Hugh MacColl.

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Font size formula asserts that the difference between a positive
infinite and a positive finite is a positive infinite.

Note 1.— A fuller discussion of the finite, the infinite, and the infini-
tesimal will be found in my eighth article on " Symbolic Reasoning" in
Mind. The article will probably appear next April.

Note 2.— The four " Modals " of the traditional logic are the four terms
in the product of the two certainties A T + A' and A f + A' + A". This pro-
duct is A^ + A^ + A^A^ + A'A"; it asserts that every statement A is either
necessarily true (A € ), or necessarily false (A''), or true in the case considered
but not always (A T A"), or false in the case considered but not always (A'A").
See § 99.

CALCULUS OF LIMITS

CHAPTER XIV

114. We will begin by applying this calculus to
simple problems in elementary algebra. Let A denote
any number, ratio, or fraction. The symbol A x asserts
that A belongs to the class x, the symbol x denoting
some such word as positive, or negative, or zero* or
imaginary, &c. The symbols A*B», A^ + B 2 ', A* : B y , A~ x ,
&c, are to be understood in the same sense as in §§ 4-
10. For example, let Y= positive, let N = negative, and
let = zero* ; while all numbers or ratios not included
in one or other of these three classes are excluded from
our Universe of Discourse — that is to say, left entirely
out of consideration. Thus we get (6 — 4) p , (4 — 6) N ,

(3 - 3)°, (f), (3 x 0)°, (3PJi* (P^/, (W, (N^f ,

,3,

(P 1 + P 2 ), p (N 1 + N 2 ) N , and many other self-evident for-
mulas, such as

(1) (AB) P = A P B P + A N B N .

(2) (AB) N = A N B P + A P B N .

(3)(AB)° = A° + B°.

(4) {Ax - B) p = Ux - B )Y = k{x - ?Y + A*(x - B

I V A/J \ A) \ A

* In this chapter and after, the symbol 0, representing zero, denotes
not simple general non-existence, as in § G, but that particular non-
existence through which a variable passes when it changes from a

positive infinitesimal to a negative infinitesimal, or vice verm. (See § 113.)

106

§§114, 115] CALCULUS OF LIMITS 107

(5)(A ,-B,={4-B)^4-By + 4_By.

(7) (ax = ah) = (ax - ab)° = { a(x -b)}° = a" + (x - b)°.

115. The words greater and less have a wider meaning
in algebra than in ordinary speech. In algebra, when
we have (.« — a) p , we say that " x is greater than a,"
whether a is positive or negative, and whether x is
positive or negative. Also, without any regard to the
sign of x or a, when we have (x — ctf, we say that " x
is less than a!' Thus, in algebra, whether x be positive
or negative, and whether a be positive or negative, we
have

(x — of = (x > a), and (x — «) N = (x < a).

From this it follows, by changing the sign of a, that

(x + af = (x > - a), and (x + af = (x < - a) ;

the symbols > and < being used in their customary
algebraic sense.

For example, let a - 3. We get

(,r-sy = (x>3), and (x - 3f = (x < 3).

In other words, to assert that x — 3 is positive is
equivalent to asserting that x is greater than 3 ; while to
assert that x — 3 is negative is equivalent to asserting
that x is less than 3.

Next, let a = - 3. We get

( x - a y = (x + 3) p = (x > - 3 )

(x - af = (x + 3 ) N = (x < - 3 ).

Let x = 6, we get

(x > - 3) = (x + 3) p = (6 + 3 ) p = e (a certainty).
Let x= 0, we get

(x > - 3 ) = {x + 3 ) p = (0 + 3 ) p = e (a certainty).

108 SYMBOLIC LOGIC [§§ 115-117

Let x= — 1, we get

(x > - 3) = (x + 3) p = ( - 1 + 3) p = e (a certainty).

Let a? = — 4, we get

(x > - 3) = (x + 3) p = ( - 4 + 3 )'' = >/ (an impossibility).

It is evident that (,/; > — 3) is a certainty (e) for all
positive values of x, and for all negative values of x
between and — 3 ; but that x> — 3 is an impossibility
(>?) for all negative values of x not comprised between
and —3. With (x< —3) the case is reversed. The
statement (x< — 3) is an impossibility (>?) for all positive
values of x and for all negative values between and
— 3 ; while (x < — 3 ) is a certainty (e) for all negative
values of x not comprised between and — 3. Suppose,
for example, that x= — 8 ; we get

(x< - 3) = (x + 3) N = ( - 8 + 3) N = e (a certainty).

