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a
If r be supposed positive, a2 will be positive and real, and con- sequently a will be real, and therefore, also, b and c will be real. Having determined in this manner the three quantities a, b, c,
we obtain the transformed equation
(x2 − a2)2 + b(x + a)2 + c(x − a)2 = 0.
Putting the right-hand side of this equation equal to y, and considering the curve having for abscissæ the different values of y, it is plain, that when b and c are positive quantities this curve will lie wholly above the axis and that consequently the equation will have no real root. Secondly, suppose that b is a negative quantity and c a positive quantity; then x = a will give y = 4ba2,—a negative quantity. A very large positive or negative x will then give a very large positive y,—whence it is easy to conclude that the equation will have two real roots, one larger than a and one less than a. We shall likewise find that if b is positive and c is negative, the equation will have two real
roots, one greater and one less than −a. Finally, if b and c are both negative, then y will become negative by making
x = a and x = −a
and it will be positive and very large for a very large positive or negative value of x,—whence it follows that the equation will have two real roots, one greater than a and one less than −a.
The preceding considerations might be greatly extended, but at
present we must forego their pursuit.
It will be seen from the preceding example that the consider- ation of the curve does not require the equation to be freed from fractional expressions. The same may be said of radical expres- sions. There is an advantage even in retaining these expressions in the form given by the analysis of the problem; the advantage being that we may in this way restrict our attention to those signs of the radicals which answer to the special exigencies of each problem, instead of causing the fractions and the radicals to disappear and obtaining an equation arranged according to the different whole powers of the unknown quantity in which frequently roots are introduced which are entirely foreign to the question proposed. It is true that these roots are always part of the question viewed in its entire extent; but this wealth of alge- braical analysis, although in itself and from a general point of view extremely valuable, may be inconvenient and burdensome in particular cases where the solution of which we are in need cannot by direct methods be found independently of all other possible solutions. When the equation which immediately flows from the conditions of the problem contains radicals which are essentially ambiguous in sign, the curve of that equation (con- structed by making the side which is equal to zero, equal to the ordinate y) will necessarily have as many branches as there are possible different combinations of these signs, and for the com- plete solution it would be necessary to consider each of these branches. But this generality may be restricted by the partic-
ular conditions of the problem which determine the branch on which the solution is to be sought; the result being that we are spared much needless calculation,—an advantage which is not the least of those offered by the method of solving equations from the consideration of curves.
But this method can be still further generalised and even rendered independent of the equation of the problem. It is suf- ficient in applying it to consider the conditions of the problem in and for themselves, to give to the unknown quantity different arbitrary values, and to determine by calculation or construc- tion the errors which result from such suppositions according to the original conditions. Taking these errors as the ordinates y of a curve having for abscissæ the corresponding values of the unknown quantity, we obtain a continuous curve called the curve of errors, which by its intersections with the axis also gives all solutions of the problem. Thus, if two successive errors be found, one of which is an excess, and another a defect, that is, one pos- itive and one negative, we may conclude at once that between these two corresponding values of the unknown quantity there will be one for which the error is zero, and to which we can approach as near as we please by successive substitutions, or by the mechanical description of the curve.
This mode of resolving questions by curves of errors is one of the most useful that have been devised. It is constantly em- ployed in astronomy when direct solutions are difficult or im- possible. It can be employed for resolving important problems of geometry and mechanics and even of physics. It is prop- erly speaking the regula falsi, taken in its most general sense and rendered applicable to all questions where there is an un- known quantity to be determined. It can also be applied to
problems that depend on two or several unknown quantities by successively giving to these unknown quantities different arbi- trary values and calculating the errors which result therefrom, afterwards linking them together by different curves, or reduc- ing them to tables; the result being that we may by this method obtain directly the solution sought without preliminary elimina-
Fig. 3.
tion of the unknown quantities.
We shall illustrate its use by a few examples.
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