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J
QUEEN'S COLLEGE
MATHEMATICAL COURSE.
JUNIOR ALGEBRA.
BY
IT. y
. lDTJI>TJrS, M". JL.,
PROFESSOR OF MATHEMATICS.
Published by John Henderson & Co.
1882.
LP
tes. 9-^011 c. Ji
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PREFACE.
The present work comprises the Junior Algebra of the
course for the degree of B.A. in Queen's University. •
Having been written for the use of students in class it is
unlike the majority of works upon algebra offered to begin-
ners. It contains no very elementary portions, as the stu-
dents are supposed to be able to matriculate into the Univer-
sity before taking up the work. It contains no lists of
exercises, since it is expected that the Teacher will select or
frame such exercises as may best suit his. immediate purpose.
The work deals mostly with principles, and the examples,
which are fully worked, are introduced for the purpose of
exemplifying these principles. For the fuller elucidation of
principles a Teacher is supposed to be available.
In the establishing of theorems and formulae the method
followed is inductive rather than analytical, as the former
method is believed to be fully as satisfactory as the other,
and much more within the grasp of beginners.
The work consists of a small edition, and its production is
somewhat of an experiment. If it is found to serve a useful
purpose it may be followed by a similar work upon the Junior
Geometry of the B.A. course. n. f. d.
ERR A TA .
<<
for 3
x — i
The following corrections are necessary
Page 12, line 5,
12, Ex. 19, line 5
16, line 12
47, Art. 52, line 4,
60, Ex. 103, line 6,
63, Ex. 106', line 3,
72, Ex. 115, line i,
92, line 3,
" 118, line 2,
" 2^ + 3r
" =1
'* «2r
« 2«
ea<
d 2
«
x'z
((
x + i
<i
a
K
2fi-sa
((
4b -a
«'
= 11
((
»P2
«
i-n*
MULTIPLICATION.
I. Wherever practicable multiplication should be performed
in one line ; by reducing the quantities tn be operated upon to
the form of binomial factors the multiplication can be readily
effected by referring it to the results obtained by the multi-
plication of well known simple forms.
Fundamental Forms.
i. {a±bY-a'^-¥b^±2ah.
ii. (a + 6)(a-6)=aa-6a.
iii. {x-\-a){x-\-h){x+c)=x^+x^ .a + b+c
-{■x.ab + bc + ca + abc.
iv. {mx-\-a){nx+b) = mnx^ -\-mb + na .x+ab.
Ex. I. (a+6 + c)2=a2 + (6-i-c)'»+2fl(6 + c)
= a^-\-b^ +c*4-2a6+26c + 2ca.
2. (;r+I±ll3)(;r+iZJ^) = (;r+i + il/3)
2 2
(x^ +ax-b) (x^ -~ax + b)=x*+{ax-b+-ax-\- b)x^
+ {ax-b){-ax + b)==x^-a'^x^-^b'^.
ia + b-c+d)ia~b+c-d) = {a + b-c~d){a~b+c~d)
= {a + b)ia-b) + ia + b){c-d)—(a-b)ic-d)~ic-d)^
= a^ -b^ -c^ ~d^ +2bc -2bd + 2cd.
3ip-^q){q + r)ir-\-p) = S{p + q){qr+qp + r^+rp)
= 3ip^q+Pq^+P^i'+pr^+q'^r+gr'i + 2pgr).
- 6 —
and (/) \ q l r)=* -:{/> \ q \ r)'^ = {p-\ q)^ \ ^i/i i q)'^,- '
.-. Mp\q){q\r){rip)'-^{/){q-\-r)^ p^ -q^-r^. v
2. Symmetry. — When an expression involves two or more
letters in exactly the same way it is said to be symmetrical
with respect to these letters. In writing such expressions we
usually ignore the alphabetical order of these letters and give
attention to their cyclic order. Thus,
a'{h~c) + h^{c~a)+c\a-b)
is symmetrical with respect to a, b and c, and in every term the
letters follow the same order, a b c a b.
a\b^c~d) + b\c + d~a) 4 c»((/-f « - b)-\-d'{a -I 6 - c)
is symmetrical in a, 6, c and d ; but
{a^-b+c■^-d? + {a^b-c-d)'^^{a■--b^c-'df
^{a-b-c + d?
is symmetrical in 6, c, d, but not in a, since a is positive in
every term, while the others are each positive in two terms
and negative in two.
The study of symmetrical expressions is of very great im-
portance for many reasons. The principal one at present is that
having the expansion of one term of such an expression the
expansions of the remaining terms may be written down at
sight.
Ex. 6. To find the value of
.s(s - 2a)(.s — 2b) + s(s — 2/; )(.s- -■ 2t) + .S(S - 2C)[S ~ 20)
— (v — 2a)(s — 2b){s — 2c) when s=^a-\ b f c.
This is evidently symmetrical in a, b and c.
H{s — 2a){s — 2b) = !i{s'^- 2S'.a-f-6 + 4a6),
.*. s(s-26)(s-2c)=:s(s* — 2S.6+C + 46C),
s(s - 2c)(s — 20) =s{s^~ 2S.C + « + 4co) ;
erv term the
— 7 —
and their sum is, si^.ab-^bc-^ca-s*).
Hut (s- - 2rt)(.s - 26)(.s- 2t) =s* - 2s'(a + 6-hc)
+ 4s(rt64 /jc-l-cfl)— 8rt/)c = s(4.a6 t bc+ca — s^)~8abc.
.'. the whole expression becomes Habc. '
7. To sliow that 8(rt + fc-Hc)»-(a I 6)» -(6 + c)8 - u+a)*
= 3(2af /; f c)(26-: f+4)(2r-l rt+6).
In cases of this kind we may either bring one of these ex-
pressions to the form of the other, or we may bring them both
to the same third expression. The latter method is usually the
simpler one, but the former is a better exercise of ingenuity.
8(a + b + c)^ = i2a + 2b+2c)^ = {a^b-{-V+c+cTa)^.
.". Denoting a + b by />, 6 + f by q, and c-f a by r,
we have (Art. i, Ex. 5)
ip^q + r)^-p^~q^
r^ = ?i{p + q){q+r){r+p)
= 3(rt + 26 + c){b + 2c + a){c -{-2a + b)
= i(2a-\-b + c){2b + c + a){2c+a-\-b).
3. Multiplication by detached Coefficients. — In multiplying to-
gether polynomials with one leading letter it is often advan-
tageous to work upon the coefficients only, and to supply the
leading letter after the completion of the work.
Ex. 8. To multiply x^ + ^x"^ -2^+1 by 2x^ -x+2
I
+ 3
- 2
t I
2
-I
+ 2
2
+ 6
-4
+ 2
— I
-3
+ 2
— I
+ 2
+ 6
-4 +2
8.
Product : 2X^ + $x* - e^x^ -I- lox^ -^x-\-2
To multiply ^x^ —x-\- 2 by x^ + 2x'^ - 3.
Here we must supply zeros for the coefficients of the
missing powers of x,
3+0-1+2
1+2+0-3
&c., &c.
10.
— 8 —
To multiply ax^ +bx^ +cx-}-d by px"^ -\-qx+r.
a + b + c + d
ap +bp +cp -{-dp
+ aq +b(j +cq +dq
-\-ar -\-br -\-cr -Vdr
x-Ydr. '
apx^ + bplx* +cpx^+ dp
aq\
bq
ar
x'^+dq
cq
br
cr
By observing the form which the product here assumes, and
the manner in which its terms are made up, we may write it
down at once in any similar case.
Thus, 2ax^ + bx"^ — cx-\- 1
f jr^ ~bx + 2
2acx^ — 2ab x* I- 4a x^ + 2bx^ ~ 2c\x+ 2
be
II. To multiply 2x+^y +z — i by x+y —2z + i.
We readily see in this case that the product must contain
the combinations of letters, x^, y^, z^, xy, yz, zx, x, y, z, and a
numerical term. Hence we may arrange as follows :
X* + 4a
-62
x^ + 2b
+ be
X^~2C.
- b
-C2
+ c
2-3 + 1-1 x^ y'
X y z 2 —3
1+1-2+1
z' xy yz zx X y z n
-2-1+7-3 +1-4 + 3-1
product is 2x^~ ^\y'^ - 2z^ — xy-\- yyz - -^zx + ;r - 43; + 32 - 1 .
4. Multiplication of Series. — It often becomes necessary to
square a series, or to multiply one series by another. In
nearly all such cases the terms of the series are arranged
according to the ascending powers of the leading letter.
Multiplication of series finds its application in the algebraical
development of functions, &c.
Ex. 12. To multiply a + bx-^-cx"^ -{■dx^-\-. . : .
by fl' + 6V+c';r2+rf';r8 +
Product aa + ab' .
ba
x-\-ac'
x^-^-ad'
bb'
be'
ca
cU
da'
x^ + .
— 9 —
By observing the mode of formation of the various co- j
efficients we are enabled to perform such multiphcations with '
great facility. For example the coefficient of x^ is formed by
taking the terms,
a h c d
J' ' L' '
a c a
multiplying each pair together and taking the sum of the
products.
Ex. 13. To square the series i •¥ax-\-bx^ -\-cx^+ ....
i+ax+bx^+cx^-{- ....
Square = i + 2ax -j- 26
a'
X^ +2C
2ab
x^^
14. To show that the square of the series j
x^ x^
I -1-4:+ — +^+ . . • •
2 6
is formed by writing 2X in the place of x.
i+x+^x^ +^x^+ ....
square = i + 2;r + 2x'^ +%x^ +
= i+(2X)+^-+ ^
2 6
(2;tr)a
+
Ex. To multiply i +x+ 2x'^ -+-4x1 + . . .hyi-x-x^-x'^-. . .
Ex. Show to three terms that if the series i -^x^ +^t^^ • • •
and x-^x^ ... be squared and added the sum is
unity.
11
lO
DIVISION.
5. Division, being the reverse of multiplication, may like
that process be carried out upon the coefficients, and when
only one letter is involved in the expressions under considera-
tion, the process becomes in this way very much simplified.
If we multiply Ax^ -j-Bx+C by ax^ +bx-\-c, we obtain for
coefficients,
aA -\-aB
hA
i-aC
bB
cA
+ bC
cB
+ fC
By observing how the terms in this product are formed we
may reverse the process and thus perform division. Thus, if
a + b+c be the coefficients of the divisor, we see that a A
divided by a gives A , the first term of the quotient : then bA
subtracted from the second term and the remainder divided
by a gives By the second term of the quotient ; and lastly,
cA +bB subtracted from the third term and the remainder
divided by a gives C, the third term of the quotient.
In this operation the only quantity by which we really divide
is a, and hence if this be unity its presence may be quite
ignored. Again, since algebraical subtraction is equivalent
to addition with a changed sign, we make our subtractions
additions by changing the signs of every term, except the first,
in the divisor.
Ex. 15. To divide ^x^ - ^x^ — ^x^-^yx - 2 by 2x^ - ^x-{- 1.
We may write the divisor in any position which is
convenient.
as, 2
+ 3
-I
4-4-5+7-2
+6+3-6+2
- 2 - 1
' or.
2 + 3-1
4-4-5+7-2
+6+3 -6+2
2+1 -200
-2-1
2 + I'
— II
and the quotient is 2x'^
The second position oft:
most convenient.
V-2. }
divisor is for some reasons the
Ex. i6. To Divide Jtr® — 3:r^ -f 6jr- 4 by 4r* — 24r + 1.
Here the coefficient of x^ of the divisor being i we may
ignore it altogether.
2-1
i + o +0 i-o — 3 + + 0+0 + 6-4
24684 0—4-8+4
—1—2—3-4—2 0+2
1+2+3+4+2 0-2-4 o O
.*. quotient = x"^ ^ 2X^ + yir^ + 4X* + 2x^ - 2X - 4.
6. If the case is one of inexact division, we must stop the
process at a certain point if we wish to obtain the correct re-
mainder. This point is of course reached when the last
obtained term of the quotient does not contain ^•
In order to determine this point we draw a vertical line to
the left of the divisor as usually written, i. e. between the first
and second terms of the divisor as completely written ; all the
terms of the quotient proper are to the left of this line, and no
term of the quotient line to the right of the vertical is to be
used in forming a partial product in getting the remainder.
Ex. 17. Divide ;r' -x^ -\- ^x^ + io;r'' -5:1^-1 by x* - 2X^ +x^ -2.
2- 1+0+2
i+o-i+0]+5+ 10-5-1
244: 4-244
-I -2i -2+4
I 2
1 + 2 + 2 + 2| +9+ 12 -1+3
.*. quotient = x^-j- 2X^ + 2X+ 2,
and remainder = gx^ -\-i2x'^ —x-\-^.
7. In cases of exact division the process may be carried out
in a somewhat similar manner when several letters are in-
valued.
Ex. 18. To divide />" +pq + 2pr - 2^' + yqr - ^r' by p - q -j- 3^.
— 12
Since the dividend is of two dimensions and the divisor of
one, the quotient must be of one. We may arrange as fol-
lows :
P"" q' r" pq pr qr
p
- 1
1-2-3 I 3 7
9 '
I
-I -2 -7
r
-3
P q
quotient = p-\-2q-r.
Here the coefficient of ^ in the quotient must be i so that
when multiplied by that of^ in the divisor it may cancel that
of/>2 in the dividend. Similarly we find that of q and of r.
Then for Pq we have -1.2 + 1.1= - i which cancels that of
pq in the dividend, &c.
Ex. 19. Divide 2!^ - ^x^y - ^x^z - ^xy^ - j.r-j* + 1 2xyz h 2y^ - zyh
— 2,y^ + 2^ by x->ry - 2Z.
The quotient may obviously contain all possible terms of
two dimensions, and can contain no others.
X^ y3
x'^y x^2 xyz y^z xy"^ xz'^ yz''
X
y
— I
— I
2
2 -3 -3
-2 4
12 -3 -3
-I 4-2
- I -I 5
— 10
3 -3
1 I
2 2
1-5 I I
xz yz . . . . possible terms of
two dimensions.
j/2 ^2 xy
quotient
2x'^ + 2y'^
z^ - ^xy-^xz-\-yz.
11-
1; ■
8. In a case of inexact division, as in example 17, if we
neglect the vertical line and its indications the quotient will
extend to an indefinite number of terms, which will follow a
certain law of formation, and it will thus become an infinite
series. This is very similar to cases of inexact division in
arithmetic when the quotient is run out into a circulating
decimal.
g. Expansion by Division. — Let it be required to divide i by
i-^x, running the quotient into a series ; we obtain,
terms of
■Z'i ysl
3 -3
I I
2 2
• •
terms of
13
I
i+x
= 1 -x + x^-x^ + x*- + . . . .
1
1
similarly =
1- X
= i+x + x^ + xr' + jii^++ ....
By means of these forms we may effect the expansion of any
expression which can be expanded by mere division.
Ex. 20. To expand
-; into an infinite series.
b + x
a _ a
b+x b '
^ =«fi- ^+^-^V-..
..V
•
X bV b^ b' b'^
b
7
Ex. 21. To expand
20^- 1
"'-i-d-
I -2a^
+. .
. .).
20^-1
• •/>
= 1+0
[2 +2(1* +4a^ + . . . .
Ex. 22. To expand
(i-.-y^ I
I +z
{i-zf
2
-^2 -
r:-(l +Z)(l + 2Z — Z^ -^ 2Z -
•f-. . .)
— 2Z + Z^
= 1+3--
^5z^ + 7^ +
Ex. 23. To expand
.x"^ - 2;i; - I
I+X + X^ '
This becomes (,
X^-2X~\){l-X + X^ + X + }? -. . .
. .),
= -I-
X+IX'^-2X'^-X*+IX^ ....
yjitr^^
€^y
/ f fiUr^'
; V,
— 14 —
SUBSTITUTION.
I s
.:,1|
li :vi
10. Substitution is the writing of one quantity for another
in an expression. Thus,
Aa^ + Ba^ + Ca-¥ D is obtained from
Ax^ + Bx^ +Cx +D, by substituting a for x.
Let A' stand for a general expression of the form
Ax^' + Bx^'-^i- Sx+T,
where n is an integer greater than unity.
If we divide this by ;r - a we will obtain another expression
one dimension lower, which we may denote by A'l and, in the
case of inexact division, a remainder R which does not con-
tain X.
Hence we may write,
Ax'' + Bx''-^ + . . . . Sx+r=:X, (x-a)+R.
Then,
i. If we substitute a for x throughout, we get
A rt" +Ba"-^ + . . . . Sa + T = R.
Hence we conclude that if we divide an expression contain-
ing only positive integral powers of x by x~a the remainder
will be the original expression with a substituted for x.
ii. If x — a divides A' exactly, R is nothing ; and substituting
a for X we have,
/4flM-B«"-' + Sa-{T = o.
Hence if x -a is An exact divisor of an expression contain-
ing only positive integral powers of x, the substitution of a for
X in the expression causes it to become zero ; and conversely,
if this substitution renders the expression zero it is* exactly
divisible by x - a.
— 15 —
II. Applications of i. and ii. I
Ex. 24. To find the remainder when ;r'-3;ir*4-2 is divided
by ;r-i.
Remainder = i' -3.1^ + 2=0. / ^
.*. ;t' - 3;t* + 2 is exactly divisible by x-i.
In a large number of cases the substitution is most readily
effected by means of the division itself.
E\. 25. To find the remainder when p^ - 2p* +3/> - 10 is di-
vided by /) - 4.
4
I -20 o 3 - 10
___ 4___8 35 128^ 524
2 8 32 131 514 = Remainder.
Hence to substitute a for ;«; in A' divide the expression A' by
X - a and take the remainder.
Ex. 26. To find the value of n'* - yi^ +2n-^ 10 when -3 is
substituted for n. Divide by n + 3.
-3
10
-266
I 00-3 2
-3_ 9 _-27_„90
-39 - 30 92 -266 .". Result = -266.
Ex. 27. Is x—2 a divisor of -t* — ^x^ — ^x"^ +2;r+ 20 ?
If we substitute 2 for x we obtain
2* - 3.2^ - 4.2'' + 2.2 + 20= o,
.•. X--2 is a divisor.
Ex. 28. Is fl — 6 a divisor of ab(b -a) + bc(c -b)+ ca(a - c) ?
Substitute b for a, and we obtain,
fe" .0 + 6c(c - 6) + c6(6 - c) = o,
.'. rt -6 is a divisor.
12. In transforming equations it frequently becomes neces-
sary to substitute a binomial expression for x in the general
expression Ax'^ + Bx^'^ -\-. . . . Sx-^T. This may obviously be
done by writing the binomial in the place of x, and then ex-
panding, as follows :
ii;
- i6 -
«
Ex. 29. To substitute jy - i for x in x^ - 3^' -|- 2,r -t- 1.
We have, {y~i)^ - jO' - i)" 1 2(y- i)-i-i,
=^'J - 6)'^ -f I ij* - 5, by expansion.
The following will enable us to perform this important sub-
stitution more readily : ^
Since x—y- i, :.y = xAr\, and the given expression is to
be put under the form
(^-f-i)3-f/v'2(4r+i)2 4-/e,(;r4i)+/e;
Where we have to determime the remainders A*, /?,, A'j.
If we divide the original expression by x-\-\ the remainder
is A'. If we now set aside this remainder and divide what is
left by ;r- I the remainder is A'l. Proceeding in this way we
obtain all the remainders. The whole operation is as follows :
-3
-I
-4
— I
-5
- 1
-6
4
2
4
6
A
II
-I
+ 1
-6
.*. (,r-|-i)^ — 6(;lr^ i)^ -|- ii(;tr-f i) - 5 is the expression :
or, >'3~6)'2.| 11;;- 5.
Ex. 30. Express 3/)^ ~p^ -V 4p'^ -\- 5/) - cS in terms oi p - 2.
6
- 1
12
4
22
5 -«L2
52 114
6
6
II
24
26
70
57
192
loC
12
6
35
36
96
142
249
18
6
71
48
238
1
24
6
119
30
— 17 —
I 249(^-2) + 106, is tho ref|uirefl expression.
Ex. 30'. In 111^ --^m'^n \ imn"^ jm^ substitute m - n for w.
.J
2
-.J
I
- 2
0,
-2
-J
I
- I
- 1
I
I
.'. (m - «)•'-«*(;;/ -«) - 3«'* is the expression.
13. The following form of substitution is of importance in
many operations.
Ex. 31. What does x* - ^j^ +2^-1 become wlien x^ 1 ^- i - o ?
This may be solver! by division directly as follows :
- I
I
I
4
2
- I
- 1
I
-2
3
- I
4
- 4
I
- I
2
-3
I
I
- I
2
_____
I
"4
7
-5
.•. 7^-5 is the result.
Or it may be done thus :
•.* x'^ -\rX -1 = 0, .*. x'-
X* = \ \ x'^ - 2X~ 2
S^'
X'* -X
rl^
2X - I
and x"^ --= x*.x^ - (2 - ^x){2X - i ) = ij.r 8.
.-. ;r' - ^x^ 1 2x~ I -yx - 5.
14. We will now extend Art. 1 1 to the case where the num-
ber to be substituted is partly a whole number and partly a
decimal.
Ex. 31'. Find the value of x'^ - ^t^^ -\ 2X + i, when ;r:= 2.85.
In this case we work through for each figure separately, as
follows :
— i8 —
I I:'.
I.
11.
in.
case
-3
2
4-2
- 2
+ 1 1 2^85
the
an e
-I
2
I
a
2..
I . . .
Ex.
2
3 .
.8
3.04
4.032
3.8
8
504
3.68
5032 . . .
4.6
8
5-4-
05
545
8.72 . .
.2725
8.9925
•449625
5.481625, result.
Kx
The above work is fully expanded in order to show the
various steps. We first work through for 2, as in fornner ex-
amples. Then we work through for the 8, remembering that
as It is in the tenths place the figures in column I. will be
moved one place to the right, two places in column II., and
three places in column III. And in like manner for 5.