Next, suppose x= — 1 ; we get

(x< - 3) = ( - 1 + 3) N = >? (an impossibility).

116. From the conventions explained in § 115, we get
the formulas

(A>B) = (-A)<(-B), and (A<B) = (- A)>( -B);

for{(-A)<(-B)} = {(-A)-(-B)} N = (-A + Bf
= (A-B) P = (A>B),

and{(-A)>(-B)} = {(-A)-(-B)} p = (-A + B) p
= (A-B) N = (A<B).

117. Let x be a variable number or fraction, while
a is a constant of fixed value. When we have (x — «) p ,
or its synonym (x > a), we say that a is an inferior limit
oix\ and when we have (x — cif, or its synonym (x<a),
we say that a is a superior limit of x. And this defini-
tion holds good when we change the sign of a. Thus
(x + a) p asserts that — a is an inferior limit of x, and
(x + cif asserts that — a is a superior limit of x.

§§118,119] CALCULUS OF LIMITS 109

118. For example, let it be required to find the
superior or inferior limit of x from the given inequality

x — 3 x + 6

Sx — > x +

2 3

Let A denote this given statement of inequality. We
get

\ 2 3

= i6 2« — — — - _ )\ = (tx—3y=(x —

3 .
Hence, — is an inferior limit of x. In other words, the

7

given statement A is impossible for any positive value

3
of x lower than -, and also impossible for all negative

values of x.

119. Given the statements A and B, in which

A denotes Sx — — — < — , and B denotes — 3x < -

2 4 3 4'

Find the limits of x. We have

A = Ux-°^- 1 -j = (12x-l() + 2x-lf

=(^-n>»4-liy=(*<n).

\i = ( 6 — -3x - j = (24-4x-36x-3)»

= (21-4tor = (4te-2ir' = (.,-^J = ( a; >|l).

Hence we get AB = ( — > x > — 1.

8 \14 40/

110 SYMBOLIC LOGIC [§§ 119, 120

Thus x may have any value between the superior

11 .... 21

limit - and the inferior limit - ; but any value of
14 40 J

x not comprised within these limits would be incom-
patible with our data. For example, suppose x = 1 .
We get

a -(s - *=i - 1Y Ya - 2 - 1 Y- /3 V • » (an im ~

\ 2 4/ '\ 4/ 'U/ ' >? possibility).

B : ( 6 ^ - 3 - ij Y| - 30 ; e (a certainty).

Thus the supposition (#=1) is incompatible with A
though not with B.

Next, suppose x=0. We get

/ ^ 1 N
A : ( ] : e (a certainty).

B : ( - — ) : n (an impossibility).

Thus, the supposition (x=0) is incompatible with B
though not with A.

120. Next, suppose our data to be AB, in which

A denotes ox — - > 4a; + -.
4 3

B denotes Qx — - < 4« + -.
2 4

We get

3 . IV / 13\ p / 13

^-4- 4;/: -3J = (^12 / ) = (^12

§§ 120, 121] CALCULUS OF LIMITS 111

Hence we get

5 13

AB = ->£> — = >i (an impossibility)

, /5 13\ /5 13\

t01 \8 >aJ> T2J : (8 > l2J : ' / -

In this case therefore our data AB are mutually
incompatible. Each datum, A or B, is possible taken
by itself; but the combination AB is impossible.

121. Find for what positions of x the ratio F is

positive, and for what positions negative, when F

2x-l 28
denotes — — .

x — 6 x

2x 2 -29a; + 84 2(x- 4)(x - 10£)
1 = x(x-3) x(x - 3)

As in § 113, let a denote positive infinity, and let /3
denote 'negative infinity. Also let the symbol (to, n)
assert as a statement that x lies between the superior
limit m and the inferior limit n, so that the three
symbols (to, n), (m>x>ri), and (m — x)\x — nf are
synonyms. We have to consider six limits, namely,
a, 10i, 4, 3, 0, (3, in descending order, and the five
intervening spaces corresponding to the five statements
(a, 10i), (10J, 4), (4, 3), (3, 0), (0, (3). Since x must lie
in one or other of these five spaces, we have

e = (a, 10£) + (10l, 4) + (4, 3) + (3, 0) + (0, (3).

