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THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
ELIMINATION
BETWEEN
TWO UNKNOWN EQUATIONS WITH TWO UN-
KNOWN QUANTITIES,
BY MEANS OF
THE GREATEST COMMON DIVISOR.
ALSO,
ANALYSIS OP CURVES,
WITH AN APPLICATION TO
AN EQUATION OF THE FOURTH. DEGREE.
rt"
FRANCIS H. SMITH, A. M.,
Superintendent and Professor of Mathematics in the Virginia
Military Institute.
NEW-YORK:
WILEY AND PUTNAM.
1842.
RESOLUTION, &c.
Resolution of two equations with two unknown quantities. —
Elimination by the Greatest Common Divisor.
1. The most general equation of the mih. degree, be-
tween two unknown quantities x and y, contains all the terms
in which the sum of the exponents of x and y does not ex-
ceed m. Its form may then be represented by the equation
3.^ + Vx^-^ -f Qa;"*-2 -I- Ra;'"-3 . . . . -\- Tx + u = 0,
in which P, Q, R, &c. are functions of y, as follows :
P represents a polynomial of the first degree in y of
the form a + by
Q, represents a polynomial of the second degree in y of
the form c + dy + ey^
R represents a polynomial of the third degree in y of
the form f ■\- gy -{- hy'' + ly\
&c. &c, &c.
the last co-efficient u, containing all the powers of y, from
zero to m.
2
J^ JffiAn equation thus formed is said to be a complete equa-
<r '.f the mth degree between two unknown quantities, and
..li any of its terms are wanting, it is called an incomplete
equation.
3. Could we solve equations of every degree, the ordinary
methods of elimination might be readily applied, to the solu-
tion of any system of m equations, with m unknown quanti-
ties ; and we should, in general, obtain a determinate num-
ber of solutions. It would be only necessary to find the
value of one of the unknown quantities in terms of the others,
in one of the equations, and substitute this value in each of
the other equations ; there would result a new system of
equations, with one less equation than were given, and with
one unknown quantity less. By continuing this operation,
we should obtain a single equation with but one unknown
quantity. This equation is called the final equation, and
serves to determine the values of the unknown quantity which
it contains, which by substitution will make known the cor-
responding values of the others.
4. If the number of equations exceeded the number of un-
known quantities, we could by the above method eliminate
all the unknown quantities, and there would result one or
more equations, containing only known terms, which would
be equations of condition necessary to be fulfilled, in order
that the given equations should not be incompatible with
each other.
5. Should the number of unknown quantities exceed the
number of equations, the question would be indeterminate ;
for by giving arbitrary values to as many of the unknown
quantities as were in excess, we might determine the values
of the others by means of the given equations, and thus'
as many different solutions as there were arbitrary ^liftaes'
assumed.
6. But the difficulty of solving equations in general, has led
algebraists to seek other methods of elimination than the one
just mentioned, so as to obtain at once a single equation in-
volving but one unknown quantity. Various methods have
been used to determine this final equation, and that method
is regarded as best, which leads to a final equation, whose
roots make known all the values of the unknown quantity
which it contains, which are compatible with the given
equations, and no other values. The method by the Greatest
Common Divisor is not free from the objection of leading to
foreign roots, but it is found to be the most convenient in
practice. We propose to explain this method.
7. Let
A = o B =
be two equations involving x and y, and let /3 be any assumed
value of y. If we substitute this value in the place of y, in
the given equations, there will result two equations,
A' = B' =
which contain only x and known quantities.
Now it is evident that ^ can only satisfy the given equa-
tions, when there exists at least one value of x, which will
reduce the two quantities A' and B' to zero at the same time,
that is, satisfy the equations
A' = B' = 0.
Let a; = a be such a value of a:, it follows, that these last
equations must have a common divisor in x^ since they will
both be divisible by {x — a), if a be a common root. When
this condition is fulfilled, the value of y is called a compati-
hk value. Hence,
Having given two equations with two unknown quantities,
a value attributed to one of the unknown quantities will he
compatible, when its substitution in the given equations causes
them to have a common divisor, which is a function of the
other unknown quantity.
S. The above principle leads directly to the method to be
pursued, to effect the resolution of the given equations. For,
since every compatible value of one of the unknown quanti-
ties, y, for example, must when substituted give a common
divisor in x, (if the two equations be determinate,) it follows,
that if the proposed equations be arranged with reference to
X, and we seek the greatest common divisor to the polyno-
mials composing them, we shall, after successive divisions,
find, in general, a remainder which contains only y and
known quantities. This remainder must be zero, if the
given equations have a common divisor in x. Calling R
this remainder, we shall by placing it equal to zero, form the
equation
R = 0,
which is the final equation spoken of. This equation ex-
presses the condition necessary for the common divisor in x
to exist. The roots of this equation substituted in the given
9
equations, will cause them to present the same values for x,
and of course to have a common divisor in x, if no foreign
roots have been introduced in the process of finding the
greatest common divisor. The method of detecting and re-
moving these foreign roots will presently be e:^amined.
9. It will generally be sufficient to substitute the values of
y deduced from the final equation, in the last divisor in x, in
order to obtain the corresponding values of x. For, if we
represent by Q, the quotient resulting from the division of A
by B, and by R, the fi.st lemainder, we shall have
A = B X Q + R.
From this equation it is evident, that if any values of x
and y reduce A and B to zero, they must make R = o also.
The equations
B = R =
will therefore make known the values of x and y, which will
satisfy the three equations
A = o B = o R = o.