The work may be very much condensed, as follows :
Ex. 32. Find the value of y*
- 4_y'* - I wher
i;'=2.i3.
I - 4
•I 2.13
2
4 8
6 20
81 2081
82 2163
83 2246
843 227129
16
1 808 1
20244
20925387
■I
.8081
1.43586161, result.
15. If we have an expression j*^ -rt and we put for jf any
particular value we have seen (Art. 14) how to find the value
of the expression. If that value is zero, then y^-a = o, and
therefore y = ^^a. Hence, if we can discover a quantity which
when put for^* makes the expression y - rt zero, that quantity
is a cube root of a. And similar reasoning would apply in the
1.85
— 19 —
case of any other root. The process of the last article supplies
the means of approximating to this value and thus becomes
an elegant means for the arithmetical extraction of roots.
Ex., 33. To find the cube root of 2299968.
result.
show the
)rmer ex-
iling that
'. will be
II., and
5-
I, result.
■ Jf any
e value
= o, and
y which
juantity
V in the
- 2299968
I
I
+ 1
2
33
36
392
3
399
507
51484
-1299
+ 1197
- 102968
+ 102968
extract the cube root of j.
-3 .1^
X
I
I
2
34
38
424
428
3
436
588
60496
62208
-2000
1744
- 0256000
0241984
..14016
132
44
We know that i is the first figure of the root ; we, therefore,
work through for one. We then find the next figure of the
root by employing 4 in the second column as a trial divisor,
and 20 in the third column as a dividend ; but as the 4 will be
increased by the subsequent operation we make a proper
allowance in the quotient figure. The principal points to be
attended to are that the number carried to the third column
must always be less than the number above it from which it is
to be subtracted, and that the remainder after subtraction
must not be greater than the last completed number in the
second column.*
After obtaining 3 or 4 figures the number of figures may be
doubled by employing the last completed number in the second
column as a divisor and the last remainder in the third as a
dividend. Thus dividing 140 16 by 6221 we obtain 225 ; hence
the cube root of 3 is 1.44225 true to the last figure. In a pre-
cisely similar manner we may extract fifth and seventh
roots, &c.
* In some special questions it may be greater by a small amount.
20 —
OF FACTORS AND FACTORING.
x6. In the expression
a(6 -c)(<i 4- 6 -f-c)(rt' F 2rt - i)(.i/' + 6c)
a is a monomial factor ; 6 -c is a binomial factor, and rt -t- /> -|-c
is a trinomial factor. These are linear factors, containing
terms of only one dimension, while a'+2a - i and ab + be are
quadratic factors, inasmuch as they contain terms of two
dimensions.
An expression may have real quadratic factors when it has
no real linear factors, e. g.
** + 3^' + 2 = (X^ + l){X* + 2)
in which neither of the quadratic factors has any real linear
factor.
17. Theoretically any expression of the form
Ax'' + Bx''-^ + Sx-^-T
may be written as the product of n linear factors containing
X, as
A{x-a){x-fi){x-r) .... U-c),
in which the values of a, /9, &c., depend upon those of
A, B, C, &c. ; but practically the discovery of the values of
«, ^, y, &c., cannot always be effected by any means at our
command, so that the actual process of factoring can be carried
out only in special cases. These are, however, frequently of
great importance. Only the simpler processes of factoring,
will be dealt with here.
18. Factoring by reference to known formulas.
The formulae more generally useful are :
i. a'-b''={a]-b)(a-b),
ii. a» + 62±2«6 = (a±6)^
— 21 —
iii. x'*-ir(a-\b)x-ial) = {aix)(l)-^x).
iv. rt' t 6' + c' -f 2(ab -f ic -t c «) = (« + 6 -+- c )',
V. a^ + b^ + c^-2ahc = {a-ib-\c){a.a-h\h^b'-c-\-c.~a).
vi. (rt i b -}- c)' - rt' - ^^ - 1» = 3(rt i '')(/; i- c){c f rt).
ICx. .55. X^ - 2X - ^= X^ - 2X + I - 4 = X - I - 2^ -^ {X - ^i)(X f- I ).
or .t^-24r-3 = 4r'^ (I -.i)4r-| I X - 3 = (;r| i)(:r- 3).
Kx. 36. m'' + 4;« -6=m' + 4m + 4- 10 = (;;;-[ 2)'- | lo'',
= {m + 2 1- |/ io)(;;j + 2 - | 10).
E\. 37. rt'^ 4- 2ab -i b'*~a -b-6= {a + bf - (a \ b) - 6
= (fl + f>)' - {a + /^) -I i - V = (" + -"^ - .JK<J I b -[ 2).
K\ . ^f. (!=' + 2ft» - 3ab^ - fl» + b^+b^- sabb
= {a + b \-b)(a.a - b 1 /) /> - b b.a h)
=^[a-\-2b)(a--b?.
\i\. 38. 6rt'6 + 3rt'^-f-i2rt/;^ I 12^6 + 3^+12/^^+0/^
= 3)2fl^(rt i 2/>)+a(rt + 2^)-| 2^(<J + 2/^) + </ + 2/^l
= 3|(a + 2/;)(2«<^ + a I 2<5'+ i)[
= 3(rt + 2/5)(2^ 1 i)(a f I).
19. If the quantity /)7r.s =0, then one of the factors must be
zero, and all may be zero. Conversely if one of the factors be
zero, then the product is zero, provided that none of the other
factors be infinite.
This principle furnishes a ready means of finding,' factors
when they are rational and not too complex.
Ex. 39. To factor a{b f be - c) + b{c +ca - a) -{ c{a 1 ub - b).
This is symmetrical in a, b, c, therefore if « be a factor /;
and c will also be factors.
To know if n be a factor put a = o. Then it reduces to
be -bc = o .'. a is a factor, and the literal factors are abc.
Since the highest term in the given expression is of three
dimensions, there can be but three literal factors ; but there
may be a numerical factor. Denote it by n, then,
O '">
iil
i
.: i
m
a{b-vbc -c) -vb{(:Arca-a)-\-c{a-^ah ~b) - n.abj *
must be true quite independently of any values which we may
give to a, b and c, that is it must be identically true. Make
then, a = I, ^ = 2, c=: 3, and we obtain i8=6», .". « = 3.
Hence, ^
a{b + bc -c) -\- b{c -i-ca-a) + c[a + ab ~b)= ;^abc.
y Ex.40. To (sictor ab{a—b)+bc{b — c)-^ca(c -a).
Putting a = 0, we find no monomial factors. Putting a-b
= 0, or b = a we find a—b, and hence from symmetry b — c and
c- a to be factors. .'. the expression is equivalent to ii.{a -b)
{b-c){c-a), and we readily find n= - i.
.". ab{a — b)+bc{b—c}-^ca{c- a) = -{a-b){b -c){c-a).
Y Ex. 4 1 . To factor 2ac{2a - c) \ 2cb{2c - ^) + 2ba(2b -a)- jabc.
We readily find this to be equivalent to
- {2a - c){2c - b){2b - a) .
f" Ex. 42. To factor ab{ll'-a}) -\- bcic" - b"") +ca{a'-c'').
This is symmetrical in a, b, c, and is of four dimensions ;
hence there are four literal factors.
We readily find that a—b, b — c, c — a are factors. And since
the expression is symmetrical in a, b, c, and can contain only
one more factor, it also must be symmetrical in a, b, c. There-
fore it can only be a + ^ + c. And the expression is equal to
{a - b){b—c){c — a){a + b + c).
V Ex. 43. ToidiCtov ab{c—d)-\-bc{d — a) +cd{a — b)-\-dii{b -c).
Putting a — 6 = o, or 6 = a,
a'^{c-d)+ac{d—a)+da{a—c)~o
.'. «-6isa factor, and from symmetry 6 — c,c-^, and
<i— a are factors. But being of only three dimensions it can-
not have four literal factors. ♦ Therefore, it can have none or
it must be identically zero.
20. Since (x -{-a){x-{- b){x + c) = x^ -i- x^{a-\- b +c) + x{ab + bc
+ca)-¥abc,
./.
mensions ;
— 23 —
we see that the independent term abc is the product of the •
three quantities a, b, c, which with x make up the linear factors
x + a, x-^rb, and x-\-c. A hke relation will be found to exist for
any number of factors. Hence in finding a rational linear
factor of a rational integral expression involving x in consecu-
tive powers it is necessary to try only the factors of the inde-
pendent term.
Ex. 44. To find linear factors of x^ - ^x^ - 3,^2 -|- 7;r-f-6.
The factors of 6 are ±1, ±2, ±3, ±6.
Substitute i for :r ; value =8 .'. x—\ is not a factor.
" — ifor;t:; " =0 .*. ;r + i is a factor.
'* zioxx', " =0 .*. ;ir- 2 is a factor.
And dividing by {x^\){x-2) we reduce the expression to
x"^ -2x- 3, whose factors are {x ^\){x- 3).
.-. The whole expression is equivalent to
(;»;+i)2(;ir-2)(;»r-3).
Ex. 45. To factor a* - ^a^ - 7a - 6.
Substituting the various factors of 6 for a we find two linear
factors, (a ■\ 2){a - 3) ; and dividing by these we obtain,
the third factor being a quadratic factor which cannot be
further reduced.
'
\
\
\
24
V
HIGHEST COMMON MEASURE.
! 1
I 1
I !
(^
21. If from the expression rtc^/ and o^i?^/ we take out the
factors common to both, viz., a and e, the product of these is
common to both, and is the highest factor which they have in
common. Hence it is called the highest common factor or
hif^hest common measure of the quantities, and is usually denoted
by H.C.F. or H.C.M.
Hence to find the H.C.M. of two quantities resolve them
into factors and take the product of all the factors common
to both.
Ex. 46. H. C. U. o{ a^~ad'^+ a"' I? - b '^ und a^ + ^aV> -l^^cJf^ + P.
a^ -ad^ + a'^6 -- b^ = {a -'r b){a -\ b){a - b),
a^ + ^a'U + :iab'^ + b^ - in -r b)(a ^ b)[a rb),
.-. H.C.M. = [a+b]'^.
22. If two expressions na, nb have a common factor n, their
sum, their difference and the sum and the difference of any
multiples of the expressions will have the samec(jmm()n factor.
For, na ± nb = n{a ± b) ;
and na.p±nb.q = n{ap ±bq).
This lies at the basis of the common method of finding the
H.C.M. of given expressions.
Ex. 47. To find the H.C.M. of 6.y3
2.v:^ + 3-^'^ -ii;r- 6.
Taking coefficients only,
a ... 6 - 7- 9-. 2
3/9. . . 6-j- 9-33-18
3/9-a . . . 16-24-16
Divide by 8, 2-3-2
.-. 2.^2 -3;r- 2 = H.C.M.
yx"^ - ox - 2 and
^ . . . 2H-3-II— 6
3a ... 18 -21 - 27 -6
3« -/9 . . . 16 — 24-16
-i-Sx . . . 2-3-2
r.
— 25 —
Ex. 48. H.C.M. of6;r8 + i5;r''»-6;t + 9and9;r8 + 6;r2-5i+36.:
Here we can divide through by 3, which will be a factor of
the H.C.M.
a. . . 2+ 5- 2+3
3a. .. 6 + 15- 6+ 9
2^. . . 6 + 4-34 + 24
r 11 + 28-15
5r- • -55+140-75
II5. . . 55 + 198 + 99
58+174
-58 1+3
.-. H.C.M. = 3(4: + 3).
^. . . 3+ 2-17 + 12
4a. . . 8 + 20- 8 + 12
diff. -i-x = d 5 + i8+ 9
5^. . . 25+90 + 45
3r- • . 33+84-45
58 + 174
58* 1 + 3
Ex. 49. H.C.M. of ioy^-\-y^ -gy + 24 and 20_y* -17^* + 483' -3
a. . . 10 + I — 9 + 24
5^. . .10-5 + + 15
e 6-9+9
88-a 6-9 + 9
j3. . . 20 + 0-17 + 48 — 3
2ay. . . 20 + 2-18 + 48
d 2- 1+ + 3
.•. H.C.M. = 2j'''' -3>' + 3.
LEAST COMMON MULTIPLE.
23. The least number of which two given expressions are
factors is their least common multiple.
If flcg/ and a^^^ be two expressions, their L.C.M. is adcdef
since this is the lowest expression which contains both.
To find the L.C.M. of two quantities we take the factors
which are common to both and those which are peculiar to
each and multiply them together. Thus a, e, are common to
both the foregoing expressions, c, /, are peculiar to one, and
if, d, to the other.
Ex. 50. L.C.M. oi x^~{a-b)x-ab and x^ -2ax-\-a^.
By factoring these become {x—a){x + d) and (x—af,
^ - ^ ^^
26
.'. (x—a) is common to both, {x + d) is peculiar to the first,
and the second (;r- a)to to the second.
.-. l^.CM, = {x-af(x-+lf).
24. If there be more than two quantities we proceed in<a
similar manner.
2
Ex. 51. L.C.M. of »''-3» + 2, n' + 2« — 3, »''-2«'- m + 2.
tO{
Factoring these become, (« - i)(« -2), («- i)(» + 3),
(m-i)(« + i)(«-2).
.'. L.C.M. = (»- i)(»- 2)(«+ £)(« + 3).
i
25. The product of any two quantities is equal to the pro-
duct of their H.C.M. into their L.C.M.
For if A =abcpq be one quantity,
and B = acprs be the other ;
their H.C.M. = a.c.^,
their L.C.M. = a.c./».6.g.r.s,
.-. H.C.M. X L.C.M. = a2c2^267Ks = .45.
Hence knowing the H.C.M. of two quantities we find their
L.C.M. by dividing their product by their H.C.M.
27
FRACTIONS.
26. The same principles of operation apply to algebraical as
to arithmetical fractions.
i. To add or substract fractions. Bring them to a com-
mon denominator, and then add or substract the
numerators, as the case may be, and write the com-
mon denominator beneath.
ii. To multiply fractions together. Multiply together the
numerators for a new numerator, and the denomina-
tors for a new denominator.
iii. To divide one fraction by another. Invert the divisor
and perform multiplication.
Fractions have such a multiplicity of forms that no general
method of working can be laid down. It is frequently advan-
tageous to factor the parts when possible.
A number of examples is here given.
^x^ +x-2_ i^x - 2)(x + 1) _ 3X — 2
2X
Ex. 52.
Ex. 53.
a
2X'
3 _
'-^-3 (2^-3)(;v + i) 2,if-3
3*^ +3^-2 _ {a^ -a-\-\){a-2) _ a-2
3a^ — 4a* + 4a
^+1 , X- I
I (a* -aH-i)(3fl- i) 3a -i
Ex. 54.
X-I X+I ^ jX + iy +{X-iy^ _ 2X^-\-2 __X^ + l
X+1 x-i~{x+i)^ -{x-i)'i
4^
2X
X-I X+I
Ex. 55.
x^ + (±. + l-)xy^y^ x^U±.^^xy^±.Xy^
^b a' ^ b a ' b a
x^ +
(t4)
\xy -y^ x^ +
(t->
a
T
a
a_ _ ax + by
a
\
!il|r-
28
Ex. 56.
-h
+
x+jf
(x-y){x-z) {y-x){y-z) {z-x)(2-y)
By putting in one factor in each denominator and arranging
in cyclic order we have,
-yiy - z)
+■
- x{s — x)
+ -
-{x+y){x-y)
{x-y){y-z){z-x) {x - y){y - z){z - x) {x-y){y-z){z-x)
_ ziy-x) _ z _ z
Ex. 57.
{x-y){y-z){z-x) {y-z){z-x) (y-z){x-z)'
a . b . c
+
+
{c-a)(a-b) {a-b){b-c) {b-c){c-a)
_ a{b -c)+ b{c - a) 4- c{a -b) _
Ex. 58.
(a-6)(6-c)(c-fl)
since the numerator is zero identically.
,i a b
+
(a-b){b~c){x-a) {b-a)ib-c){x-b)
{c-a){c -b){x -c)
The common denominator is (a -b){b- c){c - a){x - a){x - b){x - c) .
The first numerator = - a{b -c){x- b){x - c)
and by symmetry the
others are
j ~-b{c-a){x-c){x- a)
{ -c{a-'b){x-a){x—h)
.'. whole numerator - - \a{b - c){x-b){x-c)-\rb{c-a){x-c)
{x~a) + c{a-b){x-a){x-b)\.
Now a - 6 is a factor of this, and therefore b-c and c - a are
factors. To find the fourth factor which probably contains x,
let the factors be,
{m + nx){a - b){b -c){c~ a)
where n is numerical and m may be so. To find them put
a = 2, 6 = 1, c =0, and we obtain, m = o, n= 1,
.'. the factors are, x{a - b){b -c){c -a),
which reduces the whole fraction to
X
{x-a){x-b){x-c).
Ex. 59.
6c-
+
— 29 —
cd
+
da
(a~b){b-c){c-d) (b'C){c -d){d-a) (c -d){d-a){a -b)
+ _«6 ;
{d-a){a-b){b-c)'
The numerator in this case becomes,
ab{c~d) + bc{d-a)+cd{a-b)+da{b-c).
a- 6 is a factor of this, and from symmetry b—c, c- d, and
d -a are factors.
But it cannot have four literal factors, therefore it must be
zero identically ; hence the sum of the factions is zero.
27. The following relations among the terms of fractions can
often be employed with great advantage in algebraical trans?
formations. They are useful in reducing fractional expres-
sions, and they lie at the basis of the relations employed in
proportion.
I.
If
a
T
1. — = — -, for a</ = 6c in both cases by mere cross multi-
c d
plication.
2. - = — -y For — + 1= — + 1, and-r-- 1= -j--i;
a-v c —a a a a
a+b c+d J a-b c—d
•■• -r=-rf- ""'^ -r=T-
a-]-b b _c + d d
b a — b d 'c — d
a-\-b c + d
or
a- b c-d
a
ma+nc+pe
b mb + nd +pf
— 30 —
T . a c e
then a=^bz, c- de, e ^fz \
.*. ma + nc+pe^sziftib-^-nd+pf),
' 2 = A = ^ + nc -\ -pe
b mb + nd +pf'
Cor. Um=.n=p, 4- = ^-±^-±-^
b b+d+f
and this is true for any number of terms.
III.
Oca
4-
a _
c
a b
b' c
a a a^
b' b b^'
5-
^ _
d
a b
be
c a a a
d y~b'b"
a"
I !
Ex. 6o. If
For
2^ ~ S 2S~X 2X -y
X _ x-\-y-vz
, then each fraction =i.
2y-S i2_y-S)-{-{2S-X) + {2X-y) X+y+S
= 1.
Ex. 6i. If
- J' ^
a{y + s) b{s + X) c{x +y)\
then -Z^iy - z)^^{z-x)^--{x-y)^o,
a b c
Multiply the first fraction, both numerator and denominator,
v-e —(y~«)
•^ -, and it becomes JL •
by
a
yi-s2
and similarly for the other fractions.
■ "• ' ■ X
Then
iy - s)+Z.(js - x)+±{x - y)
= » p c
a{y+e)
y-*-z^ + x»-x' + x''-f
— 31 —
*
And since the denominator is necessariiv zero, the numera'
tor must also be zero in order that the first fraction may be finite.
Ex.62. If .^ = >:!^=.^=i.toshowthat
cr (r c^
{X ^y + ^)(aa -f ** + c») « a^x ^if^y-\- c^z
Multiply the numerator and denominator of the first fraction
by X, of the second hy y and of the third by s. Then summing
numerators and denominators,
3^-\-^-\-x^-ycyz
a^x^-lf'y^ch
= I
or factoring, .^yi^^'^^l-^ -yz-zx) ^ ^^
a^x f Iry -1- rz
l^ut from the original fractions,
x^-'ry^'V^ - xy -yz-zx=a^^-lf^^\■c^^
RATIO,
28. Th€ ratio of one quantity to another is the numerical
quotient which arises from dividing the one quantity by the
other, or it is the number which expresses how often the one
quantity is contained in the other. Hence a ratio is an ab-
stract quantity, and in order that magnitudes may have a ratio
the one to the other, they must be of the same kind. Thus
there can be no ratio between miles and years although there
is between numbers expressing aggregates of miles and years.
If a, b, c, denote certain lengths, a has a certain ratio to b
and to c, but a has no ratio to be, since be denotes an area.
And thus in geometrical applications of Algebra the terms of
a ratio must be homogeneous. But if a, b, c, denote numbers,
any combination of them may be employed as terms of a ratio.
The ratio of a to 6 may be expressed either as — , or a:b.
In any case a is the antecedent and b the consequent. If a is
greater than 6 it is a ratio of greater inequality, if less than b of
less inequality, and if equal, a ratio of equality. The ratio b'.a
— 3^ —
is the inverse ratio o( a.b. The ratio fl'rft' is the duplicate ratio
o(a:b; fl':6'the triplicate, and rt*:/>^ is sometimes called the
sesquiplicate ratio of a:b.
The doctrine of ratio is extremely important in modern
mathematics, for it frequently happens that the terms of a
ratio are of little or no importance while the ratio itself is
all-important.
We have examples of ratio in the circular measure of angles,
in sines, cosines, tangents, specific weight, &c.
The propositions of Art. 27 apply directly to ratios as frac-
tions. From these it is evident that a ratio is not changed
when both terms are multiplied or divided by the same
quantity.
29. Let rt:/; be a given ratio ; then dividing both terms
')V
X, — : — is the same rs a:d ; but when x becomes infinitely
XX
great each term becomes infinitely small. Hence quantities
which become infinitely small, and are thence called vanishing
quantities, may have a definite and finite ratio. This princi-
ple lies at the foundation of the Differential Calculus.
Ex. 63. What is the ratio of {a -\- x)"^ - a"^ to x when x becomes
infinitely small ?
ratio =
i.a-'rX)'^.- C^-2GX^-^
2ax-{- x^
— 2a-\-x=2a when x becomes
infinitely small.
