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which is the condition that the parameter a must satisfy in

order that the spiral whose equation is r = kā may be its own evolute.

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Substituting for co – qe its value in ternis of Po,

a* + (c' - a)
ce

P,?,

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dy

dz

IF

F., y, ~) = 0 be the equation to a curved surface, the cquation to the tangent plane at a point x, y, z is

dF
dF

df
(a' – 1')

+ (y - y) + (z'- 2) = 0,

dx where x', y', are the current co-ordinates of the tangent plane, x, y, those of the point of contact.

If the equation to the surface consist of a function homogeneous of n dimensions in x, y, i equated to a constant, the equation to the tangent plane becomes

df «F
+ y'

= nC,

dy dz F (1, y, z) = c being the equation to the surface.

P be the perpendicular from the origin on the tangent plane,

dF d F dF
+ y
dy

da
IF

dF +

dz

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+

da

If

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+

dir

2

dF 2

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+

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dy

and if the function be homogeneous of n dimensions,

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x2

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+

= 1.

The equations to a normal at a point x, y, & are
x'

x
dF dF dF
dx
dy

dx
Ex. (1) The equation to the Ellipsoid being

y?

1, a?

ca that to the tangent plane is

x x' yy

a? 62 ca The perpendicular on the tangent plane from the origin is given by the equation

1.22 y? р

6 cf If we wish to find the locus of the intersection of the tangent plane with the perpendicular on it from the centre, we have to combine the equation to the tangent plane,

xx
yy

1,

a bo c? with the equations of a line perpendicular to it, and passing through the origin

a2 x' boy cʻr

1

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+

+

+

+

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=

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a

These last may be put under the form
a x'
by ex

(a’n? + b*y*2 + oʻz-)!,
y
b

x2 y?
since

a

62 Multiplying each term of the equation to the tangent plane by the corresponding member in these last expressions, x, y, z are eliminated, and we have for the locus of the intersections

x"? + y"? + x^2 = (az." + b’y'? + cpx).

+

= 1.

This is the equation to the surface of elasticity in the wave Theory of Light.

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2

2

The intercepts on the tangents are

= 3x, yo = 3 y, , = 3z, and the volume of the pyramid included between the tangent

9.ry 9a plane and the co-ordinate planes is

The volume of this pyramid is smaller than that of any other pyramid formed with the co-ordinate planes by a plane passing through the point x, y, z.

The length of the perpendicular from the origin is given by

1

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G++

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(3) The equation to the Cono-Cuneus of Wallis is

(a* - ) cox = (,
and the equation to the tangent plane is therefore

y' x x' (a’ – xo) yy' + c'zz' = yo.
() The equation to the hélicoide gauche is

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and the equation to the tangent plane is

h (ry' ya') + 27 (x2 + y^) z' = 27 x (x + y'); and the perpendicular on it is

2 T 19 % p =

where pe? (li+ 4 px)}

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(5) The equation of the hélicoide développable is
(2T8 (202 + y - - aʼ)!

27X (x2 + y -a%)
h

h

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+ y cos

The cosine of the angle which the tangent plane makes with the plane of xy is

dF

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Now (ir cos 0 – y sin 0)' = xé cos0 + y sin 0 - 2xy sin cos 0, and from the equation to the surface 2xy sin cos 0 = a' – 2* sin 0 y cos* 0; therefore

(av cos 0 y sin 0)* = x + ye – a’. Hence

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From these expressions the cosine of the inclination of the tangent plane to the plane of xy is found to be

2 па

(h* + 47°a”)} The inclination is therefore constant, and equal to that of the helix, which is the directrix of the surface.

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