34. Define the differential coefficient of a function of a variable quantity, '
with respect to that quantity, and show that it measures the rate of increase of
the function as compared wilJi the rate of increase of the variable.
,. I . ^ ^-^



35. If y = -, prove the relation / ^



dy\ dx '""^



\/i + y* \/i + «*

36. If « = log — ^ prove that •— is of the form

°ar2 + a«-.^(^.^^)»_^^) *- <2aT

T and determine the Values of A and B, Am, A^x.B^a.

\/{x^\axY''bx

•o A^i. J. ^ I ' ^ n / . ' nA -^sin^tf + ^sin'a + C

37. Prove that ~ I sm a cos a v' i ~ <?* sm*^ | = ,

and determine the values of A, J, C, Ant. A = 3c', ^=- a (i + <?•), C= i.

38. iftt = » + I.- +~^^ "^r^?**" • • • <^»»/'-^ndthesumof

du /

the series represented by — . Ans. (i - «*)-*.

39. Eeduce to its simplest form the expression

30* d_ x(x^ + ag)i I

40. Ifsiny = :psin(«+y),provethat^ = 22^^?±i^).

ax sma

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JSxanyi>ks. 31



dv

41. Ifjf(i+y)»+y(i + a?)» = o, find-f.

ax

In this case «» (i + y) = yt (i + jr),



or «-t-y+«y = o, .-. y = -



i4-« ^ (!+«)*



43. If » and y are giTen as fanctioiis of t by the equations

<^ du F' (t)

find the valne of :r ^ ^nns of <. :r = "TTTir*

d» dx /' {t)

44. y



I +««



X + &c., tM inflnUum.

_ «» dy »

Hence y = - — . ~ - ■ .

45. a? = « 1

dP <fy logjff



■*' Hence y =



I -h logjr <te (i + loga?)«



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( 32 )



CHAPTER n.

SUCCESSIVE DIFFERENTIATION.

33. Successive Derived Functions. — In the preceding chapter
we have considered the process of finding the derived func-
tions of different forms of functions of a single variable.

If the primitive function be represented by /(a?), then, as
already stated, its/r«^ derived function is denoted \yj/{x).
If this new function, /'(a?), be treated in the same manner,,
its derived function is called the second derived of the original
function /(a?), and is denoted \yjf\x).

In like manner, the derived function of f\x) is the third
derived of /(a?), and represented ^jf'\x)y &c.

In accordance with this notation, the successive derived
functions oif{x) are represented by

AA, /'H, /» /-H*),

each of which is the derived function of the preceding.
34. Successive Differential Coefficients.

If y =/(;.), we have I =/'(^).

Hence, difierentiating both sides with regard to x, we get

then S=-^'(^)- , '

In like manner, ^( ^) i® represented by -^, and so on ;

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Successive Differentials, 33



'oe g=/"(;r), &o. ..


•2 - ^-'<')-


(0


The oxpressioiiB






dx" dj»' d^' '


• • dj^





are called Qiefirsty second^ thirds . , , rf^ differential coefficients
of y regarded as a function of x.

These functions are sometimes represented by

y\ y", r, . . . yC',

a notation whioli will often be found convenient in abbreviat-
ing the labour of forming the successive differential coefficients
of a given expression. From the mode of arriving at them
the successive differential coefficients of a function are evi-
dently the same as its successive derived functions considered
in the preceding Article.

35. Successive Dififerentials. — The preceding result admits
of being considered also in connexion with differentials ; for,
since x is the independent variable, its increment, dx^ may be
always taken of the same infinitely small value. Hence in the
equation dy =f {x) dx of Art. 7, we may regard dx as con-
stant, and we shall have, on proceeding to the next differen-
tiation,

d[dy)-d<cd[/{x)'\^[dxYr{x\

since dlf{x)']=f(x)dx.

Again, representing d {dy) by d^y^

we have d^y =/" {x) (dxf ;

if we differentiate again, we get

d^y =f" {x) {d^) ;
and in general

d^y =/(«) [x) {dx^.

From this point of view we see the reason why/^**) (x) is
caUed the n*^ differential coefficient oif(x).