Ex. 64. To find the ratio of s;!:"- 24: +2 to x^-\-x-i when x
becomes infinitely great.
Ratio =
2 , 2
3- +.
3x'-24r+2 ^ jp j^
= ^ = Z- = 3 when 4: = 00 .
X*+X-l
X x^
30. Prop. The addition of the same quantity to both
terms of a ratio of inequality brings it nearer to a ratio of
equality.
— 33 —
Let — be the ratio, which suppose greater than i.
then-r-- I = — ; —
6 b
Now add x to each side, and, ? — - - i =?-^ .
b + x b-\-x
Kut 5 is less than — j—, . . -,
b-t- X b b-\-x
is nearer unity than-^is.
b
A similar proof applies when a\b is less than i.
31. Ratios are compottnded by taking the product of the
antecedents for a new antecedent, and the product of the
consequent for a new consequent.
PROPORTION.
32. When two ratios are equal the terms taken in order are
said to be in proportion, or to form a proportion.
Thus, . if _- = __. , then a, b, c, d are the consecutive terms of
d
a proportion, which is often expressed as
a'.b'.'.c'.d.
a and d are the extremes, and b and c the means. The terms
a, b, as also c, d, constitute a couplet ; and the proportion is
read,
a is to 6 as c is to d.
If the terms of the last couplet be divided by c d, we have,
1... t . I
a.b .. -J- . — ,
d c
or a is to 6 inversely as d is to c.
33. The following variations in a given proportion are di-
rectly derived from Art. 27 ; some of them have been distin-
guished by special names.
— 34 —
ifA/iO^'*^
If a'. h::c:d,
1. a.c.:b:d, . . . . Alteinando.
2. b.a::d\c, .... Invertcntlo.
3. a + b.b.:c + ci'd,.... C'onJiionunilo.
4. a — b'.b\\c-d\d,.... Dividondo.
5. a\a-\b\\c.c id Convintfiulo.
6. a^b\a-b\\c^d\c-d,9^^^-^A]^i-^'y^'^^
If«:^::cur.:f:/,
9. a\b::a-\-c-\-e\b-¥d-\-f.
10. fl : ^ ■■ : ma + nc -j-pe : m^ -f Wf/ +/»/.
11. rt":<^"::a" + c"-f<'":^" + ^"+/".
U a.b:\b.c..c.d,
12. a: cwa^.b"^.
13. a:</::a^;^*.
If a •.*::*: f, then ac = b^, and ^ is said to be a mean propor-
tfonal between a and c.
Proportions are most readily worked as fractions.
Ex.65. U ax+cy:ay-\-cx::bx~cy:by-cx, then each ratio is
that of X toy.
P ax + c}' __ bx - cjf _ax^+ bx __{a^+ b)x ^ x
' ay+cx~by- ex ay ^- by ^a^-b)y y
Ex. 66. If a:b\:b:c, then, a + b + c:a -b + c::{a-^b-\-c)^
FoT,b^=ac .'.a^-\-b^+c^={a + c)^-b'^ = {a + b + c){a~b-^-cy,
and ia + b + c)ia^ -\-b^ + c^) = {a i- b -i- c)^{a - b+c) ;
.'. a + b + c:a-b + c::{a + b + c)^:a^+b^+c^.
— 35 —
EQUATIONS.
1 ratio IS
34. When an expression is put equal to another expression
or to zero, the result is called an equation. Thus, ax = h,
ax -t- 6 = c, ax-\-by~c = o, are equations.
Fundamentally, equations are of two kinds. Thus, 3;r — 6 = o
is true only under the condition that :r is 2, and it is consequently
called an equation of condition ; but, 2{a-x) -(a — -^) = a-{-x
is true for all values of x, and is said to be identically true.
Such equations are identical equations, or identities. Identities
are often distinguished by the sign e:^.
35. The solution of an equation of condition consists in
finding such a value for the unknown quantity as will render
the equation an identity. Thus to solve
2x-2{x^-i)- 2(1 - 24:) + 6,
we must find a number which when put for x will make the
expression an identity. We readily find 2 to be such a value.
In solving an equation we consider it as an identity and then
proceed upon the selfevident principle that if two equal quan-
tities be modified similarly and simultaneously, they must re-
main equal throughout all modifications.
36. Equations are divided into degrees measured by the
highest dimension of the Hteral symbol taken as the unknown
quantity, in the case of one unknown.
Thus, ax-\-y^ —e^ = o is of the first degree in x, of the second
in ^, and of the third in z.
Equations of the first degree are linears, of the second degree
quadratics, of the third cubics, of the fourth quartics, &c.
36
LINEAR EQUATIONS OF ONE UNKNOWN QUANTITY.
37. These, being the simplest of all equations, usually offer
no special difficulties in their solution.
Ex. 67.
3(*-2)= — ^+ — -?
4 2
... 3;r-6 = "-3^ + ^^-6=_^J-4
.-. i2;r-24 + ;r= -4
.*. I3;r=20 and x=
20
13
38. It sometimes happens that equations which are not
strictly linear can be solved as such, but these are usually
made for the occasion. Examples of modes of reduction and
solution follow.
TT. £_ 3 + 24r 2;r-3 , 2X-^ 4«^ + 7
Ex. 67. ^ + ^ = ^- — - .
2X 2;r - 1 x-2 2;r + 2
. J__^j. _ 2:r-I-2 ^2^-4-1,4^+4 + 3
2X 2X-1
.•.-^+1-1 + —^ +2
2X 2X- I
• 3 _ I - 3 _
;r-2
I
X - 2
= 2
2;r+2
3
2;r4-2
2X X—2 2X+2 2X-\
whence 144?^ - 22;r^ + 28^= \^^ — 22;r*'' - 144; + 12,
.*. 42;r=i2,
and ;r=-?-.
Zl
Ex.69. — '+^^=''-~^+i:ii
Z-2 s-6 z-i z-^
Hence, i + ~ 1-1+-' -' ^ -' ^
Z~2
I I
^-6 ^-3 Zr-l
I I
Z-2 z-i ^-5 z-V
. I _ I
U -"2)^-3) (^-5)(7_6)'
and the numerators being the same the denominators must
be equal ;
whence, xr= 4.
The principles of Art. 27 may sometimes be employed.
Kx.yo. C+in =„«
a
,2 j^ ♦-a
rt +^^ 2^^4-cvt:
zax
ex
num. denom. <'' ~ ■*'^'= -"'^,
x'-^6x-\-^ x^i^iox + 21'
Hence, /-t5f:-L5__ ^ 8^_j:i2
.r + io;r+2i 8;r + 44
;»:'+64r+5_8;r+i2.
4^+16
whence ;r = 2.
32
38-
OF INDICES AND SURDS.
39. It being understood as an elementary principle that a"
means a.a.a. &c., to n factors, we propose here to extend this
notation to negative and fractional indices and to examine
principles of working with such.
i. Since a'^^a.a.a. &c., to n factors,
and rt™ = a.rt.rt. &c., to ;;; factors,
.'. rt".rt"'=fl.rt.fl. 6cc., to ti III factors.
But a"+^^ -a.a.a. &c., to ii ■. in factors.
.-. «".»"» = »"+"' ; (A).
And, to multiply powers we add their indices.
This in short is the rule which has been assumed throughout
multiplication.
o- «"' a.a.a. &c., to /;/ factors
n. Smce — = — — ■ ,
a.a.a. ccc, to n factors
rt"
by dividing both
to n factors we
numer
obtain.
ator and denominator by a.a.a. Sec.
= a.a.a.
&c., to 111- It factors.
a"
But rt'"~" ==rt.rt.rt. &.C., to in -11 factor:
= rt'» "
(B).
And to divide one power by another, we subtract the index
of the divisor from the index of the dividend ; and this is the
rule assumed throughout division.
iii. If in ii, (5) we make iii = n we have
.*. any quantity raised to the power indicated by 7.[^xo is
equivalent to unity.
— 39 —
iv. If in ii, (B) we make m -o we have
a" _
-a-"
. . . (C).
Hence a negative index is to be interpreted as expressing the
reciprocal of the quantity expressed by the same index when
positive.
V. Since (a'")" = (rt.a. &c., . . . . m factors)"
= {a.a. &c, . . . m fact.)(rt.rt. &c, . . m fact.) &c . . n brackets,
-a.a.a. &c, to mn factors.
.-. (0" = rt"'° (D).
Hence, to raise a power to any given power we multiply the
index of the first power by the index of the power to which it
is to be raised.
vi. In V. (D) if we divide the indices on both sides by n or
innltiplv bv — we obtain,
' n
1
But a'" is the k*** root of (a"*)" ; therefore multiplying an in-
dex by — is equivalent to extracting the «*"* root.
Hence, a^ means the square root of a
JL
a
•A '<
cube
tty &c.
1 1^ I
vii. If we have «".«».(?» .... to ;;/ factors, the result
1 m
must be nn'™ = « n,
m
Hence, « » means that the «"* root of a is to be raised to
the ;«*** power.
a"
Ex. 72. If 4-.rti+»
— m —
a
1+n+m
a
1— 11— m
.and n.
Reducing to one line by iv.,
, to find a relation between m
— 4^^ —
we have ^r»' •"^^2"^+'-^" ;
hence 3» - in = mi -|-2» and ;; — ]}n.
Ex. 73. To simphfy,
2'^".V " _ 8^ ".2
II o I
,n 1
Reducing to one line,
2**" "'* — jil— 11
« = 3 evidently satisfies this condition, since we then have
2^ =30 or 1 = 1.
Ex.74. To simplify '^ =2".
4"
n 3
In this case, (2-^)^ .2"+* - 2".(2''^)>',
6
or, 2". 2"+^ ^=2". 2",
6
.-. ;;4- I = - .
and n evidentlv is 2.
40. An expression denoting;- a root which canriot be exactly
obtained, as | 2, 1^5, ike, is called a surd, or irvaiional quan-
tity. Surds are divided into orders dependinj^ upon the index of
the root to be obtained : if it be a square root wu have a
quadratic surd, if a cube root a cubic surd, ik.c. ,
The product of a rational quantits with a surd is known as a
mixed surd, but when all the factors are under the surd sign it
is termed an entire surd.
1 1.
Surds may be indicated by indices as 2"^, 5"', <SlC., and in
many cases their properties are best studied in this way ; but
in the case of quadratic surds, at least, and frequently in the
case of other surds, it is more customary to employ the
sign V .
The following principles establish the rules for the working
of surds. Let n denote any quantity whatever, integral or
fractional ; then,
i. Since a°.6" = (rti)", .*. | a.\ h-\ ah.
— 41 —
ii. Since ^=(^)\ .'.^4=1^. ^
iii. ai b~ p'a^y b = i a^b.
similarly rt]^6= ^rt^.|^7> = (<'fl^6.
IV. y a.f a=^u^ .a^ - a^ '^ —a**.
V. , p^q^} p'^.y q=p] q.
l^x- 75- I 3-1 2.V 5 = 1 30.
Ex.76. I 184 = 1 2^.46 = 21 46.
Ex.77. ':-^:^^-^ = ^ 3.
31 2 31 2 3
Ex. 78. I^54rt*-if2j,3 _ f' 2ya^y^.2ax^ =^ay^2ax^.
42. Fractional expressions with a compound surd in the
denominator are simplified by rendering the denominator
rational. The methods of doing this are shown in the follow-
ing examples :
Ex. 83
Ex. 84
Ex. 85
— X
2 - 1/2 2-1/2 2 + 12
•IZii-? = (3.-y2Mj. 2 + 1) ^ J
y' 2-1
l±J^ = i±J_2.i^^,,2.
+ 2l/2.
2-1
2 _ 41-1^2 + i/3)_
I -\ 1/2-1/3 21/6 - 4
_ 2{ I - |/ 2 + J/ 3j(2 V 6 H- 4) ^ (l -y 2 + v 3)(t/64-2)
24-16 2
_ 1/ 6 + v'2 + 2
43. The following propositions with respect to quadratic
surds in particular are important :
i. The product of dissimilar quadratic surds cannot be rational.
For, let I /> and 1 (/ be their simplest surd factors; then
neither /> nor q contains square factors, and being dissimilar
they are not made up of the same factors ; therefore, their
product cannot be made up of square factors, and consequent-
ly \/pq is not rational.
ii;
— 42 —
ii. A surd cannot be made up by combining rational quantities and
surds by addition and subtraction.
For if possible, let y'p - m ± \,^n ;
squaring, p = m^ +n± zm \/n ;
.'. |/»= ±C-— " = a rational quantity.
2m
iii. A surd cannot be made up by combinini^ two dissimilar surds
by addition and subtraction.
For if possible let V/» = Vq± \/y
squaring, p=q-\-r±2 \/qr,
.'. ^qr = ±^{p -q -r)= a rational quantitity.
But since q and r are dissimilar, \/qr cannot be rational.
iv. If x+ ^/y = a-\- 4/6, then x - ^'y = a - \/b.
For X- a=: yb - i/y... But since ^ - a is rational it cannot
be equal to the difference between two surds.
Hence x — a=o, and \/''b-\/y=:o ;
.*. x=a diXid \b = \/y ;
and X - [/y=ci- \/b.
44. To find the square root of a binomial quadratic surd.
Let Va + \^b = y/'x -f |/j.
squaring, a + y/b =x-j-y + 2 y/xy ;
.*. Art. 43, iv., x-{-y = a, and 4xy = b.
Hence (x +y)^ -b = {x -y)"^ -a"^ -b,
and .•. x—y= V a^ -b.
V>wt x-y = a,
^ .-. x = \{a + Va'^-b),
2Sid y=^{a-\/a^ -b).
Ex. 86. To find the square root of 3 + 2^/2.
Here ;r+>' = 3, and ;ry = 2.
43 —
and x-yzz\.
Hfence, x-= 2, and^ = i,
•"• 1^3+21/2= I +]/2. • .
Ex. 87. To find the square root of 23-41/15.
^+^' = 23, and 4;ry = 240 ;
.-. ;r -j/=i/(232 - 240) = 17 ;
.-. ;«r=20,3'-3,
and v/(23 - 4v'i5) = VS + 1/3-
45. In the case of trinomial quadratic surds which are com-
plete squares we may proceed as follows :
Let |//> + \/q + i/r be the root.
Then (|//» + yq + \/y)'^ z=p + q + r + 2\/pq + 2\/qr + 2\/rp.
But p = ^J^^Mlly^.
2X2y'qr
Hence if P, Q, R, denote the surd terms, taken in order,
P
PR ,.QP ,^RQ
Ex. 88. To find the square root of 54 — 41/2 +61/5-121/10.
2x6|/5
41/2.61/5 ^ . /
2X41/2
.*. I ±21/2 ±31/5, form the terms in the root, and a little
inspection shows us that the signs must be
1-21/2+31/5.
In cases of this kind the subsequent squaring of the root is
the only sure test of correctness.
44
SURD HQUATIONS SOLVED AS LIN EARS.
46. Equation? containing surds can sometimes be solved as
linears, but in all cases they involve certain peculiarities which
will be more fully comprehended hereafter.
Ex. 89. Given I a^+x^ + v'a^ -x^ = b to find x.
Squaring, 2a^ + 2l^a* —x*= b'^ :
transj.osing and squaring, 4a* - 4-r* = (6' - 2a*)^ ;
.'. x=t {a*-\{b'-2ay\'
Since the fourth root of a quantity has four values, x ihas
four \alues which will satisfy the equation ; and thus the
e(iiiatioi) although apparently solved as a linear, is in fact a
quartic.
Ex.90. Given ^a-^i/x=-i/ax to ihnd x.
Here we reduce the number of surds containing x by divid-
ing l.y I X,
and
and
a (|/a — i)^'
a
X =
(l/«-i)^'
^ ^. \/a + x , V'a + x i/'x , c j
Ex. 91. Given, — - -f 1_ =!L. to find
a x b
CLXfT-' b
.'. b[,a-\-x)^ = ax^.
Squaring and extracting cube root,
A 2
whence, x—-
a
ab'^
1 2"
45 --
• OF THE QUADRATIC r'^UATION.
47. A quadratic equation contains the second power of the
unknown quantity. If it contains that power only it is called
a pure quadratic, but if it contains the first power also it is a
mixed or adfected quadratic. This distinction is, however, of
little importance.
48. Origin uf a Quadratic. If two linear equations, with the
same unknown quantity, be multiplied together the product is
a quadratic.
Thus, (x-a=^o)(x- b=o) gives x'^ - {a-\-b)x^-ab=o.
And conversely, every quadratic can be formulated as the
product of two linears.
Thus, '\{ x"^ +px + g=o denote any quadratic,
■-i^r _i_r» — " — n ^-«^>-# — — —
:t' +px
4
4
= 0,
TO
.-. [X+±)^ -l(PL-gy2=.o,
X +
H*^T-^'}•.|--^^l<^-^)|=<'•
In which the quantities within the | i are linear equations.
Hence every quadratic may be considered as the product of
two linears.
49. Roots of a Quadraiic. A quantity which, when put for
the unknown quantity in an equation, satisfies it, or renders it
true, is called a root of the equation.
A linear has but one root ; but a quadratic, being the pro-
duct of two linears, is satisfied by the root of each linear ;
every quadratic has accordingly two roots.*
Thus, if x^ -\-px + g = (x- a){x — b) =0, where />= -(a + ^) and
g = ab, then;ir = a, or x-b satisfies the equation since either
substitution renders the expression zero, hence a and b are the
roots of the quadratic.
* This Btatemeiit 1b not \«^ithout oxct i lionp, to some of yflilch reference will be made
hereafter.
-46 -
5o. General solution of a Quadratic. The most general form
in which a quadratic can be written is
ax^ +6x+c = o ;
when a, b, c, denote any quantities whatever, and this we are
to resolve into linear factors.
Multiply throughout by 4a, and add and subtract 6^, and we
obtain
4a*;r* + 4adx + 6^-6"^ + ^ac = o.
.'. (2ax-hd)^ -{d^ -4ac)=o,
or, {2ax+d-\-Vd^-'4ac)i2ax+d- i/d^ -4ac) = o.
Whence if x^, x^, denote the two roots,
^1 =
x^ =
2a
-b-\-\/W-~^c
{A).
2a
These may be combined in one formula by using the double
sign ± , and we get,
_ -b±Vb'^ -^ac
2a
(B).
A study of this form serves for the solution of all quadratics.
Ex. 92. Let ^x^ -2;r-f4 =0 ;
then,;r = ^^^f-lg- = ^^<"^,
= i(i±v/-ii}.
Ex. 93. Let {a - b)x^ ■\-ax-\-b = o.
then,;r=--^±^^^'-4fe(^-^).
2{a-rb)
_ —a±{a-2b)
2{a - b)
- — , or -I.
b — a
- 47 -•
51. Stun and product of ! he roots.
Adding the values of the roots in Art. 50, (A), we obtain,
and nuiltiplying, we obtain,
c
a
X\*i -
Hence in a quadratic the sum of the roots is equal to the
quotient arising from dividing the coefficient of x by that of x^
taken with a changed sign ; and the product of the roots is
equal to the quotient arising from dividing the constant term
by the coefficient of ;r*.
Ex.94. Given (s/+^)'' + (cZ + a)" = y*, and s'*+c'''=i, a qua-
dratic in / to find the sum and product of the roots.
Squaring and arranging in powers of/,
.-. /j +t.. = - 2(c« + s/9),
and /i/2 = aa+/92 ^r2.
52. Nature of the roots. In the formula Art. 50 (B), since 6*
is essentially positive, and since a may be rendered positive by
change of signs, the character of the quantity under the surd
will depe.nd upon the «ign and value of c, a being positive
i. If c is negative, then b^ - ^ac is positive, and has a square
root either rational or irrational. Hence in this case the roots
are always real quantities.
Thus, \{ x^ i-4x — n = o, x has two real values for every posi-
tive value of w.
ii. Iff is positive and less than — , the quantity 6'- 4^0 is
4^
positive, and the roots are real.
iii. If c, being positive and less than — , gradually increases
4a
in value, then the two values of at, i.e. the two roots become
more and more nearly equal as \? - ^ac becomes smaller ; and
finally the roots meet and become equal in value when h^ — 4ac
-48-
becomes zero. The surd part then disappears and the qiin-
dratic has equal roots which may be botli positive or both
nejjfative, but which are always rational.
The very important condition then that a quadratic may
have equal roots is that ^ac = 6".
Ex. 95. If r^- 2drc-^(P -a^^o be ix quadratic in r, find the
condition that r may have two equal values.
Condition, 4((f - «") = 4^/^,
or,
iv. If c be positive and greater than — , b^ - ^ac is negative,
4a
and as the square root of a negative quantity has no real ex-
istence but is wholly imaginary, the roots of the equation will
be imaginary or impossible. These imaginary results are not to
be dismissed as of no consequence, as they are frequently of
very great importance. Let it be required for example to
divide 10 into parts such that their product may be 30. If x
be one part, 10 -;« will be the other, and
;r(io-;r) = 30 = lojr-r* ;
1 10 ± 1/ - 20
whence x=^ ^ ^
2
= 5±l/-5:
where the imaginary result |/ - 5 shows that there is some
impossibility involved in the question. Upon examination we
find that the largest product which it is possible to obtain from
the two parts of 10 is 25.
V. If 6 be zero, the value oi x reduces to
•,jv
In this case the roots are equal in value, but of opposite
signs. The condition that this should take place is, then, that
the coefficient of x in the first power shall be zero. If c be
positive the roots are imaginary, but if negative they are real.
Ex. 93. G'w&n {rs — aY-\-{rc—'bf — i =0 to find the conditions
under which r will have values equal in magnitude
but opposite in sign.
Expanding, r\s^ + c^) — 2r{as + 6c) + rt' + i' - 1 = 0.
.*. Condition is as + 6c = o,
or,
s
c
49
b
a
The relations devploped in the present article are of the
hijijhest importance in coordinate geometry, and in the appli-
cation of algebra to geometry.