Reciprocally, the values of x and y, which satisfy the two
last equations, will also satisfy the equation
10
The determination of the roots of the given equation is
then reduced to that of the equations
^»^' B = o R = 0.
Dividing now B by R, we shall obtain an equation
B = R X Q' + R'.
The roots of the equations
B = R = o
will therefore be found among the solutions of the equations
R = o R'=o.
And if R' be the remainder in y,
W =
will be the final equation, the roots of which substituted in
R =
will make known the systems of values which correspond
.to the equations
R' = o R = o B = o A = o
11
and of course the values which are compatible in the given
equations.
1 0. Let us apply the foregoing principles to the following
ample: .^^J^
EXAMPLE I.
A = a;'' — 37/x' + {3y' — y -{- 1) x — y' + y'—2y = o
B = x' — 2yx -\- y^ — y = o.
Following the rule for obtaining the greatest common di-
visor, we have,
First Division.
x* — 3yx^ + {3y' — y+l)x—y' + y^—2y'\x^ — 2yx-\-y'—y
x^ — 2yx'^ + {if — y) X \x — ?/ = Q,
— yx' + {2f + \)x — f+y' — 2y
— i/x" + 2i/'x — y + y^
X — 2y = R.
Second Division.
X — 2y
X — 2yx + y — y
x^ — 2yx
X = Q'
f-v = R'
In order that x — 2y be a common divisor to the twa
12
given equations, the last remainder must be zero. We have
therefore for the final equation,
♦ y" — y = o.
The roots of this equation are
y = 3/ = 1.
Substituting them in the equation
X — 2y ~ 0,
formed by placing the last divisor equal to zero (9), we find
the corresponding values of^ to be
X = o X = 2,
which determine the solutions of the given equations.
11. Should the quotient resulting from the division of A
by B, be a fraction, the denominator of which contained ei-
ther or both of the unknown quantities, the principle devel-
oped in Art. 9. would no longer hold good. For if in the
equation
A = B X Q + R,
Q, were equal to — , K containing one or both of the un-
13
known quantities, we should have
The values of a: and y, which reduce A and B to zero,
might also cause K to be zero. -— - would then become -^
K
and might have a finite or infinite value The value of R
would also be finite or infinite, and could not in either case
be zero. The roots of the given equations could not then
be found from the solution of the equations
B = o R = o.
12. To avoid fractional quotients, we adopt the same ex-
pedients resorted to, in obtaining the greatest common divi-
sor, and which consist, either in suppressing the common
factors of the dividend or divisor, or by multiplying by some
factor which will render the division possible. We shall
thus be enabled always to obtain entire quotients, in which
case the method of Art. 9 may be followed.
13. In seeking the greatest common divisor to two poly-
nomials, the suppression of common factors, or the introduc-
tion by multiplication of a new factor, does not affect the re-
sult, but it is not the case in the solution of equations. We
shall therefore examine the consequences resulting from the
introduction or suppression of these factors.
14. Let us take the equations
A = a B = o.
14
and let us suppose that the division of A by B cannot be
effected ; which supposes that the co-efRcient of the first
term of B contains factors of y which are not common to
that of the first term of A. Let D be the product of all
these factors, and suppose D common to all the terms of B.
The proposed equations will take the form
A = o B'D = 0.
These equations may be satisfied by making
A = o B' = o.
or
A = o D = 0,
We might then suppress the common factor D, provided that
to the solutions of the equations
A = o B' = 0,
we add those of
A = o D = o.
To obtain the solutions of the last equations, we find the
values of y in the equation
D = o,
15
and substitute them in
A = o;
the systems of values of x and y, thus obtained, will give all
the solutions belonging to the given equations.
15. Let us now suppose that D is not common to all the
terms of the divisor, and that an entire quotient can only be
obtained by multiplying the dividend by D. We shall then
have
AD = o B = 0.
These equations may be satisfied by either of the systems
of equations
A = 6 B = 0, or D = B = 0.
Hence the system of values of x and y deduced from the
equations
D = B = 0,
must be suppressed as not belonging to the solutions of the
given equations, if they are found existing among the solu-
tions of the final equations.
16
16. The following example will illustrate the case alluded
to in Art. 14 :
EXAMPLE ir.
yx^ + Zyx' — 2/V + (y + 1) x — y = o . . (1)
^z" + (3?/ — 1) a;^ — y'x + 3/ — 1 = . . . (2).
The result of the first division gives for a remainder
x^ -\- 2x — y.
Dividing equation (2) by this remainder, we have for the
second remainder
(y — 1) x-^ + (y — 1).
This remainder having the common factor {y — 1), we
suppress this factor (Art. 14), and proceeding in the opera-
tion, we have for the last divisor {x — y), and for the final
equation
f + \=o.
The roots of this equation are
J/ = +■/ — 1 y = —^ — 1.
17
Substituting these values in the equation formed by put-
ting the last divisor equal to zero,
X — y = 0,
we have for the corresponding values of a;
X = -f-/; 1 x = — V |7
Making now^ the suppressed factor {y — 1) equal to zero
(Art. 14), the equation
y — 1 = 0,
gives y = 1, which being substituted in the preceding divisor
placed equal to zero, viz :
x^ -\- 2x — y ■= 0.
The values of a: deduced from this equation, will give the
solutions to the given equations which were omitted in the
suppression of the common factor (?/ — 1).
17. For an application of Art. 15, take the example:
EXAMPLE III.
(y_l)a;» + 2a: — 5?/ + 3 = o .... (1)
t/a:" + Ox — lOy = (2).
18
To render the division possible, we multiply the polyno-
mial in equation (1) by y, and going through the operation of
division we have a remainder
(—ly+Q)x + bf — ly.