Taking these statements separately, Ave get

(a, 1 0+) : (x - 1 0|) p : (x - 1 0|)> - 4)> - 3) V : F p

( 1 Oh 4) : (x - 1 0|-)> - 4) p : (z - 1 Offix - 4) p (;v - 3 ) V : F K

(4, 3) : (x - ±)"(x - S) ¥ : (x - 10i) N (fl - 4) N (.v - 3)V : F p

(3, 0) : (x - 3)V : (x - 10|)> - ±f(x - 3)V : F N

( , /3) : x" : x\x - 3 f{x - 4) N (sc - 1 0i) N : F p .

Thus, these five statements respectively imply F p , F N , F p ,

112 SYMBOLIC LOGIC [§§ 121, 122

F N , F p , the ratio or fraction F changing its sign four
times as x passes downwards through the limits 1 Oi, 4,
3, 0. Hence we get

F p = («, 10*)+(4, 3) + (O,0);

F N = (10i 4) + (3, 0).

That is to say, the statement that F is 'positive is equiva-
lent to the statement that x is either between a and 10 \,
or between 4 and 3, or between and ft ; and the state-
ment that F is negative is equivalent to the statement
that x is either between 10i and 4 or else between
3 and 0.

2«-l_28
122. Given that — , to find the value or

x — 3 x

values of x.

It is evident by inspection that there are two values of

x which do not satisfy this equation ; they are and 3.

m n . 2a; -1 1 ... 28 28 . .

When x=0, we get = - while — = — ; and evi-

6 x-3 3' x

dently a real ratio - cannot be equal to a meaningless
o

28
ratio or unreality — (see § 113). Again when x=3, we

. 2re-l 5 ... 28 28 , ., fl 5

get - — = — , while — = — ; and evidently - cannot
6 x-S x 3 J

28
be equal to — . Excluding therefore the suppositions

(x=0) and (x=o) from our universe of possibilities, let

A denote our data, and let F = — — - — . We get

x — 3 x

A . F o . / 2a- 1 _ 28\°. f 2(x-<k)(x-10b) \°
\x-3 xj'l x(x-'S) J
: {( X - 4)(^-10i)} : (x- 4f + (x-10if
:(x = 4:) + (x=10i).

§§ 122-124] CALCULUS OF LIMITS 113

From our data, therefore, we conclude that x must be
either 4 or 10i.

„^ n „ , . 13j; 3 3« 6 — 7%

123. Suppose we nave given >

8 4 4 8

to find the limits of x.

Let A denote the given statement. We have

. /13a; 3 3x G - 7.A 1 ' , 1Q B B , . _ XP

A = | = (13# - 6 - 6x + - 7«) p

\ 8 4 4 8/

= l ' = v.

If in the given statement we substitute the sign < for

the sign >, we shall get A = N = >/. Thus, the state-

,, ,13a; 3 . , ,i 3x 6 — 7x . .

ment that is greater than is nnpos-

8 4 4 8 l

'tQ, - , Q

sible, and so is the statement that — is less than

8 4

3% Q — 7x TT 13a; 3 , , , J ox G — 7a?

. Hence must be equal to ,

4 8 8 4 * 4 8 '

whatever value we give to x. This is evident from the fact

2,x 6 7x

that - . when reduced to its simplest form, is

4 8 r

\2x — 6

, which, for all values of x, is equivalent to

13x 3

If in the given statement we substitute the

8 4' b

sign = for the sign > , we shall get

/13a_3_3. G-7,y = ()0 =
\ 8 4 4 8/

so that, in this case, A is a formal certainty, whatever be
the value of x.

124. Let A denote the statement x} + 3>2>x\ to find
the limits of x. We have

A = (x 2 - 2x + 3) p = { (x 2 - 2x + 1 ) + 2 } p
= {{x- l) 2 + 2}" = e.

H

114 SYMBOLIC LOGIC [§§ 124-128

Here A is a formal certainty whatever be the value of x,
so that there are no real finite limits of x (see § 113).
If we put the sign = for the sign > we shall get

A={(,e-l)° + 2}° = > h

Here A is a formal impossibility, so that no real value of
x satisfies the equation x 2 + 3 = 2x. It will be remem-
bered that, by § 114, imaginary ratios are excluded from
our universe of discourse.

125. Let it be required to find the value or values of
x from the datum x — s /x= 2. We get

(x -Jx=2) = (x - Jx - 2)° = (x v + x* + x°)

( ;>J _ J x _ 2 )° = x\x - Jx - 2)°
= x p {(x h - 2)(xi + 1)}° = A ' P (^ - 2)° = (x = 4) ;

for (x = 4) implies x v , and x° and « N are incompatible with
the datum (x - Jx - 2)°.