53. Limits 0/ positive and negative values of quadratic expressions.
Let ax^ -t- 6;r + c be a general quadratic expression. Resolv-
ing it into linear factors we find the expression to be ecjuiva-
lent to
2a za
Disregarding the factor a for the present, when the two
factors within brackets have the same sign their product will
be positive, but when they have different signs it will be nega-
tive ; and the only effect of a change in sign of a is to reverse
these results.
But the bracketed factors can have different signs only when
one is greater than ;?ero and the other less.
Suppose the first factor to be greater than zero and the
second one less ; then we must have
^>
d + 1/^' - 4rtc
2a
and <^:^-4^^ -£
2a
Between these limits for the value of x the expression is
negative for positive values of a and positive for negative
values ; and for all values of x beyond these limits the sign of
the expression is the opposite to that which it has when the
value of ;r is taken between the limits.
Ex. 94. What are the limits of negative values for the expres-
sion 3X^ + 2X- ^ ?
Tu- • • 1 .. ^ / . 2 + V^4+ 60v , , 2 - V^4 + 6o V
This is equivalent to, 3(;r+ ^ ) {x + 2 ),
o o
or 3(^+|)(^-i)-
.'. X must be less than i and greater than -f, and if any
quantity between these limits be substituted for x in the given
expression the result will be negative.
— 50 —
Ex. 95. Under what conditions will 'jx - 3;r' - 2 be negative ? '
This is equivalent to -{ix^-']x^2)= -^{x-^^x-2).
Hence the expression will give positive results for all values
of X between 2 and \, and negative results for all other values*
54. Of maximum and minimum solutions of quadratic extressions.
By dividing by the coefficient of x^ any quadratic may be put
into the form, ^
x^-\-px-\-q-o.
We know, Art. 49, that there are two quantities real or
imaginary which when substituted for x in this expression will
render it true. These are the roots. If, however, we put any
other quantity whatever for x the expression will not be equal
to zero, but to some finite quantity which we may denote by
y. The value oiy will depend upon that of the quantity sub-
stituted for x. If among all the quantities which can be sub-
stituted for X there be one which will make y frreater than
it can be made by substituting any other value for x, that
value of ;r furnishes us the maximum solution, and y or the
quadratic expression is said to attain its maximum. If on the
other hand the particular substitution renders y less than any
other substitution does, we have a minimum ^o\ut\or\ and v or
the quadratic expression attains its minimum.
55. To find the maximum or minimum solution of a quadratic.
Let x^-\-px-\-q=^y ;
then,;r=rl±J^ZlS±4V.
2
Now, in order that x may be a real quantity the expression
under the surd sign must not be negative. It is readily seen
that increasing the value of y has no tendency to make
p'i—^q^^y negative, and hence that y has no maximum. By
diminishing jy howexer the value of the whole surd expression
will be gradually diminished until it passes through zero and
becomes negative. Hence y has a minimum limit ; that value
which makes the surd expression zero.
Again, let the expression be,
-}^-\-px-Yq=y;
i-atic,
— 51 —
t
Changing signs, :i^-px~q— —y ;
. whence ;r=^A!^2±SE47
2
In this case, since y is negative, increasing the numerical
value oiy diminishes the expression under the surd ; hence y
has a maximum limit when this expression becomes zero, but
it has no minimum limit.
We infer then,
i. That every quadratic admits of a minimum or a maximum
solution according as the coefficient of 3^ is respectively posi-
tive or negative.
ii. That the maximum or minimum solution is obtained by
solving the equation for x and then equating to zero the quan-
tity under the surd sign.
In the general equation 3?-¥px-q-y^ the minimum value
of jy is ii — ^-, and the corresponding value of ;r, or the value
4
of ;r which renders the expression a minimum is ^.
Ex. 95. It is required to divide a number a into two parts such
that their product may be a maximum.
Let X be one part, and a-x the other.
Then x(a - x) -y a maximum ;
.'. ax~3?—y
a ± /a'
or x=
4y
)ression
seen
make
n. By
)ression
ro and
t value
Hence, for a maximum, a' - 4y = o, or y = t — \ ,
And the number must be divided into equal
r.ud x =
2
parts.
If a denotes a line, we see from this that for a given perimeter
the square contains a greater are than any other rectangle.
Ex. 96. To divide a given numberinto two parts such that the
sum of their squares divided by their product may
give a maximum or a minimum, and to determine it.
Let a be the given quantity ;
then x*+ia-xY = 2x^—2ax+a^ = sum of squares,
and x{a —x) -ax-x^ = prod uct.
24^ — 2ax -\-'a^
.' ■ — = y = a max. or a min,
ax -3?
From this we obtain,
2 2 \ 2+y
Whence we readily see that ^ can have a minimum value,
but no maximum.
Put I
^ =o .*. v = 2 = the min. value ; and x- —
2-Vy 2
Hence the number must be divided into two equal parts ;
and the sum of the squares of the parts divided by their pro-
duct cannot be less thah two.
This article is of particular importance in the appli-
cation of algebra to geometry.
56. Graphic representation of the quadratic.
All the prominent properties of the quadratic may be ex-
hibited graphically by means of a curve.
Take for illustration the quadratic expression ;ir^-3;r- 2.
We know that for two particular values of ;r, (the roots), the
expression will be zero, but that it will have some finite value
when any other quantity is substituted for x. Let y denote
that value ; then
x^ — ^x—2=y.
Substitute different values for x, integers for convenience,
and we obtain corresponding values of _y as follows :
if ;r= -I o I 2 ^ 4. . . . . ^'^^
y= 2 -2 -4 -4
3
2
4
2
venience,
-" 53 —
Draw two lines xxj, yYi, inter-
secting: at right angles in o. Let
the different values substituted for
X be denoted by distances meas-
ured from o along xXj, the posi-
tive values to the right of o and
the negative to the left.
Also, let the corresponding
values oiy be measured from the
linexxj parallel to the line yYi,
the positive values upwards and
the negative downwards. We
thus get a series of points «, 6, c,
d, e,f . . , . The curve which
passes through these points and
through all points similarly ob-
tained by substituting all possible
quantities for ;r represents the quadratic expression x^ — ^x—2.
i. Consiier >he points P and Q where the curve cuts the
lineXXj. t t icse points^' is zero, and they accordingly
represent tht ...juation, x^- 2^+^ = o. And the values of x
for these points i.e. OP and OQ, or the distances of these
points from O represent the roots of the equation. We thus
see that one root is positive and has a value between 3 and 4,
and the other root is negative with a value between o and i.
If both points, P and Q, were upon the same side of O the
roots would have the same sign, positive if upon the right
side and" neg.itive if upon the left.
ii Since the curve actually cuts the line XX ^ the points P
and Q are not imaginary but real, and the equation has con-
sequently real roots.
If the curve after approaching the line XX i turned and re-
ceded from it without meeting it, the roots would be im-
aginary.
iii. Suppose that the curve merely touches the line XX^ at
its lowest extremity M. This might be brought about by
moving the curve bodily upwards : but in so doing the points
P and Q would gradually approach one another and finally
meet at the point of contact, and the distances OP and OQ
would be one and the same. Hence this denotes equal roots.
If the curve were still more elevated the points P and Q would
become imaginary.
,\
— 54 —
Hence we see that if a quadratic chan,t,'es its form con-
tinuously so as to pass from real to ima:ijinary roots or vice
versa, it must pass tiirouf^h the condition of equal roots.
Compare Art. 52, iii.
iv. As the curve lies' wholly below the line XX ^ from P to'
Q, the quadratic expression X'-y--2 is ne^'ative for all
values of x between these limits, and positive for all other
values.
V. Since the curve sweeps downward to a lowest point and
then begins to ascend, the quadratic has a minimum value.
If the curve were reversed and the apex turned upwards, it
would denote the existence of a maximum value for the cor-
responding quadratic.
vi. If YY^ passed through M so that the cu.ve was sym-
metrical with reference to the line YY-^, OP would be equal to
OQ in magnitude, but would differ from it in sign. Hence
the roots would be equal in magnitude, but opposite in sign.
Art. 52, V.
Ex. The quadratic 6 +;r- ;»r"-^ has equal roots, one positive
and the other negative. It is positive for all values of a: be-
tween the roots, and negative for ail values beyond them. It
admits of a maximum but not of a minimum.
The curve described as above is known in Geometry as the
Parabola.
Of the double solution furnished by the quadratic equation.
57. When the statement of a problem involves a quadratic
equation, the two roots indicate in general two possible solu-
tions to the problem ; the double solution being sometimes
directly applicable and sometimes not.
In purely arithmetical questions it usually happens that
only one of the solutions is directly applicable, the other
becoming so only after some changes in the wording of the
problem.
Ex. 97. A man died in a year A.D. which was ^^^ times his
age : 13 years before the year was the square of his age. To
find his age at death.
Let ;tr=his age, then 33^^ .jr=:the year A.D.
and 33¥^-i3=(-«^-i3)''
Herel
the 3
Sill
a neg^
Hei
55
IS the
ition.
Iratic
aolu-
fimes
that
pther
the
Is his
To
whence x=^6 or 3^.
Here 56 is evidently the answer to the problem, but what does
the 3^ mean ?
Since 33iX3i - i3=if|i=(3^ - 13)"
.•. ^^ satisfies the algebraical condition, but 3i - 13 = - 7|,
a negative quantity.
Hence we may interpret the two solutions as follows :
\ since (after) the man was born
13' years ago the year A.D. 5
was the square of the years 1, c ^\ u
• -^ betore the man was born ....
58. It sometimes happens in even arithmetical questions
that both solutions are applicable.
Ex. 98. A man buys a horse and sells him for $24, thus
losinjj as much per cent as the horse cost in dollars : To find
the cost.
X
\{x-.
loss — X
24
the cost,-—- .X
100
Whence x=^o or 40.
And •.• both soiiitioiis satisfy the condition, the problem is
l(j a certain extent indeterminate.
59. In geometrical [)roblems and problems involving geo-
^ metrical mairniiudes, the double solution is frequently of the
ihiKhest importance, and it should not be neglected, inasmuch as
it often increases materially our knowledge of the problem in
Ihaiid.
[Ex. 99. The attraction of a {)lanet is directl/ proportional
to its mass and inversely proportional to the square
of its distance. The mass of the earth is 75 times
that of the moon, and their distance apart is
240,000 miles. It is required to find a point in the
line joining them where their attractions are equal.
Let P be the point and denote ^ ^
pP by a;. Then PM = 240000 -at. e p m q
[Attraction of 0=75 X 2 ! "^^ =lX
and these are to be equal ;
(240000 -;t)2 '
- 56 -
hence 75(240000 -x)^=x^,
whence, ;r=2i5i6o or 271330 miles.
We thus see that there are two points of equal attraction,
the latter of which lies beyond the moon at the point Q ; a
result which, when once obtained, recommends itself to our
judgment as true.
60. When a quadratic equation so involves a surd as to
necessitate the process of squaring in the course of the solu-
tion, it sometimes happens that the roots obtained are not
those of the equation proposed, but of an equation differing in
sign only from the original.
Ex. 100. Given 3^+1^30 ;»: — 71 = 5 to finder.
By the regular mode of solution we here obtain the values 4
and 2f for x, neither of which will satisfy the given equation,
they being in fact roots of the equation,
3;r-v/3o;r-7i=5.
In cases of this kind it is only by verification that wr' can
determine whether we have a correct solution of the proposed
equation or not.
Again from the equation ^x-\- U^2x-2-j, we obtain x=i^
and x=i^, of which ;r=if only will satisfy the given equation,
while ^=3 satisfies the equation ^x- \^2x- 2 = 7.
The difficulty in these cases seems to arise from the fact
that when we square a quantity we lose all trace of its oiiginal
sign, and we have afterwards no means of determining alge-
braically what sign it was at first affected by.
Thus: V^2;r- 2 = 7-3;rand - l/2;r-2=7 - 3:1:, evidently be-
come identical upon squaring, whereas they cannot possibly
be satisfied by the same quantities ; so that any solution must
give us either both roots belonging to only one of these equa-
tions, or one root belonging to each.
Whether any value of x can satisfy the equation ■^x +
V3ox-yi=$ or not we do 'not know, but if there be such a
value it cannot be found by the usual mode of solving a
quadratic.
is a
— 57 —
TWO OR MORE UNKNOWN QUANTITIES,
6i. It fre(}uently happens that the conditions of a problem
require the introduction of mor ' than one unknown quantity
in its statement. M>... *
In such cases we require tor tuw complete determmation of
the unknowns as many equations as there are unknown quanti-
ties, and these equations must moreover be independent, that is,
they must be such that any one of them cannot be obtained
from the others by any legitimate process. The equations in
such a set are termed simultaneous equations. Thus :
x+2y-{-z = 8
is a set of three simultaneous equations involving the three
unknown quantities x, y and z; and they are thus named
because the values obtained for x^y and z must satisfy all the
equations at the same time. This takes place when ;r=i,_y=2
and ^=3.
62^ If the number of independent equations be less than
that of the unknown quantities, the equation can be satisfied
by an infinite number of sets of values for the unknown quanti-
ties, and the problem is said to be indeterminate. Thus if we
have one equation with two unknowns, as 2;r-3^=io, it is
evident that if we put any value whatever for x we can find a
corresponding value for^. This species of equation is exten-
sively employed in co-ordinnte geometry, where x denotes an
abscissa of some locus Sindy ihe corresponding ordinate.
63. If the number of equations be greater than that of the
unknown quantities, then some of the equations must be incom-
patible with the others, or else they are dependent^ and hence
redundant.
Thus, if 3;r + a^' = 8
2x - y = 3
x + y = 1
be a set of three equations with the two unknowns x and y^
- 58 -
the values which satisfy the first two cannot possibly satisfy
the third, or those which satisfy the second and third cannot
satisfy tne first, &c. ; i.e., one of the equations is incompatible
with the other two.
If the third equation were * + 3^ = 5, then since this may be
derived from the other two, or any one of them from the
remaining two, one equation is dependent, and, thus giving no
new relation, is redundant.
But if the equations are literal and are to be also compatible,
some relation must exist among the literal co-efficients.
Art. ^^.
Ex.
Ex. J
LINEAR SIMULTANEOUS EQUATIONS— ELIMI-
NATION.
64. When we so coml)ine two or .more equations as to get
rid of a quantity we are said to eliminate that quantity between
the equations; and the process of solving a set of simultaneous
equations consists in eliminating the unknown quantities, one
after another, until we finally have a single equation contain-
ing only one of the unknowns.
The methods of elimination will be considered under the
following heads :
1. By comparison.
2. By substitution.
3. By cross-multiplication and addition and subtraction.
4. By indeterminate or arbitrary multipliers.
5. By determinant forms.
These modes are all applicable in any case, but they are
not all equa,lly convenient. Thus i and 2 are not often con-
venient with more than two unknowns ; 3 may be applied to
any number, and is one of the most practical ; 4 applies with
greatest advantage to three« unknowns ; and 5 applies most
profitably to three or more.
65. Elimination by comparison. This method consists in
finding the value of the same unknown quantity or some func-
tion of it in terms of the other, from each equation, and then
equating these values.
66.
tutef
its vai
Ex. I
SI
■';ih
satisfy
cannot
patible
may be
om the
zing no
patible,
JMI-
1 to get
between
taneous
ies, one
jontain-
der the
tion.
hey are
ten con-
plied to
ies with
es most
nsists in
me func-
md then
Ex. 99. Let
be the equations.
— 59 —
sy .
Then from the first, x=S--^'f
and from the second, ;r=— +^.
5 5
. 8_?^^i8 42:.
3 5 5
Whence ^ = 3, and thence x=6.
Ex. 100. Given — +- = a, ^+- = ^ tofind^-and^.
X y X y
mn n' , m'
Here, — =«» ^mb- — .
X y y
nt^-n^
.•.-(w»-»>)=m6-«a,and>' = 4— -,
y mb-na
and from symmetry *=
w'-n'
ma—nb
66. Elimination by substitution. In this method we substi-
tute for one of the unknown quantities in one of the equations
its value drawn from another equation.
Ex. loi. Given 4^±M=;r-y, and ?£z>' = l-2> to deter-
40 3
mine ;rand>'.
From the first, ^-^^y = ^ox-^oy', whence, :« = 52.. And
4
substituting this for x in the second.
2. 5Z-J/
= ^-2y
Whence ^=^,
and hence x = \.
67. Elimination by cross-multiplication and addition and sub-
traction. The following examples will illustrate this very im-
portant method :
6o
Ex.^102. ^ Let ax-by=o, x-\-y = cht the equations.
Multiplying the second equation by b and adding to the
first we eliminate y and obtain
«M;+^;r—^; whence ;r= ^
«+*'
and thence, y ^
ac
a-^b
Ex. 103. Let the equations be,
2* + 4J' + 5« = 49,
3* + 5^ -»- 6« = 64,
4* + 3^ + 4'«^ = 55 r
. . . a
. . .^
2a - r
2t + r
• • • •
S^' -f 6-e = 43 d
- 2y - 3<p = - 19 «
^ = 5;
r
whence ^==7, 5=3.
Ex. 104. Given, 9:r - 2* + « = 41
7y - $z ~ t = 12
4y - ^x +2U = 5
3^ - 4« +3^ = 7
7^ -5U = II
Since t occurs the least often eliminate it first.
d + 3/9. . . . 24>'-i5«-4M = 43 C
^ and d cannot hereafter be employed.
Next eliminate u.
C+2r Z2y-i$z-tx-$i, ....
2a- 7'....2i;r- 4y-4s = 77, . . . .
50+ « . . . . lye- z =72
To eliminate y ;
g7 + 8(? . . . . 162:1; -47:5 = 669. .... A
And finally,
47X-A — 543-* = 2715 ;
whence«=5»J' = 4. * = 3. « = 2, /=i.
7
^
— 6i —
«
Ex. 105. Given ax-^by = c, and a';r 4 ^'^^ -- c' to find_;rand>'.
Multiply the first equation by d' and the second by d and
subtract one product from the other, and we get, _
x{ab'-a'b) = b'c~bc.
b'c — be
X-
ab' - ab '
ac — ac
and from symmetry, j* =-7^ — ^.
ab — ab
We notice here that in order to eliminate y we multiply the
first equation by the coefficient of y in the second, and the
second equation by the coefficient oiy in the first ; and simi-
larly to eliminate x ; hence the term cross-multiplicaiion.
68. Elimination by indeterminate cr arbiirary multipliers, f his
method may be readily applied to the case of two equations,
or to the case of three.
Ex. 105'. Given 3x+y = y, and io;r- 2y = 2 to find x and^.
Multiply one of the equations, the first for example, by
the indeterminate multiplier / and add the product to the
other, and we have,
x{-^k + 10) +>'(A - 2) = 7^-1-2.
Now this is necessarily true whatever value may be given to
L But if k = 2, y will disappear from the equation and we
obtain 164;= i6, or ;tr= i.
Similarly if 3^ +10 = 0, ;r disappears from the equation and
t here results, ^(-^-2) = 2-^; whence >' = 4.
69. If we have a set of three equations, for example:
2.^ — 3^ + -s^ = 2
+
+
22f = I
J' -
^x + 2j/ - 3^ = 5,
it is possible to multiply them by such multipliers that when
the products are added the coefficients of two letters may
both become zero at the same time, and thus we may elimi-
nate both letters at one operation. In the example given if
we multiply the first equation by i, the second by -7, and the
third by 5 and add, we obtain io;r=: 20.
-- 62 —
To investigate a rule for finding the proper multipliers.
Let, ax H by -^ cz = d
a'x + b'y + c'z - d,'
a''x + b^'y + ^"^ = (V\
Then multiplying the first by /, the second by m, and the
third by n and adding, we have,
x{la + ma + ;/«'') +y{lb 4- ;;;6' • , nh") <■ ^(/f -f- ;«c;' + «c")
Now if y and ^^ are both to disappear their coefficients in
this equation must be zero. We must accordingly have
lb-\-mb'^nb" = o
Ic + uu' -| «c" = O
Eliminating n between these, we ohtaiii
/
;;/
^,V' - b"c' b"c - be"'
n
and from symmetry each. =
bc'-b'c'
And having three equal fractions the numerators must be
proportional to the denominators.
Hence /, m, n may be any quantities proportional to
b\
b"c\ b"c~bc'\ bc'-h'c
respectively ; and these quantities themselves are usually taken
as the multipliers.
To apply this, notice, i. That the multipliers are made up
solely from the coefficients of the letters to be eliminated.
ii. That the multiplier for any line involves only coefficients
belonging to the remaining lines.
iii. That each multiplier is the difference of two products,
these being formed of terms taken always in the same order.
Ex. 106. Given, x- y-22^ >,
2x-\- y - T^z — ii
3x-2y+ 2= 4.
We find for /, m and n, in order to eliminate y and z, the
rs.
and
the
n
cien
ts in
ve
must be
to
illy taken
nade up
ted.
•efficients
products,
order.
d z, the
- 63 -
I
values -5, 5 and 5 respectively. Then multiplying and add-
ing, we get 20;r- 60 and hence ^=3.
Ex. 106'. Given, ax ^by - as~b{a-\-b)
bx-ay+z = a-b
x-\-2y - 2s = ^-a.
To eliminate^ and e the multipliers are,
l = 2a-2, m = 2b — 2a, n=a~b,
whence we obtain, after reduction, x = a ', and similarly
y = b and s = a-b.
70. Elimination hy determinant forms.
If from the simultaneous equations,
a^x + b^y ■\-c^z = d^
a^x^b.^y-¥c^z=d^,
We eliminate J and z by Art. 68, or by any other means ^^
obtain for the value of x,
;r = '^^2f8_+'^2^3f 1 '^^3^\^2 ~^^1^3 ^'a -^2^1^'3 ~^8^2^1
^i^a^a+^a^s^i +«3^i'"2 -^i^s^a ~^2^i^3 -«3^2^i
The complex expressions forming the numerator and de-
nominator of this fraction are determinants ; and as we ne
they occur in the common process of elimination. The num-
erator may evidently be obtained from the denominator by
substituting d for a throughout ; and hence from the principle
of symmetry in order to obtain equivalent expressions for y
and z we must substitute d for b and c respectively in the above
form.
