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therefore r + pr is a constant, or a maximum, or a minimum; but it cannot be a maximum, for such a value would be inconsistent with the geometrical possibility of the problem: therefore it is in general a minimum, and may sometimes be constant; the former case is that of an ordinary caustic; in the latter the refracted rays converge to a single focus.

829.] Hence also we may prove that all caustics are rectifiable. Let £, r, be the current coordinates, and da the length of the element of the curve of the caustic, so that da* = d?j2 + tf£*;

therefore ~, ^ are the cosines of the angles made by its

da d<T b J

tangent with the coordinate axes;

dr, _ r, y d$ _ j x

"da~ r'' da~?'

therefore from (95), dr = p dx ± dr) dy; (98)


and differentiating the latter of (96),

/dr* = (f-*) (rff - dx) + (V-y) (dr, - dy);

■■■ dr'=d2(di-d*) + d£(dr,-dv),

d£dx + dr, dy = da'

= aa;

.-. ,xda = dr + pdr',
.-. pa = r + pr' + c. (99)

An expression exactly analogous to that of Art. 292, and to
which therefore a similar mode of explanation is applicable; and
therefore the length of the caustic curve is equal to that of two
straight lines increased by a constant which is to be determined
by the data of the particular problem; but in all cases, if
fi i"\ T\t ai ri r2 > represent two sets of corresponding values,
H (tra — <ra) = ra - n + y. (r2' - r{).
The law of refraction becomes that of reflexion, if /u = — 1;
and therefore the properties of caustics by refraction proved
above arc likewise true of caustics by reflexion; attention must
however be paid to an ambiguity of sign, of which no notice has
been taken in the preceding investigation.

330.] To determine the caustic by refraction of rays refracted at a plane surface; see fig. 122.

Let s be the source of light; Sp the incident ray; RPr the

refracted ray; As = a, Ap = y; therefore the equation to the

refracted rav is . e

J ri~y tanPTA.f,

T being the point where Pr intersects A sc.

And since sinSpn = p sinrrn, .•. sinpsA = /* Suipta,


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{M»a*-f 0*»-l)y»}* therefore the equation to Pr is

n-9 = - ; - ~y - -J. (ioo)

Differentiating which with respect to y, and reducing

whence by elimination, and from (100),

f*-0i»-l)M = p*a*. (102)

which is the equation to the evolute of a hyperbola or of an ellipse according as p is greater or less than unity.




331.] An explanation of the mode of generation and of the equations of such curved surfaces and curves in space as are needed for illustration in this and the following Chapters, requires more room than we can afford to give; but it is the less necessary to introduce it, as the ordinary text-books contain sufficient information. It is however desirable to explain the equations to the straight line and the plane, in the forms which we shall employ, as a familiar knowledge of them is requisite to a due understanding of our processes.

(1) To find the equations to a straight line in space.

Let £, r\, C be the current coordinates to the straight line; x, y, z the coordinates to a point through which the line passes; A, n, v the direction-angles of the line; that is, the angles between a parallel line through the origin and the coordinates axes. And let r be the distance between (w, y z) and (f, Jj, f); then the equations to the line are

(— x n — y (— z ,

= -—i = = r, (1)

cos A cos /* cos V

the last of the equalities following by reason of Preliminary Theorem I.

If therefore the equations to a straight line are given under the form ,


each of these equalities is by reason of the same Preliminary Theorem equal to

(L2 + M2-f-N2)V (3)

and therefore comparing (1) with (2) and (3),

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r (l2 + M2 + N2)*

and therefore L, M, N in (2) are proportional to the directioncosines of the line, that is, to the cosines of the angles between the line and the coordinate axes.

(2) To find the equation to a plane.

A plane is a surface generated by a straight line revolving round another straight line which is at right angles to it.

Let the origin be at the point o in the straight line Oq, fig. 123, round which the perpendicular and generating line Qp turns; and let A, p, v be the direction-angles of Oq; let f, 7j, f be the current coordinates to any point p in the line Qp which is in any position; and let Op = p, Oq = 8; then the

> v

direction-cosines of Op are -; and since Oqp is a right


angle, Oq = Op cos Poq, that is,

8 = a \ - COS A + - COS a + - COS V \ i

(p p p )

.•. f cos A + Tj cosju -f fcos v = 8; (5)

and as this relation is true for every point in Qp, and in every position of Qp, it is according to our definition, the equation to a plane; A, p., v being the direction-angles of the normal to the plane, and 8 the length of the perpendicular from the origin on the plane.

Equation (5) is evident by the theory of projections, the lefthand side of the equation being the sum of the projections of the broken line Omnpq on the line Oq.

If therefore the equation to a plane is given in the form

+ Btj + Cc= D; (6) on comparing this with (5), we have,

T^t = — = — = T = (a2 + B2 + C2)*: (7) cos A cos p. cos v 8

whence it appears that A, B, C, D are proportional respectively to the direction-cosines of the normal to the plane, and to the length of the perpendicular on the plane from the origin.

332] To find the equation to a tangent plane to a curved surface at a given point.

Let the equation to the surface be

F (*, y, z) = 0. (8)

Our present object is to shew that if a straight line is drawn through a point on the surface (x, y, z), and through a second point {x -f dx, y + dy, z + dz) infinitesimally near to it and also on the surface, the locus of such tangent lines is in general a plane; and is what is called the tangent plane. Of course it is manifest that the number of points (x + dx, y + dy, z 4 dz) contiguous to the first point is infinite, and so therefore is the number of tangent lines.

Let £, Tj, ( be the current coordinates to one of the tangent lines, and x, y, z the coordinates to the point of contact on the surface; then the equations to the line are

t=E = 1=1 = <=£. (9)

L . M N

as the line passes through the point (x + dx, y + dy, z + dz), we have ■ , ,

df=dl = dZ; (10) L M N

and therefore by division,

j-x _ n-y _ C-z _ r
dx dy dz ds1 -

r being the distance of (x, y, z) from any point (f, rj, () on the line, and dg = ^ + dyi + dz%^. (12)

that is, ds is the distance between the two points on the surface through which the line passes; (11) therefore are the equations to any straight line touching a surface at a given point.

But as the second point through which the line passes is on the surface, though it may have any position infinitesimally near to {x, y, z); so dx, dy, dz, must be consistent with the equation to the surface. If therefore at the point under consideration

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