Since we have introduced a factor in the dividend, we
must examine whether or not any foreign roots have been
added to the question. Taking the equations (Art. 15),
y = o yx^ -{- 9x — lOy = o,
we find their roots to be
y = X = o.
Should the final equations give these values among their
solutions, they must be rejected, as not belonging to the
given equations.
Proceeding now to the second division, after multiplying
the last divisor by — "7?/ + 9, to render the division possible,
we obtain for the last divisor
(— 7y + 9) z + 5^ — 7y,
and for the final equations
25y' — 70/ — 126y' + 4Uy' — 243y = o.
The roots of which are found by known rules to be
19
y = o.y = Uy=S.y = ^l±JjL^,y = ^±:Lt^.
The corresponding values of x deduced from the equation
( — '^y + 9) X + 53/" — ly = 0,
being
x = o, x = l, x = 2, x = — 5 — VJo, x = — 5 + VYo.
Here we see we have the solutions x = 0, y — 0, found
above. They must then be rejected, and as the multiplication
in the second division by — 7y + 9 has not introduced any
foreign roots, the proposed equations admit of the four fol-
lowing solutions :
Cy^l
1 \
^ —3 + 3^1^
sT 5
(x= 1
(x = — b— Vio
(3^ = 3
2^
( —3 — 3 Vlo
4^ 5
(x = 2
( x = — 5+ Vio.
If we had substituted the roots x = 0, y = o, in equatior?
(1), in the first place, we should have seen they would not
satisfy it, and we might have concluded that they formed
no part of the solution of the question.
20
18. Examples are frequently presented in which it is ne-
cessary in the process for finding the greatest common divi-
sor, to introduce as well as to suppress factors, to render the
division possible. The foregoing directions must be ob-
served, and all compatible values which are thus sup-
pressed must be joined to the solutions given by the final
equations ; while those which have been introduced must be
rejected.
19. If the given equations can be decomposed into com-
mon factors, their resolution will be very much simplified
by putting these common factors equal to zero, separately.
There may be two cases, 1st. the common factor may be a
function of one of the unknown quantities only ; 2ndv It
may contain both.
20. Let us examine the first case. Take the two equations
A = o B = o,
and suppose them to contain a common factor which is a
function of a;^ we may substitute for the equations the fol--
lowing :
f(x)xF{x,y) = o .... (1)
/(^)X? {x,y) = o .... (2).
These equations may be satisfied by making
/(^)=o.
As this equation contEiins only x and known terms, it
21
will give a determinate number of values of x, which will
satisfy the given equations independently of any determi-
nation of y.
But the given equations may be satisfied by either of the
following hypotheses, viz :
/ (a:) = and 9 (a;, ?/) = o . . . . (3)
/ (x) = and F (x, y) = . . . . (4)
or
F (x, y) = and 9 (.t, 3/) = o . . . . (5),
But the solutions resulting from the systems (3) and (4)
do not differ from those determined by the equation
/ {^) = 0,
for the values of x resulting from this equation, will when'
substituted in the equations
cp (x, y) = o F {x, y) = 0,
lead to a determinate number of values of y, which will be
included in the solutions of equation
/ (^) = o,
the roots of which will satisfy the given equations for any
values of y. Hence, to determine the remaining solutions
of the question, we have to obtain the values of x and y re-
sulting from the condition (5),
4
22
F (x, y) = o 9 {x, y) = o,
the values of which we can determine by the ordinary rule.
21. Tf now the common factor contain both x and y, the
two equations will assume the form
/ {x, y) xF (x, y)=o
f{x,y) X (?{x,y) =0.
Making in the first place
/ (^, y) = o,
the two equations will be satisfied. This equation shows
(Art. 5,) that by assuming any values of y, we shall have a
determinate number of values of x, and reciprocally, the
values of a: being assumed, those of y will be determined.
The equations therefore admit of an indefinite number of so-
lutions, resulting from the presence of the common factor,
But the hypotheses of
/ {x, y) = and 9 {x, y) — o
and
/ {x, y) = and F (x, y) = 0,
will also satisfy the given equations. These equations can-
not however give new solutions to the question, since they
23
will necessarily be included in those which result from the
condition
/ {x, y) = 0.
To determine new solutions, we must therefore take the
final conditions
cp {x,y) = o F {x, y) = o,
and determine the values of x and y, as in ordinary cases.
22. To apply the above principles, take the following
example :
EXAMPLE IV.
(a;2 ^ y2^ (yx — 6)(x—l)=0
{x^ + y') {2x — 3y){x — y)=o.
These equations give an indefinite number of solutions
(Art. 21) in consequence of the common factor {x^ + y^).
Suppressing this factor, we have the two equations
(i/x — 6){x — l) = o
{2x — 3y) (x — y) = o.
These equations are satisfied by either of the following
systems of equations :
24
yx — 6 = 2x — 3y =
yx — r Q = X — y =
X — 1=0 2x — 32/ = o
X — 1 = X — y =^ 0,
From which we deduce the following additional solutions ;
y = -f2 a:= + 3
y=—2 x = — 3
y = ±v/6 x==±^Q
y = + i a:= +1
y = + 1 a; =, 4- 1^
23. EXAMPLE V.
Let us take the equations
y {x — 1) {x -}- y) X {x + 1) {x^ — 2y — 1) = o
y (^x—1) {x +y) xy {x"^ — y^) = o.
These equations have three common factors, viz : y,
X — 1, x -\- y.