Taking the denominator then as the type form tic numera-
tors may all be derived from it by substitution.
In the case of three simultaneous equations involving three
unknowns as above, each term in the denominator is the pro-
duct of three «/^;;ie«/s or is of three dim .fusions. With four
equations each term will be of four dimensions, and so on ;
and determinants are thus divided into orders according to the
dimensions of the terms.
A determinant of the third order contains six terms, while
one of the fourth order contains no less than twenty-four terms.
For the purpose of denoting tliese expressions without writ-
ing them in full the following notation is commonly employed :
- 64 -
denotes ^162 -^2^1 which is a determinant of the
second order.
Hi 61 Cj
^2 "2 ^2
*3 ^3 ^3
denotes aid^c^ +^^^2^3^! +''3^i^2 ~^i^3^a
which is of the third order and is the same as
«l(^2^3 -^3^2) — ^2('^1^3 -^3Cl) + ^3(^l^"2 -^2^1)'
From this we see that
«1 ^1 ^1
= «!
*2 ^^2
-«2
<^1 Ci
+ «8
^ Ci
«2 ^2 ^2
<^3 ^3
^3 <^3
^2 ^2
*3 ^3 ^3
In Uke manner the determinant of the fourth order,
rti bi Ci di
= a^
«2 bi Ci di
«3 63 Cs da
at bi Ci di
bi C2 di
-«2
bz C3 dz
bt^Ci di
bi Ci di
+ «3
ba C3 ds
bi ^4 di
b^ Ci di
bi Ci di
bi Ci di
a^. b^ Ci di
bi Ci di
bz C3 t/g
These relations between determinants of different orders
enable us to expand a given determinant, or to find its value.
Ex. 107. To find the value of
342
I I 3
211
We have,
342
I I 3
211
I 3
I I
4 2
I I
+ 2
4 2
I 3
= 3(1 - 3) - (4 - 2) + 2(12 - 2) = 6
Ex. 108.
3123
= 3
021
-4'
I 2 3
-h6
I 2 3
-7
123
4021
412
412
021
021
6412
301
301
3 I
412
7301
= 3(- 8+9) -4(1-8+3)4-6(2 -12) -7(3 -16) = 50
_i
Cl
C'i
>
K Cl
d.
^2 Ci
d.
bi C4
d.
-65-
The follovvinfj principles established in works on deternii-
nants assist us in the evaluation.
i. If a column or row contains a common factor that factor
may be placed outside and each element in the column or row
divided by it.
ii. Any column may be added to or subtracted from another
column, or any row may be added lo or subtracted from
another row without changinj^ the value of the determinant.
iii. If two columns or two rows be exchanged the sign of the
(letrrminant is changed.
iv. If two columns or two rows be the same the determinant
is zero.
Applying these in evaluating the last determinant, we have,
by bringing the third column first,
which does not change the sign, it
being a double exchange ;
3123
=
4021
6412
7301
2313
2401
1642
o 7 d, T-
2313
=
2 401
2 12 8 4
0731
I - I -2
8 i, 3
7 3 I
,1
4 2
o 5
4 2
-4
3
I
= 2
I 3
I
8
1
- 1
8
7
3
2 I
—
5
4 2
£ 2
5
-4
3
5
3
■2
3
I
by dividing the first col-
umn by 2 and then sub-
tracting the first row
from the second and the
second from the third ;
by subtracting the second
column from the first :
by S'lbtracting the first row
from the last and dividing
by 4-
= 2(5X5-oX3) = 50-
Ex. 109. Given
x =
X -vy +e
{b + c)x +{c + a)y +{a+b)2
bcx + cay + abe
= o
= o
= I.
01 I
c + a a + b
^ >_^
I ca ab
III
b + c c + a a+b
be ca ab
Now if in the second of these determinants we put ^ = c we
— 66 —
obtain two columns alike and the determinant becomes zero ;
hence ^-c is a factor, and from symmetry a — b and c-a are
factors.
.*. the second determinant = - {a-b) {h-c) {c-a).
But the first =6-c; ,.
X =
{a-b){c-a) {a-b)(a-c)'
Similarly, y =
z =
(6-c) {jb-ay
I
{c-a){c-b)*
o,
o,
71. If we have a set of equations which do not contain a
constant term, we can determine only the ratios of the un-
known quantities to one another and not the unknowns them-
selves.
Let aiX tK b-iy + c^z -
a^x + b^y + c^z =
be a set of two such equations.
Put — = w, -^ = «, and they become,
z z.
ayin + b^n -f Cj = o
a^m -f 6a« 4-^2 = 0;
and we see that the unknown quantities to be determined are
m and «, i.e., the ratios oi x : z and ^ : z, or any other two
ratios which we chose to fix upon.
Now, m - — = -
■ ft, ,
C\
61
K
C\
H
62
=
b^
C2
«1
b.
«i
bx
^2
b^
«2
^2
X
z
bx
b^
C2
b.
Cx
Cj,
«1
by symmetry.
Hence x, y^ z may be any quantities respectively propor-
tional to the denominators. This result is practically identi-
cal with that of Art. 68.
67-
Ex. no. To find the ratios a . b \ c when x . y \ z ^ mb
-\- nc - la \ nc •\- la - mb : la -\- mb - nc.
X _ y _ e
Denote the ratio
mb+nc-la nc + la — mb la+mb — nc
— by
l_
V
Then, -la + mb -^ nc — vx ■= o
la - mb ■{■ nc — vy = o
la -f- mb — r.c — vz = o
and considering a, by c, v, as unknowns, we have
a b c
m n-x
-m n-y
m — n—z
-I
n-x
I n-y
l — n-z
and by expanding the determinants, we obtain
a b c
I m- X
l-m-y
I m-z
V
-I
m
n
I-
-m
n
I
m-
-n
V
y
2mn{y+z) 2nl{z+x) 2lm{x-\-y) -/^Imn
.'. a : b : c - mn{y-\-z) : nl{z-\-x) : lm{x-[-y).
X 72. Of sets in which the number of equations is greater than that
of the unknown quantities.
In order that such equations may coexist there must neces-
sarily be some relation among the coefficients. Thus if we
are to have,
ax + by = c
bx -\- ay = 2c
X + y — a -\- b + c,
we must also have {a-\-b) {a-\-b + c) — zo ;
and unless this relation exists the given equations cannot pos-
sibly coexist.
Let aiX + b^y-\-Ci = o, a^-^ b^y + Cj = o, flair -f ^aj/ + Ca = o be three
equations involving the two unknowns x and j'.
Eliminating y between the first and second, and then be-
tween the first and third, we obtain,
X - -
cx b,\
^2 *2 1 = _
Ci bi
C3 bs
«i bi 1
«8 bt
(h bx
<h bi
% I
" I
i
p
m
— 68 —
or, (cidi - Ci^i ) {aibi - a^by) = {aib.^. - aj)^ ) {cib^ - cj?{) ,
And multiplying out, rejecting terms which cancel eath
other, dividing through by /; and arranging, we have,
ciiibj:^ - baPi) + <'a(<^3'"i - ^^1^3) 1 n^ibiC^ - b.fx) = o,
whence, from Art. 70, we have,
o as the required condition of
coexistence.
ax
b.
Ci
tti
bt
Ca
«3
b.
^3
Ex. III. If the equations >> = mA;+/i, y=^miX-\-hi, jy = m2« + //2,
are to exist together, determine the condition.
Here,
o
I m h
or, )n{hx - 112) + n?i(/j2 -h)-\- m^ih - //,) = o.
INDETERMINATE ANALYSIS OF THE FIRST
DEGREE.
73. As stated in Article 62, if the number of equations be
less than that of the unknown quantities an indefinite number
of sets of values may be found to satisfy the equations.
Thus, \{ ax-\-by = c be the given equation involving the two
quantities ;r and ^ we may evidently put any quantity what-
ever for X and find a corresponding value for j/.
In practice the number of solutions is restricted by the con-
dition that the values of x and y must be positive whole
numbers.
Ex. 112. It is require dto pay three dollars in ii-cent pieces
and 7-cent pieces.
Let X denote the number of 1 i-cent pieces and y that of the
7-cent pieces.
Then, I i^r -1-7^ = 300 is the equation.
From this, ;r=3£^Lz7Z= 27 + ^ "^.T
II
II
\
;el eath
ion of
: UliX + hi,
1.
IRST
tions be
number
the two
ty what-
tbe con-
/e whole
it pieces
lat of the
-69-
As ;r is to be a whole number, the expression ^ . ~'/ , and its
II
multiple by a whole number, must be a whole number.
We now endeavor to multiply by such an integer that the
coefficient oiy may be greater or less by unity than some
multiple of II . 8 is such a number, since 8x7 = 56 = 5X11 + 1.
Hence, -xIlAZ = 2 - 5^+ -^ must be a wh. no.
II
II
and
2-y _
II
a wh. no. =/> say.
Then J/ = 2 - 11/), and putting this value in the original equa-
tion we obtain, r=26 + 7/>.
Hence, x=2b^ yp, y=2—i\p is the required solution,
where p may be any integer, positive or negative, which will
give positive values for x and y.
If p = 0-1—2-3
;f = 26 19 12 5
y = 2 13 25 37
which four setj- are all the possible positive integral solutions.
Any other integral values for p would make either *" or _y nega-
tive, which is not consistent with the original condition.
Ex. 113. It is required to find a number which when divided
by 3 leaves a remainder 2, divided by 5 leaves 3, and
divided by 7 leaves 5.
Let X be the number ; then,
f , — 1^, ? must all be whole numbers.
3 5 7
Put ^^=/> .-. * = 3/' + 2 ;
3
and writing this for x in the second fraction,
.3r ~ ^ must be a whole number.
\
/. P ^ — q must be a whole number,
.*. /> = 5^ + 2 and ^ = 159 + 8 ;
i
' . — 70 —
and this in the third gives,
-M~^, or ? — ^= whole number = r ;
7 7
.'. ^y = 7r - 3 and ^=i05r- 37,
where r may be any positive integer whatever. Making
r=i gives 68 for the smallest number satisfying the required
conditions.
74. If we have ax+dy=c an indeterminate equation of the
first degree, it is readily seen that by increasing x, y may be
made to pass through zero, and conversely by increasing y,
X may be made to pass through zero. If then negative values
of X and y are to be excluded, x cannot be greater than — nor
a
less than zero, and hence the number of solutions is necessari-
ly limited.
But i{ax-dy=c be the equation, an increase in the value of
X must be accompanied by un increase in that of y, and as
both may be indefinitely increased the number of solutions is
quite unlimited.
75. In the equation ax±dy = c, a, b and c cannot have a
common factor, for we may divide throughout by such factor
and thus get rid of it.
Again, a and h must be prime to each other, for if they have
a common factor, it must also be a factor of ax±by ) but as it
is not a factor cf c, the equation cix±by-c is impossible.
Thus 2;r--|-ioj/ = 3i cannot have an integral solution,
76. In Ex. 112 we found for values of ;rand y,
x=: 26 -\-yp, y = 2~iip.
Now it will be noticed that the coefficient of/) in the value
oix is the coefficient of^ in the original equation ; and sim'-
larly the coefficient of/> in the value of _y is that of x in ti e
original equation. This may be proved to be always the
case.* Hence if ax + by = (' be the original equation, the
values of x and y may be written, x = a± bp, y -i^ + ap, where
a and (3 are fixed quantities, wliicli bolve tlic equation when
p = 0.
DemouBtrationB of tbiH kiud beloug to on aclvoiiced courBe of Algebra.
Making
required
on of the
' may be
leasing y,
;ive values
an
c
a
nor
necessan-
le value of
y, and as
:>lutions is
)t have a
ich factor
they have
but as it
npossible.
the value
ind sirr"-
in tie
ways the
ition, the
ap, where
ion when
— 71 —
If then one solution can be determined by any means, all
the other solutions may be obtained at once.
Thus, if we find one solution of Ex. 112 to be x- 12 and
J/ =24, we have x-\2-\-jp, ^' = 24 -np as general formulae,
and by making />= - 1, o, i, 2 successively we get all the pos-
sible solutions. "
If the equation be a;r-^^= c, we have only to change the
sign of b in what proceeds.
77. In Ex. 113, the coefficient of r in the value of x is the
L. CM. of the three denominators, 3, 5 and 7. Hence if /
denote this quantity the value of x may be written,
x=Y + lr ;
and if one solution (y) can in any way be found, others will be
obtained by adding on multiples of /.
SIMULTA NEOUS QUA DRA TICS.
78. If an equation contains two unknowns, its degree is
measured by the term of highest dimensions in these un-
knowns.
Thus, 2;»r+3;ry -1-4 = is a quadratic since the second term
is of two dimensions. In like manner \fx,}f, z, be unknowns,
;jr2 -fj^ = o is a quadratic, ;r2j/ + 5^2 -(-^'^ = i? a cubic, xy'^z-\-z^
- 2xy = o is a quartic, &c.
The most general quadratic in ;r and^ that can be written
is, ax'^ -f bxy + cy"^ +dx + ey +f— ;
and the most general in x, y and z, is
cix"^ + dy^ -(- cz"^ + dxy -f exz +fyz + fx^- hy -j-kz + l =0.
79. In genera] the elimination of an unknown between two
(juadratics produces an equation of a higher order ; but if one
of the equations be linear the resulting equation will be still a
quadratic.
In any case elimination between two quadratics cannot pro-
duce an equation of a degree higher than the fourth. As a
consequence the solution of simultaneous quadratics may re-
i is I
— 72 —
quire finally the solution of a quadratic only, or of a cubic, or
of a quartic. Th« problem may, therefore, admit of two, three
or four solutions depending upon conditions.
Solution of simultaneous quadratics is often effected by in-
genious combinations and artifices rather than by any fixed
principles of elimination. These artifices are best learned by
observation and practice.
TWO EQUATIONS WHEREOF ONE IS A QUADRATIC
AND THE OTHER A LINEAR.
m
li
80. The solution of these is effected by substituting in the
quadratic the value of one of the unknowns as derived from
the linear equation.
Ex. 114. Given, att^ ^by^+cxy ■\-dx-\-ey+f=o,
and }nx-\-'ny+p=o.
From the second equation, x = - ^ — — .
m
And this value in the first gives, after reduction,
yian^ + 6W - cmn) +y{2apn - cpm - dmn + em^) -\-ap^ — cipin +fm'^
= ; a quadratic in y.
Ex.115. Given s^-2jf^-\-xy-y=i,
and 2;r-3_y=i.
Here, x = ^ — -, which in the first gives,
2
1(3^ + 1)^-2/+ -^ (3)' + i)->'="
2
From which we obtain, v = i or — 4_ ;
45
and thence, ;r = 2 or - ^ .,'
25
81. If the quadratic equation be divisible by the linear the
equations are equivalent to a pair of linears only, and x and ^
have but one value each.
— 12^ —
Ex. ii6. Given 3^''- 5^J'-2y= 17, 3
x-2y = i.
The first equation is {x- zy) (3:r-f y) = 17.
I^iit x-2y-\'^ .'. ^x+y=iy.
Whence, ^=5,^ = 2.
If we solve this l)y substitutinjj from the second equation in
the first we obtain,
x=i + 2}'; .'. 3+ 12^'+ 12^-5^- 12^=17
or ^ = 2, one value only.
82. Sometimes equations may be solved by combining them
in some simple manner.
Ex 117. Given x"^ + y'^ = 13
X + V = 5
Subtracting the first from the square of the second we have,
zxy = 12;
and subtracting this from the first, we jj:et
{x -yf = I , or ;r -J/ = I ;
.-. x = ^,y = 2..
SIMULTANEOUS EQUATIONS CONTAINING TWO
QUADRATICS.
8-^. It is not always possible to solve tl.ese as quadratics,
and experience is usually the only guide as to whether it is ()os-
sible or not.
Ex. 118. Given 2x'^ + ^xy = 26,
3/ + 2xy = 39.
Here, 2X^ + ^xy = x {2X+$y) = 26,
and 3/ -\- 2xy = y {2X+^y} - 39;
V 2 2y
.'. dividing L = -, and x = -=^.
^3 3
Putting this value for x in the first,
— 74 —
^l + 2y'i = 26.
9
Whence, ^ = ± 3 and x = ± 2.
84. If the terms involving the unknowns he homopfeneous.
we may advantageously obtain a third equation in which the
unknown quantity is the ratio of one of the original unknowns
to the other.
Ex. 119. Given, x^ -^ xy -{■ 4^ = 6,
Zx^ + 8/ = 14.
Let — -V
x = vy.
Then ^'^+^+Ay'^ - '^'^y'^^'"y^+'\y'^
_ t;^-ft>+ 4 _ J_
3^2 + 8' 7'
whence we find, v = 4 or — ^ ;
and writing x=^y in the second equation gives,
}> = ±^, and .*. x= ±2.
If we take the other value of v and write y= - 24: we obtain
;r= ±y^.
,, _ 2l/l0
^ = + "^
5 5
Hence x andy have each four values all of which satisfy the
equations.
85. If the equations be each symmetrical with respect to the
unknowns, it is frequently of advantage to employ two new
unknowns, one of which is the sum and the other the difference
of the original unknowns.
Ex. 120. Given x^ +y'^ +x+y = 8,
X +y +xy = ^.
Put x=u + v, y = u-v ; then the equations become,
2_-,2
Adding, 2u^+s^ = 9 5
whence, w = f or - 3.
— 75 —
With thise values of u we find, {
when «/ = f, v= ±^, x — z or i, y-\ or 2.
when M=-3, i;=±j/-2, x- -^±\/ -2, y- -3+v^-a.
Hence 4: and>' have each four values, which give four pairs
satisfying the given equations.
»n the present example, as in all cases where 4r and j' are
symmetrically involved, their values are interchangable.
86. The substitution of the last article may sometimes be
employed where the equations are not strictly s> mmetrical in
X and^.
u
Kx. 121. Given, x^-^^x"^ -y^ = — Wx+y +\/ x-y ).
(x+y)^ - {x-y^ = 26.
Put x+y=2s^,x-y = 2t'^.
The equations become,
l/8(s8-/3)=26 ^
From a we get at once,
or s3-/3-fs/(s_^)=8t/2 r
Substituting for .s* - 1^ from /9 in y, we get
s7(s - = -^ ^
1/2
j9H-tf gives, ■ =— ,
St 3
• Art 27 (^ + ^)'-i6 (s -/)=» _ 4 .
.*. —— = 2, and 5 = 31.
s-t
Whence we readily find, s = -^, « = — 7-, and
V 2 V^2
hence x=^, and^ =4.
■' 1 1
76-
INEQU.lLITIl'S.
87. An equation declares that there is equality hetween its
two members, but a non-equation or inequality declares that
one of its metnbers is greater or less than the other; and the
problems which present themselves in inequalities usually
require us to prove that one expression is greater or less than
another.
Since the square of a quantity is always positive, (x-y)'^ or
x"^ ^y"^ —2.vy is a. positive quantity whether x be greater or
less than y.
Hence, x"^ +y^ is greater than 2xy ; or expressed symbol-
lically.
The proof of a large number of inequalities depends upon this
principle.
K x = j^ the inequality becomes an equality
The following principles are important :
If a > ^
nb,
and
a
n
>
b
J
n
but
n
a
<
n
r
2. a + c > b + c, and a — c > b — c;
but, c—a<Cc-b.
3. If a and b be both positive,
l/a > i/b and a" > b"^,
but
a'
<b-
4. If both sides are divided by a negative quantity
the character of the inequality is reversed.
122. a^-^b^-\-c^ +
'^ ab + be + cd -\-
For, a^ + b^ > 2ab, b'^^c'^> zbc &c.
een its
es that
md the
usually
:ss than
-J/) 2 or
ater or
symbol-
pon
this
luantity
Id.
— 77 —
.'. 2rt*-|-2/'+ ... > 2ah-\-2bc-{- ... J
.-. a^ hb^+ . . . >ah + bc-\ rd-i- . . .
Ex. 123. I'or the same hasu and perimeter the area of an
isosceles triangle is greater than that of a scalene one.
Let s = ^ peri meTerot eacTiT^nd b= the commonbase.
Also, let a, c be thf sides of the scalene triangle and e the
side of the isosceles one.
Then, ^, = area of isosceles = I s(s~e)\s -6),
and A,^ " scalene - ]^s(s -a)(s-b)is -c).
.'. A^^A^ AS {s—e)^^(s-a)(s-c).
>
<
as e'-zsc -^ ac -s(a + c),
as e'
>
<
ac,
since a-\-c = 2e ;
ac.
as, a' + c" ^ 2ac.
But rt'-hc" > 2ac
Ex. 124. x ^-\-y^ > x^-\-xv_^ .
"I
x^ ■\-y^ -^ x^y+xy^.
as {x^-y^){x y) ^ o.
But if ;r >^, both factors are positive and their product is
positive and therefore > o.
And if ;r < ^, both factors are negative and their product is
positive and therefore > o.
I:::
^
f:
1 ■
if
s
■ 1
78
SERIES.
88. A succession of terms formed according to some regular
law is called a series. If the number of terms be limited the
series is finite, but if unlimited it is infinite. Series may be
formed or developed in a number of different ways, one of
which is given in Art. 9. Their study is important inasmuch
as in many cases we are compelled to employ them. We have
examples of what are the sums of the first few terms of well
known series in logarithms, sines, &c. The law of formation
of the terms of a series, or the "law of the series," may be very
simple or very complex.
The simplest series is one in which each term differs from
the one before it by a constant quantity. Such a series is
termed an equi-difference series, an arithmetic series, or an arith-
metic progression. ^^
OF ARITHMETIC SERIES.
-*
89. The quantities with which we have normally to deal in
an arithmetic series are a, the first term ; », the number of
terms ; d, the common difference between consecutive terms ;
z, the last or n^ term ; and s the sum of » terms.