126. Let it be required to find the limits of x from
the datum (x— Jx>2).

(x-Jx>2) = (x-Jx-2y = (c i '+x"+x°)(x-Jx-2y
= x p (x-Jx-2y
= cc p {(x i -2)(x i + 1)}^ = ,^- 2) p = (.> ; >4) ;

for (v>4) implies x F , and x° and re N are incompatible with
the datum (x — Jx — 2) 1 '.

127. Let it be required to find the limits of x from
the datum (x— Jx<2).

(x- Jx<2) = (x- Jx- 2) N = (.^+^-M')<>- Jx- 2) N

= (x v + x°)(x - Jx - 2) N = of(x - Jx - 2) N + x°

= x ¥ {^ - 2)(x\$ + 1)} N + x°=x*(x i - 2f+x°

= x\x* < 2) + x° = x\x < 4) + x°

= (4>^>0) + O=0).

Here, therefore, x may have any value between 4 and
zero, including zero, but not including 4.

128. The symbol gm denotes any number or ratio

§§ 128, 129] CALCULUS OF LIMITS 115

greater than m, while Im denotes any number or ratio less
than m (see § 115). The symbols g x m } g 2 m, g 3 m, &c.,
denote a series of different numbers or ratios, each greater
than vi, and collectively-forming the class gm. Similarly,
the symbols l-{m, l 2 m, l 3 m, &c, denote a series of different
numbers or ratios, each less than m, and collectively
forming the class Im. The symbol x gm asserts that the
number or ratio x belongs to the class gm, while x 1 '"
asserts that x belongs to the class Im (see § 4). The
symbol x gm ' gn is short for x gm z gn ; the symbol xP mln is
short iorx gm x ln ; and so on (see § 9, footnote).
These symbolic conventions give us the formulae

(1) x^ m = (x>m) = (x-my.

(2) x lm = (x<m) = (x- mf.

( 3 ) x gm ■ ln = x° m x ln = (x > m)(x < n)

= (x — mY(x — iif = (n> x > m).

129. Let m and n be two different numbers or ratios.
We get the formula

( 1 ) ,,:<"" • 9* = X^V 71 + X a V n

= (x > m > n) + (x > n > m).

To prove this we have (since m and n are different
numbers)

af m.gn _ ^m.gn^jn + ^m^ for ^jn + n gm _ g

= x gm x* n m 9n + x gm x ffn n 9m

= {x 9m m 9W )x <)n + {,:»P n n am )x° m

= x gm m gn + x 9 n n gm = (x > m > n) + (x > u > m ),

for in each term the outside factor may be omitted,
because it is implied in the compound statement in the
bracket, since x>m>n implies x>n, and x>n>m im-
plies x>m. Similarly, we get and prove the formula

(2) x lm ■ ln = J m m ln + aV = (x < m < n) + (x < n < m).

This formula may be obtained from (1) by simply sub-

116 SYMBOLIC LOGIC [§§ 129-131

stituting I for g ; and the proof is obtained by the same
substitution.

130. Let m, n, r be the three different numbers or
ratios. We get the formulae

( 1 ) aT fi - gn ■ gr = ,<? m 7>iP n in Jr + ,/;? W'" + aWV*.

( 2 ) rf m ■ ln ■ lr — J m ni ln m lr + x ln n lm n lr + J r r lm r ln .

These two formulae are almost self-evident ; but they
may be formally proved in the same way as the two
formulas of § 129 ; for since m, n, r are, by hypothesis,
different numbers or ratios, we have

m gn ■ or + n gm . gr + ^m . gn _ ^
m ln.lr + n lm.lr + r lm.ln = € ^

while jf n -9n.gr = x gm.gn.gr e ^ by fas formula a = ae, and
x im.in.i r= . x im.in.ir e ^ ^ ^q same formula. When we have
multiplied o:° m ■ 9n -' jr by the alternative e v and omitted
implied factors, as in § 129, we get Formula (1). When
we have multiplied x lm - ln - lr by the alternative e 2) and
omitted implied factors, as in § 129, we get Formula (2).
The same principle evidently applies to four ratios, m, n,
r, s, and so on to any number.