D

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34 Successive Differentiation,

In the preceding results, it may be observed, that, if dx
be regarded as an infinitely small quantity ^ or an infinitesimal
of the first order, (dxf being infinitely smMl in comparison with
dxy may be called a;n infinitely small quantity or an infini-
tesimal of the second order ; as also cPy, if/" {x) be finite. In
general, dJ^y being of the same order as [dxf is called an t«-
finitesimal of the n*^ order.

36. Infinitesimals. — ^We may premise that the expressions
great and small, as well as infinitely great and infinitely small,
are to be understood as relative terms. Thus, a magnitude
which is infinitely great in comparison with a finite magnitude,
is said to be infinitely great. Similarly, a magnitude which
is infinitely small in comparison with a finite magnitude is
said to be infinitely small. If any finite magnitude be con-
ceived to be divided into an infinitely great number of eqtuil
parts, each part will be infinitely small with regard to the
finite magnitude ; and may be called an infinitesimal of the
first order. Again, if one of these infinitesimals be conceived
to be divided into an infinite number of equal parts, each of
these parts is infinitely small in comparison with the former
infinitesimal, and may be regarded as an infinitesimal of the
second order, and so on.

Since, in general, the number by which any measurable
quantity is represented, depends upon the unit with which
the quantity is compared it follows that a finite magnitude
may be represented by a very great, or by a very small num-
ber, according to the unit to which it is referred. For ex-
ample, the diameter of the earth is very great in comparison
with the length of one foot, but very small in comparison
with the distance of the earth from the nearest fixed star, and
it would, accordingly, be represented by a very large, or a
very small number, according to which of these distances is
assumed as the unit of comparison. Again, with respect to
the latter distance taken as the unit, the diameter of the earth
may be regarded as a very small magnitude of the first order,
and the length of a foot as one of a higher order of small-
ness in comparison. Similar remarks apply to other magni-
tudes.

Again, in the comparison of numbers, if the fraction (one

million)** or — j, which is very small in comparison with



10"



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Geometrical Illustration,



35



unity, be regarded as a small quantity of the first order, the

fraction — -, being the same fractional part of — ^ that this is

of I, must be regarded as a small quantity of the seoond
order, and so on.

The preceding is introduced solely for the purpose of

illustration. If now, instead of the series — -g, f — ^ ) , f — -^ J ,



we consider the series -, — , —,



in which n is



supposed to be increased without limit, then each term in the
series is infinitely small in comparison with the preceding
one, being derived from it by multiplying by the infinitely

small quantity - Hence, if - be regarded as an infinitesimal



maybe regarded as infinitesi-



n

of the first order, -;, — . . . —
n^ n^ n^

mals of the second^ third, . . . r** orders.

37. Geometrical Illustration of Infinitesimals. — The fol-
lowing geometrical results will help to illustrate the theory
of infinitesimals, and also will be
found of importance in the appli-
cation of the DifEerential Calciilus
to the theory of curves.

Suppose two points, -4, JS, taken
on the circumference of a circle ;
join B to J5^, the other extremity
of the diameter AE, and produce
EB to meet the tangent at A
in D. Then since the triangles
ADB and EAB are equiangular,
we have

AB BE ^BD AB
AD" AE^'^'^AD^ AK

Now suppose the point B to approach the point A and to
become infinitely near to it, then ^JS becomes ultimately

equal to AEj and, therefore, at the same time, -jj^ = i,

I) 2




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36 Successive Differentiation,

Again, -jjz teoomes infinitely small along with ~j^, i. e.

BD becomes infinitely small in comparison with AD or AB.
Hence BD is an infinitesimal of the second order when AB is
taken as. one of the first order.

Moreover, since DE - AE < BD, it follows that, when one
side of a right-angled triangle is regarded as an infinitely small
quantity of the first order, the difference between the hypothenuse
and the remaining side is, an infinitely small quantity of the
second order.

Further, draw BN perpendicular to AD, and BF a
tangent at B ; then, since AB > AN, we get AD ~ AB
<AD-AN<DN,

''' BD ^ BDp DE'

Consequently, ^^7 — becomes infinitely small along with

AD, /. AD - AB is an infinitesimal of the third order.
Moreover, as BF = FD, we have AD = AF+BF /. AF
+ BF - AB is an infinitely small quantity of the third order ;
but AF + FB is > arc AB, hence we infer that the difference
between the length of the arc AB and its chord is an infinitely
small quantity of the third order, when the arc is an infinitely
small quantity of the first. In like manner it can be seen
that BD - BN is an infinitesimal of the fourth order, and
so on.