Having any three of these we can find the remaining two by
means of the relations which we proceed to develope.
Let a, a-hd, a + zd, a + ^d, &c., be the consecutive terms
of the series. Then it is readily seen that the n^ term is
a-\-{n — i)d;
.'. 2 = a+{n -i)d. . . . (A)
To find S.
S=a-\-(a-^d) + {a\qd)-\- , . . .(a + fT^.d).
and reversing the order of the terms,
S-{a + n — i.d) + ia + n — 2.d)+ .... +a
• /
adding, 2S= (2a +» - i.d) + {2a-i-n- i.d)+ .... to » terms,
regular
ted the
may be
, one of
lasmuch
Ve have
of well
irmation
r be very
srs from
series is
an arith-
deal in
imber of
terms ;
» two by
e terms
term is
— 79 —
= »(2«+«-i.i);
.-. 5 =— (2« + «^ J) (B)
2
Formulae (A) and (B) involve all possible relations among
the five quantities given above.
Ex. 125. Given ^=13, <i! = 3, w=5, to find S.
From (A), I3=:a-fi2 .'. a-i ;
then from (B), S= -5-(2 + i2) = 35.
2
Ex. 126. A falling body descends ^feet in the first second,
z
Al. in the second second, ^ in the third and so on ; how far
2 2
will it fall in the n*^ second ? How far in t seconds ?
Here, a — J—,i-f^
2
.'. z^ -L^[n-i)f=znf- JL^f{n-^)
2 2
= the distance in the «*'' second,
and s = A(2.i-^7i:7^y^^^^2^
\*
— distance fallen from rest in t seconds.
90. Multiplying out Art. 89, B, we have,
S = n{a - — ) +«* . — .
2 2
terms,
Hence, unless d be zero, an expression giving generally the
sum of an arithmetic series must involve the square of the num-
ber of terms ; and unless d = 2a it will involve also the first
power of that number.
Thus, — - — expresses the sum of n terms of some arith-
2 3
metic series. To find it ;
S ^ — iza + n - 1 . d) =^-— , must be true for ail values
2 23
of n since each is a general expression for the sum.
8o
Let «=i ; .'. a = ^-^ = ^,
** M = 2; .'. 2fl + ^ = 2-f = f .*. i/=I,
and the series is,
Ex. 127. The sums of two A. P." are as 11 -5« to 11 +3M, to
find the ratio of their sixth terms.
Let a, d, s denote the 1** term, the common diff., and the sum
of w terms in the first series ; and a^, d^, s^ denote Hke quan-
tities in the second.
Then, s and r.^ may be expressed by «(ii - 5w) and m(ii + 3«)
respectively.
.". — (2u +;i - I .d) = n(ii -<sn) and - (2^1 + ?? - i
2 2
= w(ii+3«).
Hence, by giving values to n as above we obtain,
a = 6, d= - 10, and a^ = 14, d^ =6 ;
and the ratios of their sixth term is,
d,)
a + 5^
50
44= _i.
«i +5^1 14 + 30 44
gi. If the number of terms be the unknown quantity we
may have a quadratic in n, and the problem then admits of
a double solution. In some cases both values of w are equally
applicable.
Ex. 128. In an A. P. a = 7, and rf= -2, to find how many terms
will make 12 when summed.
s = n2.— + w(a- — )= ~n^ +8n = i2 ;
2 2
n =
_8±l/64-48
= 6 or 2.
Ex. 128'. In the A. P." 6, 7^^, 9 . . . . and -3, -i, i . . . . ,
(i) discover if there be a common term, and if so its
value ; (2) if there be a common number of terms for
which the sum of the terms in each series is the same,
and if so find .it.5 value.
— Si-
ll) Taking the expression for the «*^ term, we have, if n is
a common term.
a -\-n - I .d=ai -\-n— I .d^,
or 6 + (;j-i;f = -3 + («-i)2. ~ ^
whence, « = 19 ; and the 19*'* term is common.
Its value is 6 + 18 X f = 33.
(2) Taking the expression for the sum, since s and n are to
be common,
s = — {2a + n- I . d) = — (2ai +n- 1 . d-,),
2 2
.'. 2a-\-n - I . d = 2a^ +11 -1 .d^,
or i2i-(w-i)|-= -6 -!-(;/ 1)2.
Whence « = 37 ; and the sum of the first 37 terms is the
same for each series.
The value is, -^if (I2H 36Xf ) = 1221.
If in the above n were fractional there could be no common
term, as the number of terms nmst necessarily be integral.
Since we divide by n, 11=0 is one solution : but this would
be excluded by the nature of the problem.
92. When three quantities form three consecutive terms of
an A. P. the middle one is said to be an arithmetic mean be-
tween the extreme ones.
If then A be an arithmetic mean between a and h we have,
A -a-b- A, and .'. A ~ — .
2
Hence the arithmetic mean between two (juantities is the
half sum of the quantities.
93. The sum of n consecutiv natural numbers counting from
unity is, — ^ '.
2
The sum of 11 consecutive odd numbers i, 3, 5, &c., is, n^.
The sum of n consecutive even numbers 2, 4, 6, &c., is
^1
f i
it
If
I'
!i
■ifi
:i'
- 82 -
94. A sum of P dollars is put to simple interest for t years
at r per unit per annum.
The interest at the end of the first year is Pr
2nd year is zPr
3rd year is ^Pr
ft
<(
<«
<l
«<
u
« • •
...
((
II
11
/*'' year is tPr,
.'. the whole amount at the end of / years is,
A = P + Prt = P{i-\-rt).
Ex. 129. A sum of P dollars is deposited yearly with a
banker to be left for t years from the date of the first
deposit. To find the accumulated amount at the end
of the period.
The first payment draws interest for t years and = P{i-\-rt)
The second " '* ** /- i years and =P(i-f-r./— i)
Payment before the last draws interest for i year =P{x-\-y)
Last payment draws interest for o years = P
.'. Amount =P + P{i+r)+ .... -\-P{i+rt),
= Pii+t)+Pyii + 2-^ .... 0,
= Pii+t)+PrJSJ^JK
2
= --(1+0 (2 +»'^).
2
GEOMETRIC SERIES.
95. When th« ratio of any term in a series to the preceding
term is a constant •'luantity, the series is called an equimultiple
series, a. geometric setip,§, or a geometric progression
The quantities w^tb w* '*jcM we have normally to ' ' are a
the first term, r the commo» ratio, n the number of terms, z
the »*' term, an/J > the sum of n terms. Any three of these
bein^ given th« remaining two may be found by the relations
now Uj be devfeioped.
t years
r with a
■ the first
t the end
i+rt)
:+r./-il
• • •
)receding
\uimultiple
' are a
terms, z
of these
1 relations
-83-
Let a, ar, ar', at^, &c., be consecutive terms of the series.
Then it is readily seen that the n*^ p^rm is (ir^'^>
.'. z = ar^ ^. . . . (A)
To find S.
Multiply byr, rS= ar + ar'+ .... ar "•-j-ai'°"^ + «y*
Subtract , S{i—r)=a—ar"
J^^-Vv^ v-***^^^***
ar"" +ar'
n-l
«.n
.'. ^~ci. . . . . {B)
I ~r
Otherwise as follows :
By division, =rt+ar + «r'-}-
I -r
ar-'+-fn
.-. s = a-\-ar +
ay"-' =
a
ar"
I - r I - r
= a.
r -r
Formulae (A) and (B) involve all possible relations amongst
the five quantities given above.
Ex. 130. The population of a city increases at the rate of
5 per cent per annum, and it is now 20000. What
was it 10 years ago ?
In this case, since the series is a decreasing one r is a frac-
tion, viz.: , a = 20000 and h = ii, as there are 11 terms
to find z.
From (A), z = ar'"' = ."^"""^"^ = 12422 nearly.
1.05
,,., 20000
(1.05)^°
Problems in Geometric series involving r or n as unknown
quanties cannot in general be conveniently solved without
logarithms.
96. If in (B) r is less than unity, r" may be made as small
as we please by taking n sufficiently great. The liwii then to
which s approaches as a becomes indefinitely increased is,
, and this expression is usually taken as the sum of the in-
I -r
finite series in which r is less than one. It must be borne in
*^
^ ^ 84 -
mind, however, that no number of terms which we could ever
take would by summation be as great as , for as the num-
ber of terms is infinite there must always be a remainder ;
but by taking a sufficient number of terms we may make their
sum approach the value of as near as we please while we
can never make that sum surpass it.
Ex. 131. To find the value of the repeater .36.
This is equal to i^jfg- + nf^j^iy 4- ... ad infinitum.
and
s =
a
I —r
= 36 _t. ^ T _ l \ ~ 3 6 y JJ) 0. - ol?
n
Ex. ^ J2. The series, i + — - +
ft'
For s =
n-t I {n + I)''
»'■' n ^- 1 ■
+
ad infin. =11 + 1
I -
n
n + i —n
— n-i 1.
n-\-i
If « = i, 1+-^ + ^+ . . . .
»=2, I+l + f + . . . .
n=3, 1+I + A+ • • •
&c., &c., &c.
= 2,
= 3.
97. In any three consecutive terms of a geometric series the
middle term is called a geometric mean between the extreme
terms. ., .
Prob. To insert a geometricmean between two given terms.
Let a and b be the given terms, and g the geometric mean
required. Then, since a 7, b : re to form three terms of a
geometric series, we must have
-^ = — and
a
g
g-=l/au
Hence the geometric mean between two quantities is the
square root of their product. (Compare Art. ^^ where it is
called a mean proportional.)
The side of a square is a geometric mean between the
-85-
sides of the equal rectanj^le. For if a, b be the sides of the
rectangle, and s that of the square, area = fl6=s^
98. Prob. To insert n terms between two given terms so as
to fornn a geometric series.
Let a and b be the terms, and let the completed series be,
^ n> ^a» 'Sj • • • • *n» *•
Then, A-^'^'3_
But
b_
a
a tx
U
= ;'°+^ there being n-fi factors.
••■ -(7)
^» \"+i
And
1^
n+l
(h \"'r-i
— ) = (fl"6)n
I
1+1
U - ar^ ■■
\ a
iy+' = (fl"-»62)n4.1
&c., &c.,
99. If a sum of P dollars be put at interest for one year it
amounts to P( I +>') dollars. If this be now taken as a new
principal and be put at interest for another year it amounts to
P{i +r){i +r) or Pii-i-r)'^. Similarly in three years it will
amount to P{i+r)^ ; and in t years to P{i+ry dollars.
Therefore if A denotes the amount we have
A=P(i+jf.
which is the fudamental formula in compound interest.
It is evident that the amounts at the ends of successive
years form the geometric series,
P(i+r), Pii-\-r)^, P(i +r)3, . . . Pii+r)\
Ex. 133. n annual payments of P dollars each are made into a
bank to remain at compound interest. To find the
total amount due at the date of the last payment.
Let R denote i +r.
>-^^^. (T
86
The I St payment remains « - 1 yrs.
2nd
n - 2
its amount is Pi?"**.
Pi?"-».
• • •
((
last " " o " .-.
/. The total amount is F{i+H-\- .... i^°-'),
7^" - I
P.
A=P.
or
K-i
p (i + y)°-i
This gives the amount of an annuity which has been fore-
borne or left unpaid for a period of n years.
To find the present value of such an annuity, or the sum
which when put to interest will produce its equivalent, we
have,
' 0- -
/?» R^ (i+r)"'
Ex. 134. A corporation borrows P dollars to be paid in n equal
annual instalments, each instalment to include all
interest due at the time of its payment. To find the
value of the instalment.
Let P denote the instalment and a, b,c, &c., the sums paid
in successive years upon the principal.
Then, ist payment =p = a + Pr,
amount unpaid = P -a ;
^ 2nd payment =p = b + {P — a)r, whence b=aR,
amount unpaid =P -a -b=P - a~aR ;
3rd payment =p = c-\-{P — a-aR)r .'. c = aR^,
amount unpaid = P-a — b -c, &c.
Similarly, n^^ payment =p = aR'"'^, * '
amount unpaid =P-a-b c- &c.,
= P-a-aR-aR^ - ... -aR'^'K
But the amount unpaid after the last payment must be
zero ; hence,
P-a(i+i? + R2+ R"-*)=o,
PR""-'.
• • •
P.
sn fore-
:he sum
lent, we
n n equal
elude all
find the
ms paid
aR,
aR^,
iR''-\
must be
-87 -
« - 1 fi^-i ii" - 1
Hence, 6 = -fl— + Pr = P.-!^.
HARMONIC SERIES.
100. A number of terms is said to form a Harmonic series
when the reciprocals of the terms form an Arithmetic series ;
so that if the reciprocals of the terms be taken in any arith-
metic series we have a Harmonic series.
Thus I, 3, 5, 7, 9, is an Arithmetic serie?,
and I, ^, ^, -f, ^, is a Harmonic series.
Let a, b, c be three terms in Harmonic Progression ;
I I I
then
a
I
T
h
I _
a
. a-b^
a
or a:c :: a
are in A. P., and consequently,
= the common difference.
c
b:b
c.
And three terms are in Harmonic progression or series, or
they form a Harmonic proportion when the first is to the third
as the difference 'between the first and second is to the difference be-
tween the second and third.
This is frequently taken as the definition of Harmonic Pro-
portion ; and a series of terms in which any three taken con-
secutively form a Harmonic Proportion is a Harmonic series.
Problems in H. P. are best solved as problems of A. P. by
means of the relation given in the first definition of a Har-
monic series.
Ex. 135. To find a Harmonic mean between A and B.
Let H be the mean. Then,
I
-^, — are to be in Arithmetic proportion,
■m
I
H
— 88
I I I ,2 I.I
"'^aTh-
J- /I V
loi. Harmonic proportion is so nameil on account of the
similarity which exists between its terms and the relative
lengths of a trinj,' which sound the harmonics in music.
Its chief application, however, is in Geometry.
Let A, X, B, Y be four points \y i ^
in aline. Then AX, AB, AY form A— — X -^ — B Y
three magnitudes which may be
taken as terms of a harmonic proportion, if AX is to AY as the
diffCi nee between AB and AX is to the difference between
AYai.dAB; i.e., if AX: AY :: BX: YB.
The points A, X, B, Y are then said to form a harmonic
range, and the line AB is said to be harmonically divided in X
and Y. The properties of harmonically divided lines is an im-
portant one in modern geometry.
VARIATION.
If^'^%;
I02. When 1 vvo quantities are so connected that a change
of value in one is accompanied by a change of value in the
other, in such a way t'lat their ratio remains constant, one of
the quantities is said to vary as the other. Variation is usually
denoted by the matk c/i , and is only a kind of geneialized
proportion.
A
If ^ to B, then — ---constant = « suppose
B
.'. A =nB.
Hence when one quantity varies as another they are con-
nected by a constant factor.
i. \i A sinB, A varies dir.fcctly as B.
ii. Ifv4 = — , /I varies inversely as B.
iii. If ^ =niBC, A varies jointly as B and C.
C-c
It of the
B relative
in music.
B
-Y
AY as the
e between
harmonic
videcl in X
s is an im-
a change
lue in the
nt, one of
1 is usually
:eneialized
-89-
iv. {{ A— m --, A varies directly as B and inversely as C. .
v./
Ex. 136. The space passed over by a body falling from rest
varies as the square of the time, and experiment
has shown that it descends 64 feet in 3 seconds.
Find the relation between the space and the time.'
S CO /'* we may write S = nt^.
But when / = 2, 5 = 64.
.*. 64 = 4» and « = 16.
d its attraction
AS the square
iiuinher of beats
Ex. 137. The earth's radius is 4,000 mil
upon a body without it varies inv.
of the distance from its centre. T
which a pendulum makes in a day varies as the
square root of the earth's attraction upon it. How
much would a clock with a seconds pendulum lose
daily if taken one mile high ?
Let ^ = the earth's attraction at its surface, and r = the
earth's radius. Then,
g (o
But if « = the number of beats per day at the earth's sur-
face, and «i at the height of one mile,
n c<o |/> Co — .', n = — , where a is a constant;
r r
.". a=rn; and «i = — , where r^ =4001;
are con-
.'. n =n — ;
and the loss = « - «i = «.-i — = 86400 x
4001
= 21.59 seconds.
- -, A , . ^ nA
103. Let C vary as-77 ; then we may write C = -r^.
B
B
Now if C is constant, A must vary as B ; and if B is con-
stant A must vary as C. But multiplying by B, BC=nA;
and therefore A varies as BC.
pi
^.
IMAGE EVALUATION
TEST TARGET (MT-3)
1.0
1.1
IttlM 125
gKi 122
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lU
■4.0
1^ ilM \m
1.6
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/.
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23 WIST MAIN STRIIT
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5r
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^
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-~ I \ — go —
Hence if ^4 varies as B when C is constant, and varies as C
when B is constant, it varies as BC when both are allowed to
change.
Ex. 138. It is proved in Euc. vi. i, that the area of a triangle
varies as its base when its altitude is unchanged ; and
similarly it varies as the altitude when the base is
unchanged ; hence it varies as the product of the
base and altitude.
If then A denote the area, 6 the base and p the altitude,
we have {S ^ bp ; and hence A = nbp, where n is an un-
known constant. Now the right-angled triangle whose sides
are each i is one-half the square of which its hypothenuse is a
diagonal, and therefore its area is ^ ; .•. « =^, and A =^^bp.
If the three sides of a triangle vary so as to keep all their
ratios constant, the triangle remains always similar to a given
triangle.
In this case /> c/» 6 and hence we may write p - mb, and
therefore /^,^^mb^ ; i.e. the area of a triangle varies as the
square of one of its sides when the triangle remains similar to
a given triangle.
PERMUTA TIONS— VA RIA TIONS-
NATIONS.
■COMBI-
104. If a number of objects be taken and formed into groups
such that the relative positions of all the objects are not the
same in two groups ; then, |if each group contains al l the -Ob-
jects concerned it is called a permutation^', but if it contains
only a certain number of objectsTless than the whole, it is a
variation.
If the groups are such that no two groups contain the same
assemBlage of objects, each group is calfea a combinations
Frequently jip distinction is drawn between variations and
permutations , and it is readily seen that the permutations are
only the variations in a particular case.
For this reason, and because the word variation has already
been used in a different sense, we shall employ the word
permutation for both.
— gi —
PERMUTATIONS.
itains
is a
same
[ready
word
105. Take two letters a and b ; the permutations which ckn
be made out of these are ab and ba, i.e. two.
Take three letters and we have, abc, acb^ bac, bca, cab, cba,
or six permutations.
Similarly four letters will give us 24 permutations.
But 2=1.2, 6 = 1.2.3, 24=1.2.3.4 ;
From analogy we infer that with n letters the number of
permutations is expressed by 1.2.3 ••..».
io6. Let there be 4 letters a, b, c, d, and let us take only
two at a time ; then we have, ab, ba, ac, ca, ad, da, be, cb, bd,
db, cd, dc, or 12 in all. But 12 = 4.3.
In like manner if three letters out of the four be taken at a
time we would find the number of permutations expressed by
4.3.2. And if we employ 5 letters, taking three at a time,
we have for the number of permutations, 5 . 4 . 3 or 60.
Hence from analogy we infer that the number of permutations
of n letters when r are taken together is expressed by
«(« - i)(«- 2) .... to r factors.
We propose to show that both of these inferences are
correct.
107. Let a, b, c . . . . n he n different letters, and let us
adopt the symbol nPm to stand for " the number of permuta-
tions of n letters with m letters in a group."
(i.) If we place only one letter in a group we can evidently
have n groups and no more ; .*. «Pi =«.
(2.) Put a aside and we have « — i letters left; and these
taken in groups of one give « — i groups. Now place a before
each one of these letters, and we have m-i groups of two
letters in which a comes first. Similarly by operating on 6
we will have n - i groups of two in which b comes first ; then
«- 1 in which c comes first ; and so on. But there are n dif-
ferent letters to come first, and each of these gives us « - 1
groups ; .'. the whole number of groups of two letters will be
«(«-i). .'. nF2-n{n-i).
\
:
'1^^*
92 —
(3). Setting a aside again we have « — i letters left; out of
these taking two at a time we may form (« — i) (« — 2) groups.
For if n2r = n{n—i), then («- i)P2=(n - i) (« - 2).
Now put « before each groug, and we have (n-i) (» — 2)
groups of three letters, with a first ; and a like number with b
first, and with c first, &c., and as there are n letters to stand
first the whole number of groups is n{n - i) (n - 2).
.*. «P3=«(m- i) («-2); and the law is manifest.
Suppose this law holds for r things in a group, then nPr
= «(» - i) (« - 2) . . . . to r factors.
Putting a aside we have » - 1 letters, and these taken r
together give (« - i)P/' = (« - 1) (» -2) . . . . to r factors. Now
putting a before each group we introduce an additional letter
and thus have r-\-i letters in a group. Hence there are
|(«-i) («— 2) .... to y fact.[ groups of y-f-i letters with a
standing first. Similarly there is the same number with b
standing first ; with c standing first; and so on. Hence there
are«|(«-i) («-2) . . . .' r fact.[ groups of r-f-i letters alto-
gether. Or
nF{r-\-i) =n{n - i){n - 2) . . . .r + i factors,
since we introduce the additional factor n.
If then the law holds for n letters taken r together, it holds
when taken r + i together. But it holds when r = 3, and
therefore for r-\-i or 4, also for 4+1 or 5 and so on for any
number. .*. generally,
nPr = nin-i) jn-z) . . . . (n-r-{-i) . -r^
Making r equal to n we have for the number of permuta-
tions of » things when taken all in each group,
tiFn, or simply P =w(» - i) (- ;)
= 1.2.3 ••••»•
3-2.1
108. The continuous product of « consecutive natural num-
bers beginning with i is called factorial n, and is indicated by
the symbol «! or | n. Thus 4! or | 4 means i .2.3.4.