Placing them separately equal to zero, we have the three
equations,
y — o X — 1=0 X -[■ y = 0,
25
The first result shows (Art. 20) that the given equations
will be satisfied by a value of ?/ = o, independently of any
determination of x; the second gives a root of x = \,y be-
ing indeterminate ; while the third shows (Art. 21) that for
the common factor {x + y), the given equations admit of an
indefinite number of solutions.
We have therefore the following solutions from these com-
mon factors,
\y = o {x=l I x = — y
i X indeterminate ( y indeterminate ( y indeterminate.
The given equations present in addition the following sys-
tems of equations :
y ^= o X + \ =
y = o x^ — 2y — l=o
x'^ — y^ = X -{- 1 =
x^ — y^ = o x^ — 2y — \ — o,
which give the solutions
1.
y =
x = — I
2.
y =
X = + 1
3.
y = o
X = — 1
4. y = + 1 X = — I
26
5. y = — 1 ^ = — !•
6. y = 1 + v/2 a; = 1 + Vi
7. 3/ = 1 + ^2 ^=— 1 — v/2
8. ?/ = 1 — V2 a; = 1 — V2
9. 3/ = 1 — ^2 ^ = — 1 + ^^2'
Solutions (1), (2), and (3) are included in the solution
y = X indeterminate,
resulting from the common factor, y, while solutions (4), (7),
and (9) are given by the equation
X = —y.
We have therefore the following new solutions only,
y = _l ,?/ = !+ V2 (y = l— ^2
x = — 1 \ X =\ ■\- ■v/2 f ^ = 1 — ■\/2.
24. EXAMPLE VI.
ar^ + (8y — 13) a: + 3/2 — 7?/ + 12 =
a^ — • (43/ + \) X -\- y"^ ■\- hy = 0.
27
First Division.
x^ + (8?/— 13)a; + ?/2 — 7y + 12
a:^ — {\y + 1) ar + 3/^ +5^
(12y — 12) :c— 122/ + 12.
This remainder can be decomposed into factors, as follows :
12(y— l)(a;— 1).
The question is thus reduced to the soKition of the fol-
lowing system of equations :
1. ^3/-!="
Lr2— (4?/ + 1) a; + y2 ^ 5r/ = o
2.
X — 1=0
ap- — (4?/ + 1) a: + ?/^ + 5y-= o-
The solutions of which are readily found to be
2/ = 1 ar = 3
y = 1 a: = 2
y = a: = 1
y^—\ x= I
25. When the final equation is independent of y, and con-
tains known terms only, which do not of themselves reduce
to zero, the given equations are contradictory, and cannot be
28
satisfied by the same values of x and y ; for, the condition'
of their having common values, requires that they should
have a common divisor (iVrt. 7), and this common divisor
cannot exist where the final equation is not satisfied.
The following example will illustrate this principle.
EXAMPLE Vir.
x^ — i/ + 3 = 0.
First Division.
yx^ — {y^—3y — l)x + y
yx' — (/ — 3y) X
X + y.
Second Division.
x^ _ y2 _^ 3'
yx
T^ — 3/^ + 3 ^ + y
X +xy x — y
— xy — y^ +3
— xy — y^
+"3.
The last remainder being 3, the final equation is
But this equation is absurd, since 3 cannot be equal to
zero. The proposed equations have therefore no common
divisor, and are consequently contradictory.
29
26. Should the final equation reduce to zero, of itself, the
given equations will contain a common divisor independently
of any determination of y. If this common divisor contain
only one of the unknown quantities, x for example, the equa-
tions would be satisfied by a definite number of values of :r,
y being indeterminate (Art. 20) ; while they would admit of
an infinite number of solutions if it contained both x and y
(Art. 21). Take the following equations :
EXAMPLE Vlir.
x"" — 2yx'^ + 2y'^x — 5.r- +\Oyx + 6.c —y'' — rnf — Qy = o
x^ — byx"^ + Sy'^x — x — 4?/^ -\- y = o.
After the third division we obtain for the common di-
visor, there being no remainder,
(y *— 1 0^'+35/_50 // + 2i)x--z/ + 1 0y'—35y'-{-50y^—24y.
The proposed equations have therefore a common factor,
and by putting the above common divisor equal to zero, we
have
X = y.
Hence x — y is a common factor to the two equations,
and they therefore admit of an indefinite number of solutions
(Art. 21.)
If we divide the given equations by this common factor,
we shall have the two equations,
5
30
x^ — {2y -\- h) X -{- y"^ + by -\- Q =
x^ — Ayx + Ay^ — 1=0.
Operating upon these equations by the ordinary rule, we
shall have for the last divisor in x,
(2y _ 5) a; + 5y — 3y2 + 7,
and for the final equation,
y* — iQrf -}- 35?/2 _ 50?/ + 24 = o ;
from which we deduce the following solutions,
y = 1 a: = 3
y = 2 .-c = 5
y = 3 X = 5
y = 4 X = 7.
27. The substitution of the values of y deduced from the
final equation in the divisor of the first degree in x, may
cause the values of x to assume either of the following
forms, viz :
(1) X = a,{2)x= 0, (3) X = 6,{4)x = -.
28. Represent the divisor in x by Aa: — B, A and B be-
31
ing functions of y. In the first case, if 3/ = B be the value of
y, which by substitution in the equation
kx — B =0,
gives x = <x, the given equations admit of but this value x,
corresponding to the value of y = /3 ; since the equation
from which the value of a: is obtained is of thej^rs^ degree
only, and can give but one solution. This is also the case
when the value of y gives a; = 0.
