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which is the condition that the parameter a must satisfy in
Ꮎ order that the spiral whose equation is r = kā may be its own evolute.
Substituting for co – qe its value in ternis of Po,
a* + (c' - a)
F., y, ~) = 0 be the equation to a curved surface, the cquation to the tangent plane at a point x, y, z is
+ (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
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
and if the function be homogeneous of n dimensions,
The equations to a normal at a point x, y, & are
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,
a bo c? with the equations of a line perpendicular to it, and passing through the origin
a2 x' boy cʻr
These last may be put under the form
(a’n’? + b*y*2 + oʻz-)!,
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 = (a’ z." + b’y'? + cpx).
This is the equation to the surface of elasticity in the wave Theory of Light.
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
(3) The equation to the Cono-Cuneus of Wallis is
(a* - ) ” – cox = (,
y' x x' – (a’ – xo) yy' + c'zz' = x® yo.
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)}
(5) The equation of the hélicoide développable is
27X (x2 + y -a%)
+ y cos
The cosine of the angle which the tangent plane makes with the plane of xy is
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
From these expressions the cosine of the inclination of the tangent plane to the plane of xy is found to be
(h* + 47°a”)} The inclination is therefore constant, and equal to that of the helix, which is the directrix of the surface.