Taking the formula nFr = n{n- i)(»- 2) . . . . (»-r + i), and
multiplying and dividing by i . 2 . 3 . . . . (» -r - i)(» -r), we
have,
t; out of
) groups.
I) (» - 2)
er with b
to stand
then Mpr
e taken r
rs. Now
inal letter
there are
rs with a
sr with b
;nce there
iters alto-
■, it holds
= 3, and
for any
permuta-
iral num-
icated by
3 •4-
+ i), and
-r), we
'jv
-vn
/l-z
O"*""^
'^tJt^' -HUx-^M^'^
93
„Py- »*(«-i)(»-2) {n-r^i){n-r) ....3.2. 1 .!
I . 2 . 3 . . . . (» — r)
n
or «Py = — = . . . . B.
n-r
Making r = « in Art. 107, A, we have for the number of per-
mutations when all the articles are included in each group,
P =
n.
n
But making r = « in B, we have P = , — .
o
Hence we must
interpret | o as meaning unity.
Ex.139. If »P4:(« + 2)P5::3:56, find n.
«(» — 1)(«-2)(«-3)
— TS 5
(n 4- 2)(» + i)n{n - i)(m - 2)
••• 56(w-3) = 3(« + 2)(« + i) ;
whence « =6 or gf.
Of which, although both numbers satisfy the condition, the
integer only will apply to articles.
log. // u of the article^i be alike. If the u articles were all
different they would give rise to | u permutations, each of
which could be combir.ad with each permutation from the re-
maining articles, and this would give the complete number of
permutations of n different objects taken all together.
If we denote the number of permutations of » articles taken
all together, of which u are alike, by P(w) we have
P(w^ . |_w=P= |_«; and .'. P(m) =
Similarly if v other articles be alike,
P(«) (v) = _ '— .
n
u
r
u
V
Ex. 140. How many permutations can be made from the
letters in the word Ontario ?
Here « = 7 and « = 2, since there are two O's ;
|.-. P(2)= 7:5:5^^^ = 2520. //
|fi-.i
ill :1
M
11 i
III
94
COMBINATIONS.
.1 ^'
1 10. Let «Cr denote the number of combinations of n things
taken r together. Then from the definition of a combination
each one would give rise to | r permutations. For abed forms
only one combination however you arrange the letters, while
it can give 1.2.3.4 different permutations. lIHence, (tfcLe^nunL-
ber of combinations} X (the number of jgermutations whi ch can
^ made from eac^h^ combinat ion ) = tITe totaF number
tations
that is, nCrX I r =
n
n -r
; Art 108, B.
n
'. nCr =
ii-
\ r \ n-r
r
• • • • v^
This may be put in another form ;
I n _ n{n - i) . . . . («->'+i)(» ->•) .... 2.1
n—r
(» -y) .... 2.1
= n{n- i) . . .. . (n—r + i)
= n{n— 1) .... to r factors ;
And I r = 1.2.3 • • • • to r factors ;
. to r factors . . . . D.
r- n n-1 n — 2
.if
From this it appears that the product of any n consecutive
integers is divisible by factorial m, since »Cr must necessarily
be an integer.
Ex. 141. How many different guards of 4 men can be chosen
from a company of 1*0 men ?
Here « =? 10, r=4 ; .*. ioC4 = — .^ . - . ^ = 210.
12 3 4
III, If in Art. no, C, we make n — r—p, we have r=n-p,
(^■..V 9:
llo-
(V^vvyt^ -'Jll_
^— ''^/--'/j
K^.S.
Yi I
n things
nhinaiion
*cd forms
rs, while,
lie_num:
;hi ch can
if permn-
.D.
nsecutive
ecessarily
V
95
and nCn —p =
n
n-p\P
and substituting r for p,
' n
nCn -r =
I r I n — 1
nCr.
)e chosen
j
.
J
r = n-p,
^JL-Yl 1
V
Hence the number of combinations of n things taken r to-
gether is the same as that of « things taken n—r together.
This must necessarily be true for the following reasons : —
When from n things we take out r to form a combination, we
leave another combination of » - r things, and therefore the
number of each must be the same. These are called supple-
mentary combinations.
Thus 6C2 = A.A=i5 : 664= — .-5-.-4..J_ = i5.
12 1234
112. Forming the combinations of 6 articles i at a time, 2
at a time, &c., we have,
6Ci = 6, 6C2 = i5, 603 = 20, 604 = 15, 665 = 6.
Hence if n is an even number the largest number of combinations
ft
can be made by taking — articles at a time.
Again, forming the combinations of 7 articles i at a time, 2
at a time, &c., we have,
7Ci = 7, 702=21, 703 = 35, 7^4 = 35» 7^5 = 21, &c.
Hence, if n is an odd number the maximum number of combi-
nations occurs when the articles are taken or — I^ at a time.
2 2
In this case there are two greatest terms.
113. To find how often any one thing occurs in the combi-
nations of n things taken r together.
If from all the combinations containing a we take out a we
will have left the combinations of » - i things taken r—i to-
gether. Hence in the combinations,
«0r, any one thing occurs n - iCr—i times.
Similarly any two articles will occur together n - zOr — 2
times, &c.
-96-
li:
■ H i
i'! 'f
Ex. 142. The number of combinations of n letters 5 together
in all of which a, b, c occur is 21. Find the number
when taken 6 together and in all of which a, h, c, d
occur. <n-^ ^i-H - C^
Here w -3C5 -3 = 21, and « -406-4= ?
» - S w-4 ,
— A J = 21, whence n—io,
I 2
and io-4C2=f .f =15.
Ex.143. If the combinations of w + i things taken «-i to-
gether be 36 ; find the permutations of n things alto-
gether.
From Art. no, C,
I « + 1
« + iC« - 1 =
«- I
jw-i I (w + i) -{n-i) 2
n-
.'. « = 8, and P= 18 = 40320. ,
Combinations find their application in the Binomial
theorem, in Probabilities, &c.
BINOMIAL THEOREM.
114. The Binomial theorem is a formula by which we are
enabled to write down the expression of a binomial to any
power without the actual labor of multiplication.
We have, {i-\-ax){i-\-bx){i-^cx){i^-dx)-i -^r {a -^ b + c + d)x
+ {ab-\-ac-\-ad + 6c + bd-{-ed)x^ + {abc-\-abd-\-acd + dcd)x^
+ adcd.x*.
Now a, d, c, d are the combinations of 4 things taken i at a
time, .*. the number of terms in the coefficient of ;r=4Ci.
ad, ac, ad, be, bd, cd are the combinations of 4 things taken two
at a time, .*. the number of terms in the coefficient of
;p2=4C2.
\
together
number
a, 6, c, d
n- I to-
ings alto-
I
-I
Binomial
h we are
ial to any
+ d)x
en I at a
= 4Ci.
taken two
icient of
— 97 —
Similarly, the number of terms in the coefficient of *'=4C3 ;
and the number of terms in the coefficient of ;r* =464= i.
.*. making a = ^ = c = rf=t, we have
(i+r)* = i+4Ci.* + 4C2.;ra+4C3.:r84-4C4.;r<.
In like manner by starting with 5 factors we may show '
that,
The regularity of these expressions suggests at once that
(i-{-x)'* = i+nCi.x + nC2.x'^ ^nC^.x^ + . . . . +»CM.;r", . . . .A.
which is one form of the Binomial theorem.
115. Putting for mCi, nC2, &c., their values in factorials,
^we have,
1.2 1.2.3
which is a second and commoner form of the Binomial
theorem.
Ex. 144. Find the 5th term in the expression of (i4-«)"
... n(« — 1)(« - 2)(« - 3) A
5th term is -^^ ~ — -^i.it* .
1.2.3.4
Ex. 145. Show that when x is very small, (i +;r)^* = i + io;r
/ approximately.
(I +ir)i = I + lox+^^x"^ + ,
1.2
But X being very small, x^ and all higher powers of x may
be rejected in compaiison with i and lox
y
fi6. In B, Art. 115, write -^ for x, and we have,
a
(x+i).= , +„ . _i+l(''JiiI . ^ +«(" rlHlrl) . ^,+ . . .
a a 1.2 a^ 1.2.3 ^
Hut (i+~)»=i-(a+:r)"
.*. multiplying both sides by a^
(a+;r)" = «" + wa"-*;r +
. ^(^ - ^) ^ n-aya ^ n{n - i)(» - 2)^ „.3^
• • •
1.2
I. 2.3
W 'I
:
f.ii
i
1;'
:i
- 98 -
And this is a third form in which the Binomial theorem is
written. >^
117. Dividing both sides in C, Art. 116, by | n, we get,
![i±fl°=^-i-
n
n
|« — I ' I r l» - 2
— -fp^ — -i- D.
I 2 «-3 3
A fourth and very symmetrical form of the theorem.
118. We have drawn these expressions for the Binomial
theorem from the expansions of (i 4-;r)* and (i +;•;)'. We shall
now prove that if the theorem is true for (i +4:)", it is also true
for (n-;r)'^+^
Putn-i-i=w, then n = m-i; and writing this for n in B,
Art. 115,
1.2
+ (w-i)(w-2)(m -3)y3_|_
I . 2u 3.
Multiplying both sides by i ■\-x, using detached co-efficients ;
1.2
(w-i)(w-2)(m-3)
+ ^ —
1.2.3
+ ; x(i + i)
1.2 1.2.3
Hence the formula is true for m ; and m = n-\-i, .'. &c.
But the formula is true for 71=4 as we have seen, .•. it is
true for » = 5,' and if for « = 5 then for m = 6 and so on ; i.e., it
is generally true when n is any positive integer.
We have thus proved that the Binomial theorem holds when
n is any positive integer. It may also be proved that it holds
when n is any quantity whatever, but the general proof is be-
yond the scope of this work.
119. The following generalizations are readily drawn from
the form of the theorem.
i. If « be a positive integer the series is finite and consists
of » + 1 terms, •.* n terms contain x and one term is without x.
01 em IS
. . . D.
Binomial
We shall
also true
n in B,
tfficients ;
&c.
.•. it is
; i.e., it
ds when
it holds
of is be-
.wn
from
consists
ithout X.
— 99 —
ii. If n be not a positive integer the series can never termi-
nate, as reducing n by units can never give a factor equal to
zero.
iii. If n be a positive fraction and x negative, all the terms
after the first are negative. .
iv. If n be negative and x negative, all the terms are posi-
tive.
Ex.146. /T+^=(i+:r)^=i+i^+i^^^*'-f-
^ X ^' 2"^ ~~' ' " ' •• • ' ■■ — ■ • ^ i~ • • • •
4 X.2
= , + ^-i .(?)' + -L3.. (?)•- + ....
2 1.2 ^2^ 1.2.3 ^2'
Ex. 147.
^ ^ ^ 2 I.2V2'' I.2.3V2'
Ex. 148.
I -X
1.2 1.2.3
= i+x+x^+x^+ .... (Art. 9.)
Ex. 149.
a^/ ( \a/ 1.2 ^a^
= a~
2X^
• • •
3a* I.2.3^rt'*
120. The Binomial theorem may be used for the expansion
of the power of a trinomial or polynomial.
Ex. 150.
{i-^ax-\-bx^)^ = i-^n(ax+dx^) + ^^^^^^^{ax + bx^)^'\-
1.2
= I + nax + nb
n(',
nin-i )^^
1.2
;r2'+«(»-i)
2ab
+
1.2
n(n- 1)0; -2 )^8
1.2.3.
a*
^ "y" • • • •
!l
IOC)
Ex. 151. }^l+X+X^:s{l+X-\-x'*y
=» I -t iix+x^) -f ilri^ [X x"^ )'*
1.2
^J^~8 re
[X x')' f .
121. The Binomial theorem may sometimes be employed
to approximate to the roots of numbers.
Ex. 152. Required the fifth root of 12.
12 = 32 -20 = 2*(l -ft)
.-. l/l2 = 2{l-ft)^ = 2|l-^.|- -t^(ft)2- ±1^.{^)^-. . . .}
2-5'
2.3-5'
= 2|i-i -^-jh- } =i-^5 nearly.
LOGARITHMS.
We propose to deal here with the nature and use of Logar-
ithms, and not with their development.
122. Take the equalities, 2° = I, 2^ = 2, 2^ = 4, 2^ = 8, 2*= 16,
2" = 32, 2' = 64, 2^ = 128, &c. ; the quantities i, 2, 4, 8, 16, 32,
64, 128, &c., are numbers ; the indices of 2, i.e., o, i, 2, 3, 4, 5,
6, 7 are the corresponding; logarithms, and 2, the number raised
to the several powers, is the base.
By tabulating these, as in the margin, we
have a table of logarithms to the base 2. In
like manner we may form a table of logarithms
to the base 3, or to any other base which one
may choose.
For common purposes the base employed is
10, for being at the same time. the base of our
numeral system, it possesses certain practical
advantages over every other number.
To illustrate the practical applications of
logarithms we may employ a table to any base
TABLE.
No.
Log.
I
2
I
4
2
8
3
16
4
32
5
64
6
128
7
&c.
&\i.
lOI —
wliatever, for Ihc general properties of lu<^arithins are the saute fbr
all bases, 'lakinjj tlie table above, then, let it be required (i)
to multiply i6 by 8.
log. 16 = 41 . J
m ployed
Number of which 7 is the log. = 128, .*. 8X16= 128.
(2) To divide 64 by 4.
log. 64 = 6
log. 4
= 61
btract
• • • • r
if Logar-
2«=l6,
16, 32,
> 3» 4» 5»
)er raised
fVBLK.
Log.
O
I
2
3
4
5
6
7
Ac
Number of which 4 is the log. = 16, .•. 64-^-4 = 16.
We thus see that imiltiplicHtioii of numbers corresponds to
ailiiitjon of logarithms, and division of numbers to subtraction
of logarithiis. This will be shown more geneially hereafter.
123. The above table is not complete, even as far as it goes,
sirce the numbers do not follow each other in order. Thus it
lav'ks the numbers 3, 5, 7, 9, &c. To find the logarithm of
ono of these numbers we notice that the numbers in our table
are in geometric progression while the h garithms are in arith-
metic progression. Hence the geometric mean between two
numbers must correspond to the arithmetic men between
their respective logarithms. Thus 3] is tli» logarithm of
I 8X16 or II 3136 ....
This may be readily shown as follows :
27=128 = 8x16; .-. 2''^'"- 1/8X16, (jr 2=*^» = 11.31 .. .
.'. 3i = log. II. 31 . . .
By this means we may calculate the logarithm of 3.
1. I 2X4 = 2.8284; i(i + 2) = i.5 .-. 1.5 = log. 2.8284,
2. J 4X2.8287=5-6568 : ^(2 + 1.5)= i.7j = l-'g- 5-6568,
3. 1 '2X5^6568'= 3.363 ; ^(1 + 1.75) = 1-375 = lpg- 3.3630,
4. 1/2.8284 X 3.3630 = 3.0842 ; ^11.5 +1-375) =1-4375
= log. 3.0842 ;
And by continually approximating towards 3 we at last find
102 —
« ■
log. 3 = 1.585 . . . approximately. And in this way, although
exceedingly operose, the logarithms of the prime numbers were
once calculated.
We infer then that 2^ **^=3, i.e. 2**^"" =3, or 2
1586
_ olOOO
Of course we have no means of proving this except
through logarithms themselves.
124. The Base. In the computation of logarithms by means
of series, we come naturally upon a system having the strange
number 2.7182818 . . . , j^enerally designated by e or e, as a
base. These are called natural logarithms, Napierian logar-
ithms, and sometimes hyperbolic logarithms.
This system is usually employed in mathematical analysis.
The only other system in use is the one having: 10 as a base.
These are common or decimal logarithms.
Let a denote any base ; then,
*.• a" = I, the logarithm of i is always zero.
If rt > I, then ci" > i^and «"" < 1.
And, since a is greater than r in both systems of logarithms,
the los^arithm of a qiiantiiy greater than i is positive, and of a
quantity less than i, nef^a'.ive.
Thus log 3 is a positive quantity ;
but log .3 is a negative "
Since a'* = — ^ = o .*. log. o = -00. Hence the logarithms
of all proper fractions lie between o and - 00 . And since
fl* = 00 , the logarithms of numbers above unity lie between o
and + 00 .
Since if a is positive no power of a can be negative it fol-
lows that negative quantities have no special logarithms.
125. The number which we found for log. 3 to the base 2 is
composed of two parts, an integer i called the characteristic,
and a fractional part .585 . . . called the mantissa.
In decimal logarithms the distinction between these parts
is important.
126. Ihe characteristic. Since, io"^ = .ooi, lo"^ = .oi, 10'*
s= .1, 10® = I, 10^ = 10, lo^ = 100, 10^ = 1000, &c., we have,
number, .001 .01 .1 i 10 100 1000 &c.
logarithm, -3-2-101 2 3 &c.
although
bers were
lis except
by means
le strange
r €, as a
ian logar-
i analysis.
LS a base.
ogarithms,
, and of a
logarithms
nd since
between o
ve
it fol-
base 2 is
iracteristic ,
lese parts
.01, 10
have,
1-1
[C.
EC.
— 103 —
We see from this that the characteristics are the logarithms
of numbers made up of unity and ciphers only.
Also, for a number between 100 and 1000, log = 2 + a decimal
10 •' 100, log = 14-
I " 10, log = +
.1 " I, log= -1 +
" .01 " .1, log = -2+ "
&c. &c., &c.
Hence we may write down the characteristic of the logar-
ithm of any given number at sight by the following rule :
// the number is a decimal the characteristic is negative and
greater by unity than the number of ciphers to the right of the deci-
mal point.
If the number is integral or contains an integral part the charac-
teristic is positive and less by unity than the number of figures in
the integral part.
Or by the following rule :
Call the units place zero and count from it to the significant figure
farthest upon the left. The number of that figure is the character-
istict, positive if counted leftward, negative if rightward.
E.x.
153- To find the characteristics of, .00000734, 386.5,
943007.0162.
123456 210 643210
0.00000754 386.5 943007.0162
units place.
.-. -6
units pi.
units pi.
••• 5
For reasons now readily seen the characteristic is not usual-
ly written in tables of common logarithms.
127. The Mantissa. Let log 425 be 2+m, where m is the
mantissa or decimal part.
Dividing 425 by 10, we must subtract the log of 10 from
that of 425, (Art. 122). .•. log of 42.5 = 1 + w.
Dividing by 10 again, log of 4.25=0 + w.
Dividing by 10 again, log of .425 = — 1+ in.
&c., &c,
it
— 104 —
We notice that the mantissa remains constant, the mi'y
change being in the characteristic. Hence we may sum up
the significance of the parts of a lopfarithm as follows :
The mantissa is connected with the group of figures and their
arrangement; //i^ characteristic, with the position of the decimal
point.
128. A table of decimal los:arithins re;^nsters only mantissae ;
and since these start from zero at every power of 10, the table
extends only bi^tween two consecutive powers of 10. For
7-place logarithms, i.e., for those with 7 decimals in the man-
tissae, the usual extent is from 10* to 10'.
We give below a portion of a table of 7-place logarithms
taken from Hutton's tables as published li\ Chambers.
No.
1
2
3
4
5
6
7
8
9
D.
2397
379C680
CG86
7043
7224
7405
7586
7707
7918
8130
8311
181
98
8492
8673
8854
9oa5
9216
9397
9578
9759
9940
0121
99
3800302
0484
0665
0846
1027
1208
1389
1370
1750
1931
2400
2112
2293
2474
2655
28:^0
»)17
3198
3379
3500
3741
01
3922
4102
4283
4464
4645
4826
50O7
5188
5368
6549
02
5730
5911
6092
6272
6153
6631
6815
0095
7176
7357
03
7538
7718
7899
8080
8261
ai4i
8622
8803
8983
9104
04
9345
9525
9706
9887
0007
0248
0428
(kiOi)
0790
0970
OS
3811151
1331
1512
1693
1873
2031
2231
2415
2595
2776
D H
181 P.
18
36
54
72
91
109
127
145
163
129. The workings; of a table of loj^arithms consists in two opera-
tions the converse of one another, viz : (a) ^iveii an arrange-
ment of figures to find the cories[)ondinLr mantissa, and (6)
given a mantissa to find the corresponding arrangement of
figures ; for the characteristic n'>t being registered has no im-
mediate connection with the table.
(a) Given an arrangement of iigures to lind the correspond-
ing mantissa.
The table above mentioned gives the mantissae of all arrange-
ments of 5 figures at sight; lour of these are found in the
column of numbers marked No'., and the filth in the horizontal
row at the top.
i. When the arrangement contains 5 figures.
To find the mantissa of 23987, start at 2398 in the first
lie at) y
sum up
nd their
decimal
ntissae ;
he table
o. Fol-
ic man-
arithms
9
D.
Ul
181
lai
)31
741
349
157
104
)70
m
LC3
o opera -
;lnall?e-
and (6)
ment ot
no im-
lespond
111 range-
Id in the
Irizontal
he first
— 105 —
column and pioc^e orizontally until in the column marked 7
at the top. To t. . figures 9759 there found prefix the 379
which the first colunin shows to be common to several rows.
We thus have, mantissa of 23987 =3799759.
ii. When the arranf;ement contains leas than 5 figures.
Add ciphers or suppose them to be added to raise the num-
ber of figures to 5, and then proceed as in i.
Thus, mantissa of 24= mantissa of 24000 =3802112.
And, log 24=1.3802112.
iii. When the arrangement contains more than ^figures.
To find the mantissa of 2403872.
We find the mantissa of the first 5 figures to be 3808983.
In order to show what is to be done with the remaining
figures 72 we shall explain the column and row of the table
marked D and P respectively.
Mant. of 24038 = 3808983) D = difference of
" " 24039 = 3809164) mantissae= 18 1.
Now the 72 occurring here is -^-^ of the difference between
24038 and 24039. .'. we should add to 3808983, 3^X181.
But33jrVXi8i=7XW + iV(2XW).
