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Application of this method to the solution of the equations 205-1=0; x-1=0; 11-1=0; 213—1=0.
(Lagrange, Equ. Num. Note 14; G. A. 77; Leg. 435–74.)
GENERAL SOLUTION OF EQUATIONS.
(51.) The method of symmetrical functions is inapplicable to the solution of equations of more than four dimensions, because the reduced equation is of 1.2.3... (n—2) dimensions, if n is a
1.2.3...n prime number, and of
dimensions, if n (p-1)p(1.2.3...9)” is composed of two prime factors p, and q; both of which quantities are > n, if n > 4.
(Lagrange, Equ. Numer. Note 13.) Wronski has given what he asserts to be a general solution of equations of all degrees.
(Résolution générale des Equations.) Torriani has published a Memoir, the object of which is to shew that Wronski's solution is incorrect.
(Hist. da Acad. de Lisb. Tom. 6.) A general solution of equations of the fifth degree cannot be obtained.
(Abel, Bulletin Univ. des Sciences; Ann. 1826.) A general solution of equations of any degree superior to the fourth cannot be obtained.
(Ruffini, Theor. delle Equaz. Cap. xiii.)
EQUATIONS OF THE FIRST DEGREE.
(52.) Solution of one equation containing two unknown quantities.
Let the equation be ax + by=è.
If a and 6 have a common factor c, this must also be a factor of e, otherwise the equation is impossible in integers ; a will therefore be considered prime to b.
The above equation is always possible if c > ab-a-b.
Let the value of y obtained from (1) be
e - ac
e - a
f-gw being the integral part of the fraction
from this last equation we obtain a value of un-1 which is an integer for all integral values of un; and thence an integral value of Un-2, and so on, until we obtain the values of ,
X = A, Un + B1,
If a and 6 have different signs, the number of solutions is infinite; in this case 4, and A, are both positive.
If aab, the number of solutions is finite; in this case A, and A, will have different signs. Suppose A, negative, then the number of possible solutions cannot exceed the number
of values of un
A (W. 367-9; E. Pt. II. 1-23; Bour. 1229.)
Second Method.  Let the equation be aa— by=+1. Convert the fraction into a continued fraction, and let
be the converging fraction immediately preceding
then (Art. 30.) a Bm - bAn= +1,
If ax - by=1, and a Bon-Am= -1, the given equation will be satisfied by assuming
If positive values only of w and y are required, we must have
If aw — by=-1, and a Bm -- bam=1, the values of , and y may be similarly obtained.
If aw — by aBm-b Ama
it will be sufficient to assume
x=nb + B, y=na + Am
 Let the equation be
and as before let
* be the converging fraction immediately
preceding. The general values of w, and y are
w=nb+cBin y=na+Cam; + or according as a Bm-b Amy and c, have the same or different signs.  Let the equation be
a x + by=c.
id=CBm — nb, y=na - CAmi n being so assumed, that w and y may both be positive.
The number of solutions equals the integral part of
unless CBb, in which case the number of solutions will be one less than the above number.
(B. 159-61; G. A. 24; Leg. 12—4.) (53.) Solution of two equations containing three unknown quantities.
Eliminating , and reducing the resulting equation to its lowest terms, we obtain
a w + by=e.
Let the values of x and y in this equation be
X = Ayu + B,
y= 4,0 + B, ; substituting these values in (1) and reducing, we have fx +gu=h;
(4) obtaining the values of x and u in terms of a new indeterminate, V, and substituting for u its value in those of X, and y, the final result is
x=C,v + EN
x=C2v + Ez. The conditions of possibility of the equations (1), (2), may be determined from the relations that must exist between the coefficients of (3) and (4), in order that those equations may be possible, (Art. 52.)
If a,=b,=c, =1, and a; > > >ce, then e, must be >c,en, and <a,e,, in order that it may be possible to obtain a positive result.
(B. 164; E. 24–30; Bour. 136_9.) (54.) Solution of one equation containing three unknown quantities. Let the equation be
a w + by +0x=e, in which at least two of the coefficients, as a, and b, are prime to each other.
being, as before, the nearest converging fraction to
Bm the values of and x=(e-cx)B,m— nb, y=na
y=na-(e-c%) Ami subject however to the following conditions,