Again, if AB represent an elementary portion of any
continuous* curve, to which AF and BF are tangents, since
the length of the arc AB is less than the sum of the tangenta
AF and BF, we may extend the result just arrived at to all
such curves.

* In this extension of the foregoing proof it is assumed that the ultimate
ratio of the tangents drawn to a continuous curve at two indefinitely near
points is, in general, a ratio of equality. This is easily shown in the case of
an ellipse, since the ratio of the tangents is the same as that of the parallel
diameters. Again, it can he seen without difficulty that an indefinite numher
of ellipses can he drawn touching a curve at two points arhitrarily assumed on
the curve ; if now we suppose the points to approach one another indefinitely
along the curve, the property in question immediately follows for any con-
tinuous curve.



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Geometrical Illustration. 37

Hence, the difference between the length of an infinitely
small portion of any continuous curve and its chord is an
infinitely small quantity of the third order ^ i. e. the difference
between them is ultimately an infinitely small quantity of
the second order in comparison with the length of the chord.

The same results might have been established from the
expansions for sin a and cos a, when a is considered as
infinitely small.

If in the general case of any continuous curve, we take
two points Ay B, on the curve, join them, and draw BE
perpendicular to ABy meeting in E the normal drawn ix)
the curve at the point A ; then all the results established
above for the circle still hold. When the point B is taken
infinitely near to A^ the line AE becomes the diameter of
the circle of curvature belonging to the point A ; for, it is
evident that the circle which passes through A and -B, and
has the same tangent at ^ as the given curve, has a contact
of the second order with it. See " Salmon's Conic Sections,'* .
Art. 239.

Examples.

1. In a triangle, if the vertical angle be regarded as infinitely small, tbe
other angles remaining finite, prove that the difference between the sides is
infinitely small in comparison with either of them ; and hence, that these sides
may be regarded as ultimately equal.

2. Id a triangle, if the external angle at the' vertex be very small, show
that the difference between the sum of the sides and the base is a very small
quantity of the second order.

3. If the base of a triangle be an infinitesimal of the first order, as also its
base angles, show that the difference between the sum of its sides and its base
is an infinitesimal of the third order.

This furnishes an additional proof that the difference between the length of
an arc of a continuous curve and that of its chord is ultimately an infinitely small
quantity of the third order.

4. If a right line be displaced through an infinitely small angle, prove that
the projections on it of the displacements of its extremities are equal.

5. If the side of a regular polygon inscribed in a circle be a very small
magnitude of the first order in comparison with the radius of the circle, show
that the difference between the circumference of the circle and the perimeter of
the polygon is a very small magnitude of the second order.



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38 Successive Differentiation.

38. Fundamental Principle of the Infinitesimal Calculus. —
We shall now proceed to enunciate the fundamental prin-
ciple of the Infinitesimal Calculus as conceived by Leibnitz :*
it may be stated as follows : —

If the difference between two quantities be infinitely
small in comparison with either of them, then the ratio of
the quantities becomes unity in the limit, and either of them
can be in general replaced by the other in any expression.

For let a, j3, represent the quantities, and suppose

i •

Now the ratio ^ becomes evanescent whenever i is infinitely

small in comparison with j3. This may take place in three
different ways : (i) when /3 is finite, and i infinitely small :
(2) when i is finite, and (5 infinitely great; (3) when /3 is
infinitely small, and i also infinitely small of a higher order i

thus, if i = A'j3^, then j^ = A]3, which becomes evanescent along-

with j3.



* This principle is stated for finite magnitudes by Leibnitz^ as follows : —
'' Cseterum eequalia esse puto, non tan turn quorum differentia est omnino nulla^
Bed et quorum differentia est ihcomparabiliter parva." ..." Scilicet eas
tantum homogeneas quantitates comparabiles esse, cum Euc. Lib. 5, defin. 5,
oenseo, quarum ima numero sed finito multiplicata, alteram superare potest ; et
qufld tali quantitate non dififerunt, aemialia esse statuo, quod etiam ^chimedes-
siimsit, aluque post ipsum omiies." Leibnitii Opera, Tom. 3, p. 328.