131. If, in § 130, we suppose m, n, r to be inferior
limits of x, the three terms of the alternative e v namely,
m gn-ir i n gm -° r , r gm - an , respectively assert that <m is the
nearest inferior limit, that n is the nearest inferior limit,
that r is the nearest inferior limit. And if we suppose
m, n, r to be superior limits of x, the three terms of the
alternative e 2 , namely, m ln - lr , n lmAr , r lmAn , respectively
assert that m is the nearest superior limit, that n is the
nearest superior limit, that r is the nearest superior limit.
For of any number of inferior limits of a variable x, the
nearest to x is the greatest; whereas, of any number of
superior limits, the nearest to x is the least. And since in
each case one or other of the limits m, n, r must be the
nearest, we have the certain alternative e 1 in the former
case, and the certain alternative e 2 in the latter.

§§ 131-133] CALCULUS OF LIMITS 117

It is evident that mP n may be replaced by (m—n) v
that m ln may be replaced by (m — ?i) N , that m lnAr may be
replaced by (m — n)"(m — r) N ; and so on.

CHAPTER XV

132. When we have to speak often of several limits,
x x , x 2 , x 3 , &c, of a variable x, it greatly simplifies and
shortens our reasoning to register them, one after
another, as they present themselves, in a tabic of reference.
The * symbol ® m >, n asserts that x m is a si^erior limit, and
x n an inferior limit, of x. The* symbol x m , n , rs asserts
that x m and x n are superior limits of x, while x r and x s
are inferior limits of x. Thus

aW.n means (x - x m f{x - x n J or (x m >x>x n ),
x m'.n'.r. ■ means (x - x m f(x - x n f{x - x r ) p (x - x s f,

and so on.

133. The symbol os m . (with an acute accent on the
numerical suffix m) always denotes a proposition, and is
synonymous with (x — x m y, which is synonymous with
(x<x m ). It affirms that the m th limit of x registered in
our table of reference is a superior limit. The symbol
x m (with no accent on the numerical suffix), when used as
a proposition, asserts that the m th limit of x registered in
our table of reference is an inferior limit of x. Thus
x m means (x-x m ) p .

* In my memoir on La Logique Symbulique et ses applications in the
Bibliotheque du Congres International de Philosophic, I adopted the symbol
x™ (suggested by Monsieur L. Couturat) instead of iy„, and .v m r " instead
ofx m > n t rs . The student may employ whichever he finds the more con-
venient. From long habit I find the notation of the text easier ; but the
other occupies rather less space, and has certain other advantages in
the process of finding the limits. When, however, the limits have been
found and the multiple integrals have to be evaluated, the notation of the
text is preferable, as the other might occasionally lead to ambiguity (see
§§ 151, 150).

118 SYMBOLIC LOGIC [§§ 134, 135

134. The employment of the symbol x m sometimes to
denote the proposition (x — x, m ) v , and sometimes to denote
the simple number or ratio x m , never leads to any
ambiguity ; for the context always makes the meaning
perfectly evident. For example, when we write

X —I — \x — x z ) — x 3 ,

it is clear that the x s inside the bracket denotes the
fraction -, which is supposed to be marked in the table

of reference as the third limit of x; whereas the x 3 ,
outside the bracket, is affirmed to be equivalent to the
statement (x — x 3 Y, and is therefore a statement also.
Similarly, when we write

A = ( 2,,; 2 + 8 4 > 2 9x) = (x - 1 0|) p + (x - 4) N
= {x — x^f + (x — x 2 Y = x l + Xg,

we assert that the statement A is equivalent to the alter-
native statement x l + x^, of which the first term x 1 asserts
(as a statement) that the limit x 1 (denoting 10|) is an
inferior limit of x, and the second term »_, asserts that
the limit x 2 (denoting 4) is a superior limit of x. Thus,
the alternative statement x -\-x 2 > asserts that "either x l
is an inferior limit of x, or else x 2 is a superior limit
x.

135. The operations of this calculus of limits are
mainly founded on the following three formula? (see §§
129-131):

( 1 / x m . n = ''m\ x m ~ x n) "" x n\ x n ~ x m) •

(Z) x m , n > = x m \x m x n ) + x n \x n — x m ) .