29. In the third case, when we find x = ^, for the value
of y = /3, the two equations are contradictory ; for the
equation
Kx — . B =
can only give x =^, when the substitution of the value of
y makes A = and B equal to a finite quantity.
But when A = 0, we have from the nature of the above
equation, B = o also. Hence the equation
kx — B =
is absurd, for a: = ^ and y = /?. Further, the number B is
the common divisor which the substitution of y = /3 causes
the given equations to acquire, and if all the values of y pro-
duce in the same manner a numerical common divisor, it is
evident no values of x and y can satisfy the conditions of the
questions. The proposed equations are therefore contra-
dictory.
32
30. Finally, if y = [3 reduce A and B to zero, at the same
time, the value of x becomes - or indeterminate.
o
This result shows that the equation formed by placing the
divisor of the first degree equal to zero, does not make known
all the values of a:, which will satisfy the proposed equations
for the value of y = 13, since this equation reduces to zero, by
the substitution of this value of y, independently of any deter-
mination of X. It is therefore indeterminate, and if the value
of y be substituted in the given equations, it will cause them
to have a common divisor in x, of a higher degree than the
first. The degree of this divisor will depend upon the number
of multiple values of x, which correspond to the same value
of y. It will be of the second degree if there be two values
of a; to one ofy, of the third, if three, &c. When therefore
the divisor in the first degree becomes indeterminate, by the
substitution of a value of y, deduced from the final equation,
we make the substitution in the next superior divisor. If
this divisor be of the second degree in x, and admit of solu-
tion, there will be ticn values of a: corresponding to one value
ofy. If this equation also be indeterminate, we proceed to
the next superior divisor, and, in general, to that divisor
which does not reduce to an indeterminate form.
31. If we knew apn'on*, from the composition of the given
equations, that they contained multiple values of .r, for the
same value of y, we might at once substitute the value of y
in the divisor of the degree corresponding to the number of
multiple roots ; since its substitution in a divisor of an infe-
rior degree would lead to an indeterminate result.
32. If all the values of y gave multiple values of x, the
operation for obtaining the greatest common divisor would
33
necessarily stop at a divisor of a degree, corresponding to
the number of tliese multiple roots ; as is shown by the fol-
lowing example.
EXAMPLE IX.
x' + 2yx' + (2.y2 + 1 ) o;^ + (y^ + Qy^ + y ^ 8 1 ) a: + 2/' = o
x' + 2yx2 + 2y'^x + y' + 9?/2 — 81 = o.
First Division.
2-^+27/x3+(-2(/2-fl)c3+(2/3+9,/2-f7y— 81).r+3/2
2:3+2y.r24-2y2.T+y3-f 9^/2—81
x'+y-i:-\-y^
Second Division,
x^ + 2yx^ + 2y'^x -\- if + 9y^ — 81
x^ + y:?:^ -j- y-x
x^ -\-yx-{- y^
X +y
yx'^ + ?/^x + y^
yx^ + ylr + y^
~^f — ^ ^-
In this example the operation stops at a divisor of the se-
cond degree in x, the final equation being
9/ — 81=0,
so that for each of the values of y - ± 3, deduced from this
equation, there are two values of x.
34
33. The degree of the final cannot exceed the product of the
numbers which represent the degrees of the given equations.
M. PoissoN demonstrates this principle in the following
manner:
Let
a:"' + Px—^ + Qa:'"-2 + . . . + Tx + m = o
a;" + Fa:"-' + Ofx"'^ + . . . + Tx + u' =o,
be the two given equations ; the coefficients P, Q, P', Q', &c.
being functions of y of the most general form (Art. 1), as
follows :
V = a + hy,V' = a' -\- b'y, Q = c + ^y + ey\ &c. «Sz;c.
If we substitute for P, P', &c., their values, the above
equations become
x-^+ia^ by) x^-' + {c + dy + ey^ 3:^-2 . . . ^ y^ =
a:" + {a' -f b'y) a:"-* + (c' + dly + e^) a:"-^ . . . + y" ^-r 0.
But the degree of the final equation will not be dimin-
ished, if we reduce the coefficients of these last equations to
the term which contains the highest power of y, since the
degree of the two equations will not be changed by this
operation. We shall then have
a;"' + it/a;""-' + ey^x'"'^ . . . + 2/"' =
35
x" + b'yx"-' + e'y^x'"-^ . . . + y" = o;
which may be placed under the form
If we regard I -I as the unknown quantity in these equa-
tions, and represent by a, [3, y, &c. the roots of i - I in the
first, and by a', ^', /', &c., those in the second equation, we
shall have
or
(a; — ay) (x — (3y) {x — yy) &c. = o . . (1)
{x — ci'y) {x — (S'y) {x —yy) &c. = o . . (2).
36
If now we substitute in equation (1) each of the roots of a:
deduced from equation (2), viz :
x = a.'y X — /S'y X — y'y, &c.,
we shall have n equation of the following form,
rj" (a' _ a') (a' — /3) (a' — y) &C. =
y-{^'~a) (/3'_^) (/3' — 7)&c. = o
3/" (7' — a) (j' — (5) (7' — 7) &C. ~ 0, &LC. &C.
each being of the mi\\ degree, and giving m values of y.
The whole number of values of y will therefore be m x n,
which will represent the degree of the final equation. The
degree^ of the final equation cannot therefore exceed this
number.
EXAMPLES.
1.
yx — y^ — y — 1 =
yx — ■y'^ — 1 = 0.
Final equation, — y — 0. Common divisor, yx — t/^ — 1.
Equations contradictory. (See Art. 29.)
