2. The principle of continuity, by which the differences of velocity and displacement of points indefinitely near to one another on opposite sides of the surface of separation of the media are infinitesimally small.

3. The principle of sudden change at the surface of separation, by which it is assumed that the change from the incident to the reflected and refracted ray takes place immediately.

4. The principle of the constitution of the ether, in which it is assumed that the pressure of the ether is the same in all media, and that the difference of the velocity of light is due to the difference of the densities of the ether, so that the velocity of light in a medium is inversely proportional to the square root of the density of the ether.

If a denote the amplitude of the incident, b of the reflected and c of the refracted ray, where b and c are estimated positively when the direction of the vibration of the reflected or refracted ray coincides with the direction of vibration of the incident ray when the reflected or refracted ray is turned so as to be in the prolongation of the incident ray (fig. 80), and if i, be the angles of incidence and refraction; then, by principles 1 and 4,

If i+90°, then taniμ, and the light which is reflected at the surface will be completely polarized in the plane of incidence.

If the light fall on a series of parallel plates, then, since i+i' = 90°, the light after any number of refractions, internal or external, will always be incident at the polarizing angle, and the reflected light will be completely polarized in the plane of incidence.

1. IF a binary quantic contain a linear factor a times and not more, prove that the Hessian will contain the same linear factor 2a 2 times and not more.

Find the conditions that a binary quartic may be a perfect square, and considering the coefficients as being each of the order unity, shew that the order of the system is equal to 4.

The repeated factor may without loss of generality be taken to be x, the quantic is then xp, and it is to be shown. that the Hessian contains the factor x-, and not any higher power of x. The first differential coefficients of x

are

Hence the Hessian contains the factor 2-2, and it only remains to show that it does not contain any higher power of x; this is so if the coefficient of x2-2 does not vanish

= 0.

for x=

BB

Suppose

is of the order n, and write A, B, C for its

second differential coefficients, we have

But the quantic xp contains the linear factor a (and not more) times, hence does not contain the linear factor x, and consequently its second differential coefficient C does not vanish with x; and the remaining factor

a+n-1

does not

n (n − 1)2 vanish for any positive integral values of a or n; hence the Hessian contains a non-vanishing term in 2, or it contains the linear factor 2a-2 (and not more) times.

If a binary quartic contains a linear factor twice, then the Hessian contains the same factor twice; and hence if the quartic is a perfect square (that is, if it contains two linear factors each twice), the Hessian will contain the same factors each twice, or it will be a mere constant multiple of the quartic. Taking the quartic to be

these being, of course, equivalent to a two-fold relation between the coefficients (a, b, c, d, e).

The order of the system is equal to the number of solutions obtained by combining with the foregoing a number of arbitrary linear relations sufficient to render the system determinate; that is, 2 arbitrary linear relations. The conditions express that there exist quantities (a, B, y), such that (a, b, c, d, eXx, y)* = (ax2 + 2ßxy + vy2)2

identically, viz. that we have

2

a, 4b, 6c, 4d, e=(a23, 4aß, 2ay +4ẞ”, 4ßy, y2).

Imagining these values substituted in the arbitrary linear relations, we have for the determination of the ratios a: B: Y two equations of the form (a, B, y) = 0 giving 4 systems of values of a, B, y, and therefore also 4 systems of values of (a, b, c, d, e); or the order of the system in (a, b, c, d, e)

is 4.

2. State and prove Sturm's theorem for determining the number and position of the real roots of an equation.

If

prove that

nX1 = (2n − 1) xX11 − (n − 1) X-29

and hence shew that the roots of the equations X1 = 0, X=0,... are all real and between 1 and 1.