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student may easily verify it by writing, in accordance with the results of Chapter V,

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and the results just given follow immediately.] 291.] That the equation

p = $(t, u) = .af (t, u)

represents a surface is obvious from the fact that it becomes the equation of a definite curve whenever either t or u has a particular value assigned to it. Hence the equation at once furnishes us with two systems of curves, lying wholly on the surface, and such that one of each system can, in general, be drawn through any assigned point on the surface. Tangents drawn to these curves at their point of intersection must, of course, lie in the tangent plane, whose equation we have thus the means of forming.

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where the brackets are inserted to indicate partial differential coefficients. If we write this as

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293.] As a simple example, suppose a straight line to move along a fixed straight line, remaining always perpendicular to it, while rotating about it through an angle proportional to the space it has advanced; the equation of the ruled surface described will evidently be p = atu (ẞ cost + y sin t),

where a, ẞ, y are rectangular vectors, and

ТВ
TB = Ty.

This surface evidently intersects the right cylinder

p = a(ẞ cost + y sin t) + va,

γ

in a helix (§§ 31 (m), 284) whose equation is

pata (ẞ cos t + y sin t).

(1)

These equations illustrate very well the remarks made in §§ 31 (7), 291

M

as to the curves or surfaces represented by a vector equation according as it contains one or two scalar variables.

From (1) we have

dp[a-u (B sint-y cos t)] dt + (B cost + y sin t) du,

so that the normal at the extremity of p is

Ta (y cost-ẞ sin t)- uT32 Ua.

Hence, as we proceed along a generating line of the surface, for which is constant, we see that the direction of the normal changes. This, of course, proves that the surface is not developable.

294.] Hence the criterion for a developable surface is that if it be expressed by an equation of the form

p = $t+u↓t,

where pt and yt are vector functions, we must have the direction of the normal V {'t+u\'t} &t

independent of u.

This requires either

Vчtч't = 0,

which would reduce the surface to a cylinder, all the generating lines being parallel to each other; or

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This is the criterion we seek, and it shews that we may write, for a developable surface in general, the equation

Evidently

p = $t+up't.
p = $t

(1)

is a curve (generally tortuous) and 't is a tangent vector. Hence a developable surface is the locus of all tangent lines to a tortuous

curve.

Of course the tangent plane to the surface is the osculating plane at the corresponding point of the curve; and this is indicated by the fact that the normal to (1) is parallel to

Vo'to't. (See § 283.)

To find the form of the section of the surface made by a normal plane through a point in the curve.

The equation of the surface is

$2
2

1= p+ sp+ -p" + &c. +x (p' + sp" + &c.).

The part of -p which is parallel to p' is

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And, when A=0, i.e..in the normal section, we have approximately

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295.] A Geodetic line is a curve drawn on a surface so that its osculating plane at any point contains the normal to the surface. Hence, if v be the normal at the extremity of p, p' and p" the first and second differentials of the vector of the geodetic,

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which shews of course that p is confined to a plane passing through the origin, the centre of the sphere.

For a formal proof, we may proceed as follows

The above equation is equivalent to the three

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from which we see at once that 0 is a constant vector, and therefore the first expression, which includes the others, is the complete integral.

Or we may proceed thus

0

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- pS.pp'p" + p′′S.p2p'=V.Vpp'Vpp" =V.Vpp'dVpp',

whence by § 133 (2) we have at once

=

UV pp' const. = 0 suppose,

which gives the same results as before.

297.] In any cone we have, of course,

Svp = 0,

since p lies in the tangent plane. But we have also

Svp=0.

Hence, by the general equation of § 295, eliminating v we get 0 = S.pp'Vp'p" = SpdUp' by § 133 (2).

Integrating

C = SpUp' — [ Sdp Up' = SpUp' + f Tdp.

The interpretation of this is, that the length of any arc of the geodetic is equal to the projection of the side of the cone (drawn to its

extremity) upon the tangent to the geodetic. In other words, when, the cone is developed the geodetic becomes a straight line. A similar result may easily be obtained for the geodetic lines on any developable surface whatever.

298.] To find the shortest line connecting two points on a given surface.

Here Tdp is to be a minimum, subject to the condition that d lies in the given surface.

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where the term in brackets vanishes at the limits, as the extreme points are fixed, and therefore dp = 0.

Hence our only conditions are

= 0, and Svop = 0, giving

SS.ôpdUdp = 0,

V.vd Udp=0, as in § 295.

If the extremities of the curve are not given, but are to lie on given curves, we must refer to the integrated portion of the expression for the variation of the length of the arc. And its form

S.Udpop

shews that the shortest line cuts each of the given curves at right angles.

299.] The osculating plane of the curve

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and is, of course, the tangent plane to the surface

p = $t+up't.

(1)

(2)

Let us attempt the converse of the process we have, so far, pursued, and endeavour to find (2) as the envelop of the variable plane (1). Differentiating (1) with respect to t only, we have

S.p'p"" (p) = 0.

By this equation, combined with (1), we have

or

which is equation (2).

-p ||V.Vp'p"Vp'p"" || p′,

@=p+up = &+up',

300.] This leads us to the consideration of envelops generally, and the process just employed may easily be extended to the problem

of finding the envelop of a series of surfaces whose equation contains one scalar parameter.

When the given equation is a scalar one, the process of finding the envelop is precisely the same as that employed in ordinary Cartesian geometry, though the work is often shorter and simpler. If the equation be given in the form

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where is a vector function, t and u the scalar variables for any one surface, v the scalar parameter, we have for a proximate surface

P1 = (t1, 1, 1) = p + y2, dt + y2 ́u du + 4 ́„dv.

t

Hence at all points on the intersection of two successive surfaces of the series we have

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which is equivalent to the following scalar equation connecting the quantities t, u, and v;

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p = 4 (t, u, v),

enables us to eliminate t, u, v, and the resulting scalar equation. is that of the required envelop.

301.] As an example, let us find the envelop of the osculating plane of a tortuous curve. Here the equation of the plane is (§ 299), S.(ap) ptp′′t = 0,

or @ = $t+x☀'t + yo′′t = √(x, y, t),

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or

or

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S.p′t p′′t [p't+xp′′t+yp′′′′t] = 0, yS.p't p′′t p′′′′t = 0.

Now the second factor cannot vanish, unless the given curve be plane, so that we must have

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the developable surface, of which the given curve is the edge of regression, as in § 299.

302.] When the equation contains two scalar parameters its differential coefficients with respect to them must vanish, and we have thus three equations from which to eliminate two numerical quantities.

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