2. ..2
x^+ (8y— 13) a; + 7/2 — 73/ -I- 12 =
(4?/+ l)a; + ?/2 + 5?/ = o.
Common divisor, ~ [y — 1) (12x — 12). Final equation,
{y~\){y^ + y)^o.
37
Solutions.
y = \, 2/ = l, y = o, y = —i
0^ ^^ ^f CC ^^ Oj 2^ =— Ij CC ^^ L»
I x'—3yx'-\-3x''+3xf—6yx—z--y^+Si/+i/—3=o
^' \ x'+3yx'—3x'+3xy'—6yx—x+y'—3y'—y+3=o.
The remainder of the second degree in x is divisible by
(y' — 1), which we suppress ; after this suppression, the re-
mainder of the first degree is divisible by y^ — 2y. Final
equation then becomes y^ — 2y — 3 = o, and the common di-
visor x. (See Article 14.)
Solutions.
y=^l,y=l,y=l, y=o, y=o,y =— l,y = 2,y = 2, y = S
x = o, X = 2, x = — 2, x—1, x = — 1, x=o,x—l, x= — 1, x = o.
J 3a;2 — 5ya;2 — {3y^ — 30y) x + 30y^ = o
( 6z2_ 102/2 + Uxy=o.
Final equation in x, 113a:^ — 1310:i-' + 1800:r' = o.
Solutions.
a: = o, X = 0, a; = 10, a; = — -
1 1 O
72
y = o, y =0, y = 15, y = — _^
38
x^ + y — 5 =
x^ -\- f a-y ■\- y^ = o.
Solutions.
y=2, y=— 2, y= 1, y = — 1
x = — 1, X = I, x = — 2, a: =2
6.
( x^ + 2xy -\-y^ — 1 = o
( x^ — y^ — Qy — 9 = 0.
Solutions.
y = — \, y = —2
a; = 2, X = 1.
„ ^ x^ — 2yx -\-y^ — 1 = o
i x"" +2{y — 5)x-\-y~—\(iy + 2\=n.
These equations can be placed under tlie following form
\x-{i,+ \)\ \x-(y-\)\=^o
\x-{^ — y)\ \x — {l — y)\=o.
Solutions.
y = l, 3/ = 3, 3/ = 2, y = 4
x = 2, x=4, x=l, x = 3.
3d
? a;2— 33/a; — 5y + 2y2_ lly _ 6 = o.
These equations may be written thus :
\x — {y + l)\ \x—{3 + y)\=o
la: — (2y— 1)^ \x — {6-^y)\=o.
Putting these factors two and two equal to zero, we have
x = 3, z = 7, 1 = 6, 3 = 6,
y = 2, y = 4.
The two last results are absurd. Final equation, y^ —
6y -{- 8 = 0.
x^ — 4yx + 4y' — 1 = o
Which may be placed under the following form
\x — {2i/+l)\ \x — {2y—l)\=o
)x — (2y + 3)( \x — {2y + 2)\=oi
which furnish the following absurd results :
40
1 = 3, 1=2, —1=3, —1=2.
If we had applied the ordinary rule for determining the
final equation, we should have found a numerical remain-
jdei (see Art. 25.)
Suppressing the common factor {x —y), Art. 14, we have
the following systems of equations :
(1) x-\-y — 1=0 and x-^y — 3 = o
(2) a; + 3/ + 1 = o and x — y — 3 = o
(3) X -{-y — 1=0 and a: + y + 3 = o
(4) X -'ry + \~o and x + y -{-2 = 0.
Equations (3) and (4) are contradictory.
11.
a;' — 2yx + 8 = o
:r' — 2y' + 14=0.
Final eq.uatiop, y^ — 8y^ -^^ 9 = o. Common divisor, yx-
f + 3.
Solutions.
y=3, y = — 3, y= + v/3i; y = — y^—i
x=2, X =.— 2, X - 4y/ZIY, * = — 4 \/HX
41
^{y — \)x'+y{y-\-\)x'+{^f-\-y—2)x-\-2y = o
'■■ \ra
)x'+y{y+i)x + 3f — l = o.
Final equations, y' — 1 = o. jDommon divisor, {y — J )
X + 2y.
The value of y = 1 must be rejected (Art. 29,) since it re-
duces the common divisor to 2.
Solutions.
y = — I, a; = — 1,
ANALYSIS OF CURVES.
1. We have seen in Analytical Geometry, that every
equation between two indeterminates, may be conceived to
express the relation between the abscissas and ordinates of
the curve, which this equation represents. By giving par-
ticular values to either of the variables, the corresponding
values of the other may be deduced, and all the points of the
curve determined. F(;llowing the course therein defined,
we may ascertain whether or not this curve is symmetrical
with respect to either or both the co-ordinate axes ; we
may also define its limits when any exist, by determining the
points at which the tangents are parallel to the axes, or by
ascertaining the existence and position of its asymptotes.
2. Beyond this, however, the powers of Analytical Geom-
etry end, and we are compelled to resort to those means
which the discovery of the science of the Differential Cal-
culus has placed at our command. By this we may not on-
ly verify the results of the geometrical analysis, but we may
trace with the most exact certainty, the course of any curve
however irregular, and define its properties however pecu-
liar. The sole difficulty consists in solving the algebraic-
equation which defines the curve. If this difficulty be re-
moved, we may readily trace its course. For, suppose that
the equation of the curve has been solved, and that X, X',-
44
X", &c. represent the roots of y, these roots being functions
of x; the question is at once reduced to an examination of
the particular curves, which are expressed by the separate
equations
1/c.X, y=X', 3/=X",«fec.