The row marked P (proportional parts) gives the multiples
of ^j^ from I to 9. Thus under 7 at the top we find 127
which is 7X-^]^ to the nearest unit. Under 2 v.e have 36,
one-tenth of which is 3.6 or 4 to the nearest unit. Hence the
mantissa of 2403872 is 3808983+127+4 = 3809114.
iv. As a special case let it be required to find the mantissa
to the arrangement 24044.
Referring to our table we find the first cipher overlined 0~.
This indicates that the three figures to be prefixed to the four
there given change at this point from 380 to38i. The mantissa
is accordingly 3810067.
(6) Given a mantissa to find the corresponding arrangement
of figures.
Take the mantissa 3806745, for example.
The highest mantissa in the table capable of being sub-
tracted from this is 3806634 ; and we proceed as follows :
Ill
'■
*■
\
— io6 —
Mant. given . . . 3806745
tab. mant 3806634 arrrang't = 24025
Diff. of mant.
highest subtractive number
from P . . . .
Ill,
109
number
2,
I.
Diff. . . .
Subtract number from P
after dividing by 10 . . . 1.8 .. . number .
.'. arrangement = 2402561.
130! To find the logarithm of a number. Find the mantissa
of the arrangement without any reference to the decimal point,
and then prefix the characteristic according to rule Art. 126.
To find the number answering to a given logarithm. Find the
arrangement corresponding to its mantissa and then fix the
decimal point by means of the characteristic and rule Art.
126. t
131. To perform multiplication by logarithms.
Let a" = m, then x = log m ;
«*'=«, then _y = log «.
Then, mM = a*.a^ =«*+^;
and log mn=x+y = \ogm-\-\ogn.
.'. to multiply numbers we add their logarithms and take the
number answering to their sum.
Ex. 154. To multiply 23.974 by .024056.
log 23.974 = 1.3797405
log .024056 = 2.3812234
log .57671 = 1.7609639, sum.
•'• -57671 is the product.
As in this example, the negative sign of the characteristic is
placed above it to save room, and it must be borne in mind
that although the characteristic may be negative the mantissa
is always positive.
132. To perform division by logarithms.
with the notation of Art. 131,
107 —
nantissa
lal point,
t. 126.
Find the
n fix the
ule Art.
take the
eristic IS
in mind
mantissa
m
n
av
tn
and log — = x -y sslog tn -log n.
n
.'. To divide one number by another subtract the logarithm of the
divisor from that of the dividend and take the number answering to
the difference.
Ex. 155. To divide 1.4936 by .007453.
log 1.4936 = 0.1742343
log .007453 = 3.8723311
log 200.4025 = 2.3019032, difference.
.'. 200.4025 is the quotient.
133. To raise a number to any power.
a*=» ; .-. nv =(,a'^)y=a'y \
and log nv = xy =_y log n.
.'. To raise a number to any power multiply the logarithm of the
number by the index of the power required, and take the number
answering to the product.
Ex. 156. To find the 21'* power of 1.06.
log 1.06 = 0.0253059
21^= index.
log 3-39957 = 0.5314239
.-. (1.06)21 = 3.39957.
Ex. 157. Find the value of (.4726)*.
log .4726 = 1.6744937
8
log .00248857 = 3-3959496
.-. (.4726)* = .00248857 . . .
In this example the mantissa being positive we have, upon
multiplying, -8 + 5.3959496= -3 + -395 • - •
134. To extract any root of a number.
Since a" =«,
1 1 «
(«)!' = (a*) v = «v.
\
- io8
'■ y
.'. To extract any root of a number, divide the logarithm of the
number by the number denoting the root to be extracted, and take
the number corresponding to the quotient.
Ex. 155. Find the value of (.017325)4^
log .017325 = 2.2386732,
Divide by 7 gives, i. 7483819,
corresponding number = .56025.
In this case having a negative characteristic we make it
evenly divisible thus :
2.2386732=7 + 5.2386732, which divided by 7 gives the quo-
tient found. This is the equivalent to
— Hf + .034 . . . = - I +^+.034 . . . = - I +.748 ...
Ex. 159. To find the value of —5^ '- — 7;, being given the logs
2^X(2l6)*
of 2, 3, 5 and 7.
1 1,3
Numerator = 3^X3^X7^X.o5<'=3 2 X7^X.o5« ;
.-. log num. = Ji^ log 3+6 log 7 + 6 log .05=0.36569.
Denominator = 2^ X (2^ X3^r =2^. X2^X3^ = 2X3*;
.*. log denomr. = log 2 +f log 3=0.58730.
.*. log of the value = .36569 - .58730 = 1.77839 ;
and value = .60033.
Those who make very great use of logarithms, as astrono-
mers and navigators, do not usually employ negative indices
for the logarithms of fractions, but make use of a system
much more convenient in practice, although probably more
difficult to master at first.
An explanation of that system, as well as of other conven-
tions in logarithmic practice, can scarcely find a place in this
work.
■t:-;i
u
— log —
im of the
and take
make it
s the quo-
;n the logs
6 .
>
,569.
= 2X3^;
astrono-
[e indices
a system
jbly more
conven-
[ce in this
EXPONENTIAL EQUATIONS.
135. An equation in which the unknown quantity is involved
as an index or exponent is called an exponential equation.
These usually require the application of logarithms in their
solution.
Ex. 160. In how many years will a sum of money double itself
at 3 per cent, compound interest.
From Art. 59, A =P(i+r)*.
But A=2P .'. (i+r)* = 2.
And going to logarithms,
t log (1.03) = log 2, .'. t = -
log 2
log I. 03 =^3+ years.
Ex. 161. Given a'^ + a'^ = b to find x.
Multiplying by a', a"^' - ba^= - i ;
. , b±y¥^
.'. a'-
And X'-
log((!) ± v^6'- 4rt) - log 2
lot? a
n^ X *W5— a
ml =6 to find X
a )
nx
— lo^ a = {nx — a) log b ;
m
•. x(—\o^ a-n log b\= —a log 6 ;
X =
ma log b
»(mlog6-loga)
° 1
— no
CONTINUED FRACTIONS.
iq6. Let us take for illustration the fraction^^,
"^ lOI
Then, 45 _ i _ i
lOI lOI ^ , II "
24-
45 45
Again, ii _ i _ i
45"5 4 + ^-
II II
.*. 45 _ I
lOI
2 +
This latter expression is usually written i
or more
2^- ±
^ II
compactly- - — and ia called a conliniied fraction, which is
^ •^2 + 4 + 11
rational when the number of terms is limited, and irrational
when not limited.
137. Toconvert any fraction into a continued fraction.
In the example of the precedinp^ article we divide 10 1 by 45
with a quotient 2 and a remainder 11 ; we then divide 45 by
II with a quotient 4 and a remainder i. And this beinp; iden-
tical with the operation for finding:: the G.C.M. of loi and 45,
we deduce the following rule : Proceed as in findin<:^ the G.C.M.
of the numerator and denominator of the given fraction ; the
quotients taken in order form the denominators of the terms of the
continued fraction.
Ex. 163. To convert f|^ into a continued fraction.
Proceeding to find the G.C.M. we obtain
the quotients i, 2, 3, i, 6, i, 2, 2 in order. 2
.'. continued fraction is, i
iiiiiiiii
1+2+3+1+6+1+2+2 2
472
681
I
54
209
3
7
47
6
2
5
2
I
00
ir .
or more
whicli is
rrational
|oi by 45
Je 45 by
linpj iden-
and 45,
G.C.M.
\tion ; the
\ms of the
I
9
5
I
3
6
2
00
— Ill —
4
I
If we proceed to divide by the remainder o we get ^= oo ,
and the corresponding term of the continued fraction is i,
which is zero. But as the process of finding the G.C.M. of
any fraction must finally give a remainder o, the equivalent
continued fraction must always be limited or rational. Hence
any fraction may be converted into a rational continned fraction.
13S. If we take the values of one, two, three, four, &c.,
terms in the continued fraction of Ex. 163, we have,
1 I I_2 I I i_ - I I I 1_ 9 I I
' \ 1 ^ = |^,&c.
4-6 + 1+3
The quantities i, f, ^, -j^, |^, &c., are successively closer
approximations to the value of the original fraction. They
are consequently called convergents to the fraction |-|^.
Thus the successive differences are :
(I) i-W=W; (2)i-W=-*;
(3) tV-W = M; (4) i^F-W=-M nearly;
(5) U -|if = ^ nearly ; &c., &c., &c.
We thus see that the 5th convergent differs from the origi-
nal fi action by only ggf^o or -rrkw^ nearly.
We see moreover that the odd convergents are too great and
the even ones too small, so that the successive convergents are
alternately too great and too small, the true value of the
fraction always lying between those of any two consecutive
convergents.
To find the convergents.
I I I I
a + l)-\-c-\-d-\- &c. be the conveying fraction.
Then ist conv. =-. For the second convergent we must
139-
Let
put a 4- i for a in the first; for the third convergent we must
h
put 6 4-- for b in the second ; &c., &c.
c
112
We thus get,
1st convergent = - which denote by
a
2nd
"+*
ba-+ I
3rd
((
'-J
^^•+1
N,
.b+i
a{b + -
which denote by
N
3
4th
i«
We thus see that every convergent after the second is form-
ed from the two proceeding convergents according to a fixed
law, which may be stated as follows : Calling a, b, c, d, &c.,
partial quotients, the numerator of the 7j*'' convergent is formed
by multiplying the number of the [n - i)*'' convergent by the
tr^ partial quotient and adding the numerator of the (» -a)"^
convergent. The denominator of the «"* convergent is formed
from the denominators of the (n- if^ and (« — 2)"** convergents
in a precisely similar manner.
The operation may be carried out as in the following ex-
ample.
Ex. 164. Find the convergents to the fraction, ^^.
The partial quotients are 2, i, 3, 2, 4.
Assume ^ as the first convergent ; then ^ is the second con-
vergent.
I 4 9 40
^^132 4
I 2 3 II 25 III
) + a,
^3
Da"
2
C.
is form-
a fixed
rf, &c.,
formed
by the
(«-2)",1
formed
^^ergents
/ing ex-
:>nd con-
— 113 —
Write these two convergents in order and the rein.iining
partial quotients in a row following them.
Then starting with the partial quotient i as a multiplier,
iXi (the numerator of ^) + o(the numerator of ^)= £, which
write above for the third numerator.
1X2 (denominator of |^) + i (denominator of ^) = 3, which
write below for the denominator of the third convergent.
Next starting with 3, 3X1 + 1=4 for numerator, and 3X3
+ 2=11 for denominator, &c.
We thus find the convergents to be ^, ^, ^, ^ and finally
the fraction itself -j^.
Or the working may be arranged as in the
margin, where the various steps are readily
made out without any additional explana-
tion.
o
I
I
4
9
40
I
3
2
4
I
2
3
II
25
III
Ex. 165. To find approximate values for 3.14159.
Tai<e tlie reci;)rocal J^^ for which we find the partial
quotients 3, 7, 15, i, 25, T, &c.
.•. the convergents are, o
o 2Ji 338 355 9280 fi,p ,
7
106
7
I
25
I
I
3
22
333
355
9230
&c.
113
2931
&c.,
7: being the latio of the circumference of a circle to its
diameter is approximately 3.1415926. The approximate value
^^^ was discovered by Archimedes, and ^ by Metius.
140. The difference between two consecutive convergents
is equal to unity divided by the product of the denominators of
the convergents.
Taking the convergents to Ex. 164, i — i=i; 5-^ = 7^5
y^j- - ^ = 77-5- ; and this may be proved to be universally true.
Hence we may solve the following problem.
— 114 —
Ex. i66. To find multiples of 71 uml iji vvliicli sliull differ by
unity.
The converpents to ^ arc, |, ^, ^^^ M' iVr-
And 24X71 - ijX IJI = 1.
This principle may be employed in solutions like that of
Ex. 112. Applying it to that example, we find j as a multi-
plier adapted to the question in hand.
Ex. 167. Two wheels, A and B, of a clock being geared to-
gether should move with the relative velocities of
1401 and 1945. No more than 120 teeth being al-
lowed in any wheel, to find the numbers to be em-
ployed. After 100 revohitions of A. how much will
B be either in advance of or behind its true place?
The convergents to \^ are \, |, f , f ||, ff , -,8j\, ^, &c.
.*. 85 and 118 are the numbers to be emp'oyeil.
Now when A makes one rev. B should make \^^ rev.
But when A makes i rev. B does make ^*j^.
/ 85 _ 14^ i\ - 700 _ I
\ii8 1945/
lOOi
nearly.
1945X118 328
Or B would be before its true place by ^^fyth of a revolution.
If the greatest number of teeth allowed were 100. our
convergents give us 18 and 25 as the numbers to be employed.
67 and 93, however, give a closer approximation. The true
solution of a question of this kind can only be obtained in all
cases by the use of what are called intermediate converging
fractions, the theory of which is beyond the scope of this work.
141. To develope the square root of a number into a con-
tinued fraction.
Let the number be 15 for example,
1/15 = 3 + /15 -3; and|/i5-3 = ^^-i5.
3_
V:i5±3 = i + vllilJ. and
6 6
/i5-3_.
1/15-3
I
1/15 + 3
v/15-3
1/15 + 3-
- 115 - ' ,
•*• Ki5='3+ >- - 7 ad infinitum.
1+6+1 + 6-1-
1/15+3 = 6 + 1/15-3; eind 1/15 }=&c., as in the upper line.
Ill the above, 3 beinj? the largest integei in 1/15 we put
v/15 =3 + »/i5 -3» so that 15 -3 may be fractional. Then
6
In the third line v 15 +3 = 6 + 15 -3. and this beinff the
same as tlie first line, the quotients i and 6 will be continually
repeated.
Ex.168. v/7=2 + v'7-2; and 1/7-2 = — i— .
l/7j+2
3
|/7-^2_v/7-i+3-., 1/7-1. .^^l/7-i_ I
3 3 3 3 v/7+'
2
Ki±i=i+l^li;and l^-lzi^—i
2 2 2 I -^7+ I
3
l^i±' = i+V^lzi; audK:7zJ = __J
3 3 3 1/7 + 2
1/7 + 2 = 4 + 1/'7 - 2, &c., as in the first line.
1 + 1 + 1 + 4 + &C., ad mf.
SERIES OF SQUARE NUMBERS.
142. The squares of the natural numbers are called square
numbers, and the series of square numbers is accordingly
I 4 9 16 25 .... n^.
To find the sum of n terms we do as follows :
(«+-i)8= «» + 3«« + 3n +1
«* = (^^+i)3=(«-i)8+3(«-i)''+3(w-i) + i
ii6
{n- r)8 = („_2 + i)3 = («-2)3-|-3(«-2)2+3(w-2)-i- I
3' = (2 f if = 2' + 3.2'' + 3.2 4-1
23=(i + i)^ = iH3-i''+3i + i
l3=(o + 1)3 = 0^ + 3.0^3-0+ I.
Adding, the quantities in the first column upon the right
cancel all upon the left except the first term.
Putting Hn' to denote the sum of the square numbers to n
terms, and £n to denote that of the natural numbers to the
same extent, we have,
(n + i)3 = 32V + 32'« +«4-i ;
w(« + i)
32'«2=(« + i)3-3
-(« + !) Art. 93.
2
Or, In'^n^-
+ i)(2n + i)
6
16
If objects be arranged in squares iip'>ii :i
plane surface, as in the margin, the whole
number of objects in any square l)lock will
be the square of the number upon the siMe.
If a number of balls be piled in the form
of a pyramid with a square base, each layer
contains the square of the number of halls
forming its side, and the sides of two con-
secutive layers differ by unity.
Hence the balls in the layers give the
series of square numbers begining at the
top where there is but one ball ; and the
whole number of balls is the sum ot the square
numbers from 1 to Ji^, n being the number of
Jmlls on the side of the basal layer.
Ex. 169. How many balls are in an unfinished square pyra-
midal pile, the basal row having 22 and the top row 14 ?
If the pile were complete there would be
2a(22H-l)(2X22 + l) _ 2 2X23X45
■ ' ^ 6 ~ 6 •
But the number required to finish the pyramid is
* *
* *
* * *
* * *
* * *
* * *
* * *
* * *
* * *
*
-+ 1
— 117 —
i3(i3H-i)(2Xii-|-i) _ 13X14x27
'. wholenumberinthepile = ^^^^3X45 13X14X27 ^^^^
:he right
bers to «
rs to the
\rt. 93-
* I
* 4
*
* * *
* * *
16
luare pyia-
)p row 14 ?
SERIES OF TRIANGULAR NUMBERS.
2
143. If objects be arranged in equilateral triangles upon a
plane surface, the number required to
form a complete triangle, as in the mar- 1*1
gin is called a triangular number. With
I object upon a side we have i as the
first triangular number. With 2 ob-
jects upon a side it requires 3 to com-
plete the triangle ; there being one row
with one in it and a second row with
two. With 3 upon a side we have 3
rows, of I, 2 and 3 objects respectively;
i.e., 6 in all. With 4 upon a side we
have four rows of i, 2, 3 and 4 objects,
or 10 in all, &c.
Hence the series of triangular num-
bers is I, 3, 6, 10, 15, 21, &c.
» *
* * *
¥e * *
6
10
« « «
The numbers are evidently the successive sums of the series
of natural numbers beginning at unity.
Thus, 1 = 1,3 = 1 + 2,6 = 1 + 2 + 3,10=14-2 + 3 + 4
15 = 1 + 2+3 + 4 + 5, &c.
144. To find the sum of « terms of the triangular numbers.
Let Hn denote the sum of n terms of the series of natural
numbers, 2!n^ that of the series of square numbers, and 2t the
sum of n terms of the series of triangular numbers.
Then, 2w = i +2 + 3 + 4 ... .«,
2'«2 = 1 + 4 + 9+ 16 n"^,
In-^ In^ = 2 + 6 + 12 + 20+ ... .{n^+n).
= 2(1+3 + 6 + 10+ '^'*
)
— ii8
= 2iV.
. Vi i/v I V2\ I'Kw+i) , n(n-\-i)(2n-\-i) . _^ „
.-. 2.t = ^{2n-\-2^) = ^-~ ' -{■ —^ — '—-^ ■ — ' Arts. 93&142.
, _n{n-\-i){n-\-2)
If a number of balls be piled in a triangular pyramid, the
numbers in the successive layers will be the series of tri-
angular numbers, and the whole number of balls in the pile,
commencing at one upon the top, will be the sum of the first
n triangular numbers, n being the number of balls in a side of
the basal layer.
Ex. 170. How many balls in a complete triangular pyramid,
the basal layer containing 10 upon a side.
Here w=: 10, and, , ,• -
i'/ =
10. II . 12
= 220.
INDETERMINATE COEFFICIENTS.
145. The truth of the statement that
= I -{-x+x^ + .... ad inf.
I -X
is not limited to any particular value of x, but holds for all
values, arithmetically if .«: is less than one, and algebraically if
X is any quantity whatever.
In other words, the expression is an identity, and must,
therefore, be true, quite independently of any particular values
given to the symbols employed.
146. Proposition. If we have the identity
A +Bx-\-Cx^ . . . .=a + bx-\-cx'^+ ....
Where A, B, C .... a, b, c\ . . . are constant coefficients,
and .r is variable, then, A =a, 3 = 5, C=c,&c., i.e., the co-
efficient of any particular power of x upon one side of the
identity is equal to the coefficient of the same power of x
upon the other side.
For,
— 119 —
A -a = (b- B)r + (c - C)x^ +
But the second member chanp;es value as x changes its
value, v^'hile the first member is constant.
Hence there cannot be equality unless each member is equal
to zero. .'. ^ -a = o or A=a, and by rejecting A and a as
being equal and dividing by x we obtain in like manner B -b
= o, or B =b, &c
The coefficients A, a, B, b, &c., are called indeterminate or
undetermined coefficients, and the proposition now proved
states the principle of indeterminate coefficients.
The principle of indeterminate coefficients is one of the
most prolific in algebraic analysis. Some of its simpler appli-
cation will be illustrated by a few examples.
Ex. 171. To expand
ing powers of x.
l-irX
into a series according to ascend-
Put ^"'"^ ^a+bx+cx'^+dx^-{- . . . .
(l-;ir)2
then i-^x={i-2X+x'^){a + b->tcx'^ -\-dx^-\-
= a
+b
x^c
x'^^-d
-2a
-2b
~2C
+ a
+ h
and equating coefficients of like powers of x,
a = i ; h — 2a=i .'. 6 = 3,
c-2b-^a = o .'. c= 2b -a = $,
d-2c-\-d = o .'. d = 2c-d = y,
■i\ &c., &c.
.•■ ^^^ ^ = i+^x+Kx^+yx^-\- . . . .
/ (1-^)2 ^ . V ■■
pompare this result with Ex. 22.
Ex.r 172. To expand the square root of i i-x+x^.
Put ■/i+x-\-x^= a-\-bx-\-cx^ +dx^+ ....
Squaring, i+x-\-x'^ =a^ + 2adx + 2ac
x^ + 2ad
zbc
I20 —
Equating coefficients,
a^ = i .'. a = i
2ac-\-d^
= 1 .'. C s= =-#
2a
8
2ad-\-2dc=o .'. d—
_ _^ ^ __3^
i6
.-. |/i +^4-jr'» = I + |;r+|jr2 - j^j;< . . . .
i6
Ex. 173. What relation must exist among the quantities p, g,
y, s in order that x^+px-k-q and ^ + rx^s may have a
common factor.
Let the common factor be x+a, then the expressions may
be written
\x-{-a){x+^\ and (;r + a)(;r4-l),
a a
since the last terms in the products will evidently be q and
s as they should be.
Then we must have,
a+l-p, a-|-- = y.
a a
.'. c^ — ap— — q, a^-ar= -s.
And eliminating a^ and a by determinants.
9 I
s I
«= \p I
r 1
pq
and a' =
r s
I r
p — r
p~r
.'. (p-r){qr-ps) = iq-s)^,
is the necessary relation.
tides/, qy
ay have a
sions may
be q and