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Alpe Amr2 in pol
Whence x =

" po? - a?! 9 72 - 62! * pos - ca. Multiply by l, m, n, and add. Then by the original conditions and dividing by loo?,

[2 mo na

a+ moet – bu + monde - m2 = 0, a quadratic equation for determining r?, and consequently r.

This is the equation in the Wave Theory of Light by which the velocities of a wave propagated in a crystalline medium are determined. The surface pot = aor? + by2 + c modo is called the surface of elasticity. See Fresnel, Mémoires de l'Institut, Vol. vii. p. 130, and Herschel's Light, Sect. 1012.

(16) To find the area of a section of the ellipsoid,

as tūtent = 1, made by the plane lx + iny + nx = 0.

By the same method as in the last example we obtain as the equation for determining the principal axes,

a'la bm cana

pode - a* * mo 12 * pode - a* * The last term of this when arranged according to powers of poe is


alo + bonn' + c’n?' and this being equal to the product of the roots, the area of the section is


(a? 12 + b2 m2 + con”)! (17) To find the volume of the ellipsoid whose equation is

ax® + a'y' + a" x + 2 byz + 26'x x + 26"xy = c.

As in the preceding examples we have first to find the value of the principal axes, or rather of their product; and if this be aßy, then the volume of the ellipsoid will be

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To eliminate l', y, x, multiply the first of these by

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the third by

and add, then y and x disappear, and w dividing out, there remains 4- a) (-a) (-a") - (0 - a) – 0" (5) - a')

- 3** (- a") – 2666" = 0; a cubic equation in If it be arranged according to powers of pl, the last term with its sign changed will be equal to the product of the roots, that is, to the product of the squares of the principal axes; and its square root is the quantity which we seek. Multiplying it therefore by **, we find that the volume of the ellipsoid is equal to

3 (aa'a" - abi - a'b'? - a"6"2 + 2b6'6")!


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(18) To find the least ellipse which will circumscribe a given triangle.

Let ABC (fig. 10) be the triangle. Take C as the origin, CA, CB as the axes of x and y. AC = a, BC = h, ACB = 0. · The general equation to an ellipse is

Aw* + Bxy + Cy® + D x + Ey + 1 = 0, which involves five arbitrary constants; three of these may be determined by the conditions that the ellipse shall pass through the three points A, B, C. Instead however of directly expressing the undetermined coefficients in terms of those which are determined, it will conduce to the symmetry of our analysis to assume two indeterminate quantities of which the coefficients of the equation are functions which can be determined by the conditions of the ellipse passing through the three given points; and then the actual values of the indeterminate quantities may be found by the condition of the minimum. The two quantities which we shall assume are the co-ordinates of the centre

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