This examination will be effected by giving to x every
possible value, as well negative as positive, which the func-
tions X, X', X", &c. admit of, without becoming imaginary;
and the curves which result will be the different branches of
the curve represented by the given equation. The extent
of each of these branches will depend upon the different so-
lutions which correspond to its particular equation. If any
of the equations
y = X, y = X', y = X",
exist for infinite values of x, it follows that these branches
extend indefinitely in the direction of these values. Let us
apply these principles to the analysis of the
Lemniscate Curve.
8. Take the equation
y* — 9Ba2y2 _^ looa^x' — x' = o.
I'his being a quadratic equation, its solution is efTected by
45
the ordinary rules for such equations, and we find the values
of y to be
y - ± \/48a2 ± V 2304a' — lOOa^^z;^ + x'
or putting
2304a* — lOOa^x^ + x* = N,
the four values of y become
y = V 48a2 + v/N . . (1) 3/ = ^48a^ — ^/N . . (2)
= — \/48a2+%/N . (3) 3/ = — 'y/48a2_v/N . (4)
It is required now to ascertain each of the curves which
these equations represent.
We see in the first place, that the values (3) and (4) only
diflTer from those of (1) and (2) in the sign, and consequently
must represent similar branches whose position with respect
to the axis of x alone differs. Further, as the quantity N
contains even powers of x only, its value will not be
changed by substituting a negative for a positive value of x.
The parts of the curve which lie on the right of the axis of
y, are therefore similar to those which lie on the left of this
7
46
axis. Hence the curve is divided by the co-ordinate axes
into four equal and symmetrical parts.
4. Let us now examine more particularly the values (1)
and (2).
They can only be real so long as the quantity N is posi-
tive ; the Umit to the real values of y will then be found by
making
N = a:* — lOOa^^s + 2304a* = o.
But this equation can be decomposed into the factors
X — 6a, X + 6a, x — 8a, x -\- 8a,
and the values of y for equation (1) will be
y = V 48a2 -j- v/(.y _ 6a) {x + 6a) {x — 8a) {x + 8a)
For any values oix greater than 6a, but less than 8a, the
values of y will be imaginary, since the factor {x — 8a) un-
der this supposition is negative. No part of the curve then is
embraced within the Umits
X = Qa, X = 8a ;
but for values of x greater than 8a, the factor {x — Sa)
becomes positive, and the values of y always real.
47
The values of y which correspond to the three values of a:,
X = 0, X — Qa, X = 8a,
will be found from equation (1) to be
Equation (1) gives then, 1st, a part DF (see Figure 1)
"which extends from the point D, taken on the axis AC, to
the point F, whose abscissa AE = 6a; 2ndly, a part HX,
which beginning at the point H, whose, abscissa AG = 8a,
extends indefinitely in the angle BAG.
5. Equation (2), which, when the factors of N are intro-
duced, becomes
y = V 48a2 — V (^ _ 6a) [x + 6a) {x — 8a) {x + 8a),
will in like manner give imaginary results between the limits
a: = 6a, a: = 8a ;
but for the values
X = 0, X = Qcux = 8a,
48
we get
y = 0, y= "v^isi?, y = ^^iSo^,
which show, 1st, that equation (2) gives a part AF, which
unites with the part DF given by equation (1) at the point
F, for which the two ordinates are equal ; 2ndly, beginning
at the point H, equation (2) gives a part HK, in which y
decreases until VN = 48a^, when it becomes zero, and cor-
responds to the point I. For N greater than 48a^ the quan-
tity under the radical becomes negative, and y imaginary.
The Branch of the Curve corresponding to equation (2)
does not therefore extend beyond the point I. The abscissa
of this point is evidently determined by making y — o'ln
equation (2). We find
X = ± o, X = ± 10a.
The two first values correspond to the point A, the others
to the points I and I'.
6. We might continue this discussion, which is in every
respect analogous to the general discussion of an equation of
the second degree in analytical geometry, and ascertain
whether this curve has asymptotes ; but as the differential
calculus abridges this investigation, we will at once apply it
to this purpose, and then proceed to the determination of the
singular points of the curve.
7. An examination of the four values of y, Arts. 3 and 4,
Fig. 2.
50
has already shown that the curve we are discussing has in
each angle of the co-ordinate axes an indefinite branch. Let
us see whether these branches have asymptotes.
We know that if any curve MX (Fig. 2) have an asymp-
tote RS, the tangent MT approaches more and more a co-
incidence with the asymptote as the point of tangency is re-
moved from the origin. Under this supposition, the points
T and D in which the tangent intersects the axes, will con-
tinually approach the points R and E, in which the asymp-
totes intersect the axes ; so that AR and AE are limits to
the values of AT and AD. Hence, to ascertain whether a
curve has asymptotes, it is necessary to determine whether
the expressions AT and AD, which represent the distances
from the origin to the points in which the tangent cuts the
co-ordinate axes, have limits for infinite values of x and y.
If they have, these limits being constructed will give the
points D and E, through which, if the line RS be drawn, it
will be the asymptote sought.
8. The expressions for AT and AD may be deduced at
once from the equation of the tangent line. The equation of
the tangent line is
ax
^ being the tangent of the angle which it makes with the
dx
axis of X. We may now obtain the distances AT and AD,
by making y' and x' separately equal to zero. By the first
supposition we have
51
a;'=AT=a; — y —^
in which the quantity y—> which is the expression for the
dy
subtangent PT, is taken negatively, since it is counted in an
opposite direction from the abscissa x'. Making now x' = o,
we have
v' = AD = V — X — ^.