The foregoing can be identified with the fundamental principle oi Newton^
as laid down in Ms Prime and Ultimate Ratios, Lemma I. " Quantitates, ut et
quantitatum rationes, qusB ad sequalitatem tempore quoyis finito constanter
tendunt, et ante finem temporis illius proprius ad inyicem accedunt quam pra
dat& quayis differential, fiunt ultimo aequales."

All applications of the infinitesimal method depend ultimately either on the
limiting ratios of infinitely small quantities, or on the' limiting yalue of the
sum of an infinitely great number of infinitely small quantities ; and it may
be obseryed that the difi^erence between the metbod of infinitesimals and that of
limits (when exdusiyely adopted) is, that in the latter method it is usual to
retain eyanescent quantities of higher orders until the end of the calculation,
and then to neglect them, on proceeding to the limit ; while in the infinitesimal
method such quantities are neglected from the commencement, from the know-
ledge that they cannot affect the Jlnal resttlt, as they necessarily disappear ijk
the limit.



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Principles of the Infinitesimal Calculus, 39

Accordingly, in any of the preceding cases, the fraction
^ becomes nnity in the limit, and we can, in general, substi-
tute a instead of /3 in any fmiction containing them. Thus,
an'i^finitely small quantity is neglected in comparison with a
finite one, as their ratio is evanescent, and similarly an infini-
tesimal of any order may be neglected in comparison with
one of a lower order.

Again, two infinitesimals a, 0, are said to be of the same

3 ... 6

order, if the fraction - tends to a finite limit. If ^ tends to

a finite limit, ]3 is called an infinitesimal of the w'* order in
companson with a.

As an example of this method, let it be proposed to deter-
mine the direction of the tangent at a point {x, y) on a curve
whose equation is given in rectangular co-ordinates.

Let iP + a, y + /3, be the co-ordinates of a near point on
the curve, and, by Art. 10, the direction of the tangent de-
pends on the limiting value of - . To find this, we substi-

a

tute x-va for a;, and y + /3 for y in the equation, and neglect-
ing all powers of a and /3 beyond the first, we solve for -,

and thus obtain the required solution.

For example, let the equation of the curve be o:^ + y^ = ^axy :
then, substituting as above, we get

aj^ + s^^a + y^ + 3y*/3 = saxy + ^axfi + ^aya :

hence, subtracting the given equation, we get the

limit of ^ = ^Jl^^.
a ax - y^

39. Subsidiary Principle. — If Oi + 02 + as + • • • + On re-
present the sum of a number of infinitely small quantities,
which approaches to a finite limit when n is increased indefi-
nitely, and if j3i, jSj, . . . /3n, be another system of infinitely
small quantities, such that

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40 Successive Differentiation.

where ci, £2, . . . €n> ar© infinitely small quantities, then the
limit of the sum of /3i, /32, . . . i3«, is ultimately the same as
that of ai, 02, . . . an'

For, from the preceding equations we have

/3i-f j32 + . . .+j3» = ai + a2 + . . . +a» + ai£i +0262+ . . . + a«6«.

Now, if »j he the greatest of the infinitely small quantities,
€1, £2, . . . £n, we have

/3i + /32+ . . . +j3n- (01 + 02+ . . . +a„) < ij (oi + aa . . . + a*),

but the factor oi + 02 + . . • + on has a finite limit, by hypo-
thesis, and as IJ is infinitely small, it follows that the limit of
/3i 4^ 182 + . . . + /3n is the same as that of d + 02 + . . . + ©n.
This result can also be established otherwise as follows : —

The ratio /3. - ^^A...±P»



Ol + 02 + *•.+««



by an elementary algebraic principle, lies between the greatest
and the least values of the fractions

) 5 • • • >

Oi 02 On

it accordingly has unity for its limit under the supposed con-
ditions : and hence the limiting value of /3i + /32 + . . . + /3f» is
the same as that of Oi + 02 + . . • + an»^

40. The principles of the Infinitesimal Calculus above
established lead to rigid and accurate results in the limit,
and may be regarded as the fundamental principles of the



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