\° ) x m'.n ''- m' .n\' vi '' n) '

In the first of the above formulae, the symbol x m n means
/„,./„, and asserts that x m and x n are both inferior limits

§§ 135, 136] CALCULUS OF LIMITS 119

of x. The statement (x m - x n f asserts that Xm is greater
than x n and therefore a nearer inferior limit of x ; while
the statement (x n -x m Y asserts, on the contrary, that
x n and not x m is the nearer inferior limit (see §§ 129,
131). In the second formula, the symbol x m > n . asserts
that x m and x n are both superior limits of x. The state-
ment (x m - xj" asserts that x m is less than x n and there-
fore a nearer superior limit of x ; while the statement
(x n — x m ) K asserts, on the contrary, that x n and not x m is
the nearer superior limit. The third formula is equiva-
lent to

■' m .n • \ x m x n) >

and asserts that if x m is a superior limit, and x n an inferior
limit, of x, then x m must be greater than x w

13G. When we have three inferior limits, Formula (1)
of § 135 becomes

% m .n.r = x m « + Xnfi + X r y,

in which a asserts that x m is the nearest of the three
inferior limits, ft asserts that x n is the nearest, and y
asserts that x r is the nearest. In other words,

a — \ x m ~ x n) \ x m ~ X r)

p = {x n x m ) (x n — x r )
y=(x r -x m f(x r -x n y.

When we have three superior limits, Formula (2) of § 135
becomes

x m'. W. ? = x m' a + x n'ft + x r'7>

in which, this time, a asserts that x m is the nearest of the
three superior limits, ft asserts that x n is the nearest, and
y asserts that x r is the nearest. In other words,

a = (x m x n ) \X m x r )

ft=( x n- X mf( x n- x rY

y = (x r — x m f(x r — x n ) s .

Evidently the same principle may be extended to any
number of inferior or superior limits.

120 SYMBOLIC LOGIC [§§ 137, 138

137. There are certain limits which present themselves
so often that (to save the trouble of consulting the Table
of Limits) it is convenient to represent them by special
symbols. These are positive infinity, negative infinity, and
zero (or rather an infinitesimal). Thus, when we have
any variable x, in addition to the limits x v x 2 , x 3 , &c,
registered in the table, we may have always understood
the superior limit x a , which will denote positive infinity,
the limit x Q , which will denote zero (or rather, in strict
logic, a positive or negative infinitesimal), and the always
understood inferior limit x p , which will denote negative
infinity (see § 113). Similarly with regard to any other
variable y, we may have the three understood limits y a ,
y , y p , in addition to the registered limits y v y 2 , y 3 , &c.
Thus, when we are speaking of the limits of x and y, we
have x a — y a = a ; x = y = (or dx or dy) ; x fi = y p = - a.
On the other hand, the statement x a ,_ m asserts that x lies
between positive infinity x a , and the limit x m registered in
the table of reference; whereas x m , p asserts that x lies
between the limit x m and the negative infinity x p . Simi-
larly, x m , tQ asserts that x lies between the superior limit
x m and the inferior limit ; while ;% n asserts that x lies
between the superior limit and the inferior limit x n .
Thus, the statement « m ,. implies that x is positive, and
x ff n implies that x is negative. Also, the statement x Q , is
synonymous with the statement X s ; and the statement x is
synonymous with the statement x p . As shown in § 134,
the employment of the symbol x Q sometimes to denote a
limit, and sometimes to denote a statement, need not lead
to any ambiguity.

138. Just as in finding the limits of statements in pure
logic (see §§ 33-40) we may supply the superior limit n
when no other superior limit is given, and the inferior
limit e when no other inferior limit is given, so in find-
ing the limits of variable ratios in mathematics, we may
supply the positive infinity a (represented by x a or y a or
z , &c, according to the variable in question) when no

§§ 138, 139] CALCULUS OF LIMITS 121

other superior limit is given, and the negative infinity (3
(represented by .^ or y p or z p , &c.) when no other inferior
limit is given. Thus, when x m denotes a statement,
namely, the statement (x — x^f, it may be written x a , m ;
and, in like manner, for the statement x n >, which denotes
(x — x n y, we may write x n , tP (see § 137).

139. Though the formulae of § 135 may generally be
dispensed with in easy problems with only one or two
variables, we will nevertheless apply them first to such
problems, in order to make their meaning and object
clearer when we come to apply them afterwards to more
complicated problems which cannot dispense with their aid.

Given that 7a?— 53 is positive, and 67 — 9a; negative;
required the limits of x.

Let A denote the first datum, and B the second. We
get

TABLE

A = (7x-5Sy = (x-~X=x 1 =x a ,, 1

B = (67-9*) N = (9,:-G7) p = (^-y )

Hence, we get

AB = av. x x a ,, 2 = x a ._ j

By Formula (1) of § 135, we get
a5j _ 2 = Xjlfa — x 2 ) p + x 2 (x 2 — x^f

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