^ ^ dx
9. To ascertain whether the given curve has asymptotes,
we must substitute the values of -^ and — , deduced from
dx dy
the equation of the curve, in the expressions for AT and
AD, and see what these expressions become when x and y
are infinite. We find, the first differential co-efficient of the
given equation after dividing by 4, to be
dx _ y' — ^Sa^y
dy x^ — bOa^x
Multiplying this value by y, and subtracting the product
from a;, we have after reducing.
^ Jy x' - SOa^x
By a simple transposition of the terms of the fraction,
52
which forms the value of — , we deduce that of -^, and
dy dx
we have
AD = V — a: ^ = !/^-48gy-a:^+ bQa^x\
^ dx y' — 4Sa^y
Putting in these expressions the value of y*, they become
x" — 50a^x
y^ — 48a^y
These values of AT and AD, continually diminish as x
and y increase, and when x and y equal ± infinity, they be-
come zero. We conclude, then, that the curve has two
asymptotes, which pass through the origin of co-ordinates.
Their angle is determined by seeing what -^ becomes when
X and y equal ± infinity. We have
dy _ x^ — 50a^x
dx y^ — 48a^y
when X and y are infinite, the first powers of y and x may be
neglected, and we have
53
ax
which shows that one of the asymptotes makes an angle
with the axis of x of 45° ; the other an angle of 45° + 90°
= 135°.
10. Let us now examine the singular points of the curve.
We find the first differential co-efficient to be
dy _x^ — 50a^a:
dx y' — 48a^i/'
To determine the points at which the tangent is parallel
to the axis of x, make -^ ± o. We find
dx
x^ — SOa^x = Of
which gives
X = o a; = + VSOa^ x = — VdOa^
The value of x— o, when substituted in the given equa-
tion, gives
y = ±0 y = ± y/9Qa^.
But when x =■ o and a: = o, we have
8
54
dy _x^ — 50a'ar _ o
dx y^ — 48a^y o
■which indicates a multiple point at the origin of co-ordinates,
(see Boucharlat's Differential Calculus, Art. 138.)
To determine the value of -^, we must pass to the second
dx
differential equation, which becomes, when x - o and y = o,
— 48a%2 + 50a^dx^ = o.
Hence
I=*v1
^50
dx ~ V 4q'
It follows from these values, that at the point A, the curve
is touched by two straight lines, which make angles with
the axis of x, the tangents of which are
+ v^^, and -v/-5
^48 ^48
the point A is therefore a multiple point.
11. The values of y = ± V9Ga^ correspond to the points
D and D', at which the tangent is parallel to the axis of x.
12. If we deduce the 3d differential equation, we find by-
making X = o,y =^ 0,
— 48a2 dy d^y — QGa'dy d^y = o,
55
or
-f 144a^ dy d}y =o;
from which we conclude that
The second differential co-efficient being zero, let us find
the third differential co-efficient from the fourth differential
equation. This equation, when we make x — o,y = o, and
—^ = 0, reduces to
— 4.48a2 dy d'y -f Qdy' — 6dx* = o
from which we deduce
dx dx' dx*
Hence
dx'
\48/
^ 48*
56
when -^ is replaced by its value ± v/ —
dx ^ 48*
But we have found (Boucharlat, Art. 123), for the dis-
tance between the tangent and the curve,
dx'\.2 dxn.2.2
When we make in this expression
dx"
and
> 148/
r50\«
dx'
^ 48
it becomes
_ . Vis/ h'
— 1
5 = ± ^'^^^ V ^^^ + &C.
7^ 1-2.3
48
which shows that the branch of the curve touched by the
straight line AL, which corresponds to the positive value of
57
-^, is above the straight line on the side of the positive ab-
scissas ; and below it, on the side of the negative abscissas.
The reverse takes w^ith respect to the tangent AL'. Hence
each branch of the curve undergoes an inflexion at the
point A.
13. If in the 2nd differential equation
3fdy^+y3d^y—'^a^di/—48ah/(Py4-b0aPdx^-\-50a^xiPx—33^dx^—x3(Pz = o,
we make x = o, and y = ± v^QGa' and -7- = o, we have
^ = — (± V96ay — 48a2 x ± ^/96a^
which gives a negative value for — ^ for the value of y, cor-
dx^
responding to the point D, and a positive value for D', which
shows that at D, the ordinate is a maximum, while at D' it
is a mitiimum. The ordinate at D' is regarded as a mini-
mum, because every increment to a negative ordinate is
equivalent to a decrement with respect to positive ordinates.
14. To find the points at which the tangents are perpen-
dicular to the axes of a;, we must put ^ = 6, or what is
dx
equivalent to it, place the denominator of its value equal to
zero. This gives
58
y" — 48a'y = o,
from which we obtain
y = o y = ± s/48a2.
The first value substituted in the equation of the curve
gives
lOOaV— a;* = o.
Hence
X = ±0 X r= ± 10a.
The roots x = ± o indicate the multiple point A, the
two others belong to the points I and I'. The values
y = ± V48a'' correspond to the points whose abscissas are
X = ±6a X = ± 8a.
One of these results makes known the points F, F', the
other the points H, H'. At each of these points the tangent
is perpendicular to the axis of a:, and as there are no points
of the curve between these limits, we can readily see in
what direction the curve is turned towards its tangents.
59
15. We might continue this discussion, and by analogous
means ascertain the singular points belonging to the branches
represented by the equations (3) and (4) ; but as we have
already seen that these curves are in every respect identical
with those just discussed, the number and character of the
singular points may be regarded as known.
J. p. Wright, Printer, 18 New Screet, N. T.
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