Let ya x be the equation to the parabola D A D',

.. z2 + a x = a2 is a parabola on x z.

Let A Ba, A C = a, and let the ordinate C D = a.

To trace the curve, we have the three equations on the co-ordinate planes,

If x = 0, y = 0, and z = a, .. the curve passes through B; as r increases, y increases, and z diminishes;

When xa, y = a, and z = 0, therefore the curve decreases in altitude from B down to meet the parabola in D. This gives the dotted branch BD. If a is greater than a, z is imaginary; therefore the curve does not extend beyond D.

But since z = ±√a (a) there is another ordinate corresponding

to every value of x between o and a; hence there is another branch, equal and opposite to B D, but below 'the plane ry. This is represented by D B'.

Again, since when y is negative, the values of z do not change, there is another arc, B D' B', represented by the double dotted line, which is exactly similar to B D B'.

Therefore, the curve is composed of four parts, B D, D B', B D', and D' B', equal to one another, and described upon the surface of the parabolic cylinder, whose base is D A D'. These branches form altogether a figure something like that of an ellipse, of which the plane is bent to coincide with the cylinder.

540. Ex. 2. Let the circle, whose equation is a + y2 = a2, be the projection of the curve of double curvature on xy; and the curve, of which the equation is a2 y2 = a2 22 — y2 2, be the projection on y z, to trace the curve.

Let B C B'C' be the circle on x y whose equation is x2 + y2 = a2; then the equation on y z being a y2 = a2 ~2 — y2 ~, the equation on

is

If x = 0, ya, z infinity, therefore the vertical line C L through C is an asymptote to the curve. As x increases, y decreases, and z decreases, therefore the curve approaches the plane of x y. If x = a, y = 0, z = 0, therefore the curve passes through B. If x is greater than a, y and z are each impossible, therefore no part of the curve is beyond B: for any value of y there are two of z, therefore for the values of y in the quadrant A CB, there are two equal and opposite branches, LB, BL.

Similarly there are two other equal branches, K B, B K', for the quadrant BAC'; and as the same values of y and z recur for a negative, there are four other branches equal and opposite to those already drawn, which correspond to the semicircle C B'C', and which proceed from B'.

These two examples are taken from Clairaut's Treatise on Curves of Double Curvature; a work containing numerous examples and many excellent remarks on this subject.

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ERRATA.

Page 7, line 1, read Let x =

√3 = √2 + 1. In the last figure let A B, B C, and CD each be equal to the linear unit, then A D = 3.

y= ±

1 +

read

and if the axis coincide with A Z, x2 + y2 = r2, ≈ = 0;

249, line 1, for rc read z, and for c read ≈ - c.