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that is, 0 4- = 180°, and the opposite angles of the quadrilateral figure are together equal to two right angles, and the quadrilateral is such as can be inscribed in a circle.

Ex. 5. To determine the maximum and minimum values of the central radii vectores of an ellipse, and their relation to a pair of conjugate axes.

Let the ellipse be referred to a pair of conjugate diameters, whose lengths are 2ax and 2bx, as the coordinate axes; and let o> be the angle between the axes, and r be the length of any central radius vector; then the equation to the ellipse is

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therefore employing an indeterminate multiplier, we have

I + A (x + y cos o>) I dx + I + A (y + x cos <■>) | rfy = 0;

and equating to 0 the coefficients of dx and dy, x v

^1 + A(# + y coso>) = 0, ^2 + A(y-( a?costo) = 0;

whence, multiplying the former by x and the latter by y, and adding, 2 2

+ o + A(^2 + 2a^cosa) + y2) = 0;

.-. 1 + Ar2 = 0, .-. A=--L;

and substituting,

/ 1 1 \ COSW _ COSO) / 1 1 \

whence, by cross multiplication,

. •. r* - (d2 + fij2) r2 + a!2 (sin to)2 = 0.

Let a and b be the greatest and least values of the radii vectores, so that a2 and b2 will be the roots of the last equation; then by the theory of equations,

a2 + A2 = flj2 + V, a2 62 = fll2 V (sin o>)2;

by means of which the values of a and b may be easily determined.

By a similar process may we determine the analogous relations of the principal axes of an ellipsoid to any system of conjugate axes.

Ex. 6. To inscribe in a sphere the greatest parallelepipedon. Let x2 + y2 + z2 = a2 be the equation to the sphere; and let u be the content of the parallelepipedon.

.•. u = 8xyz;

and taking the logarithmic differential,

Dm dx dy dz
— = 0= — + + —;
u x y z

also 0 = xdx + ydy + zdz;

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and the volume of the parallelepipedon = ——

Ex. 7. To find a point within a triangle, such that the sum1 of the lines drawn from it to the angular points may be a minimum." Fig. 20. Let p be the required point; and let Ap = x,

BP = tf, CP = Z, BC = C, CA = b, AB = C, BPC = 0, CPA = <f>,

APB = f; u = sum of the required lines; wherefore we have

u = x + y + z;

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170.3 In the present Chapter I shall take the following to be the type of an algebraical expression of the nth degree, and for convenience of reference shall symbolize it by f(x),

z"-p1xn-l+piz"-2-p3xn-3+ ... ±pn-xX+pn = Q=f{x), (1)

the upper or lower sign being taken according as n is odd or even; n being integral and positive; and no fractional or negative powers of x being involved in the expression; px, p-i, ... p„ being constant coefficients, real or imaginary, and the coefficient of the highest power of x being unity.

A root of such an expression is a value which, when substituted for the unknown quantity x, makes the whole to vanish; thus, if a is a root of f(x),

which, as was explained in Art. 60, always admits of being put

the latter pair of imaginary factors being called conjugate to each other.

Now of such general algebraical expressions as (1), almost all the properties which are proved in the usual text-books on the Theory of Equations arise from considering them in their resolved or analytical form; that is, as made up of factors of the form, x a, x b, ... x a + fiv —1, x—a—fi V—1 ... j such are the relations which exist between the roots and the coefficients, between two equations the roots of one of which are symmetrical functions of those of the other, &c. But a

/(«) = 0.

An imaginary or impossible expression is of the form a + b»J1;


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general treatment of the subject requires the expressions to be considered in their synthetical or uuresolved state; and we propose in the following Articles to exhibit such properties of them in their compounded state, as fall within the grasp of the preceding principles.

In the first place I must remark on the continuity of such expressions; and I shall hereby be led to shew that every equation, be its dimensions odd or even, has a root. In other words, I shall prove that a value, real or imaginary, exists, which when substituted for x in f(x) makes f(x) = 0.

If it is thought that an unfair assumption is made in the following Articles in the extension to imaginary quantities of processes of differentiation which have been hitherto applied to possible quantities, let it be borne iu mind that the principles of differentiation as unfolded in the first and second Chapters require only the law of continuity to be satisfied in the functions to which they are applied, and that imaginary quantities satisfy that law as well as possible ones. And I would also observe that logarithmic and circular functions are related to each other by means of the exponential equivalents, and that these involve symbols of impossibility; and as we have introduced and operated on each of these in the previous Chapters, our processes have been applied to impossible as well as to possible quantities.

171.] Positive real values may be assigned to x in (1) so great that the corresponding value of xn shall be greater than the sum of all the following terms.

Let us take the case which is most unfavourable, that namely in which all the terms are positive, and in which consequently their sum is greater than it would be if some of the terms were negative. Let pk be the greatest of all the positive coefficients —pi, Pi, -- +Pni then

piX"-1+p2x"-2... +pn is less than pk{x"~l+x"-1+ ... +x

- - - - - - - - - is less than pk -—^,

x l


- - is less than pk =-,

x l

and therefore is less than x", if pk = x — 1 ; that is, if x = pk +1, xn is greater than the sum of all the subsequent terms; and

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