Gears have their teeth cut by a machine so as to conform to certain shapes which will bring about smoothness of running when the gears are in mesh. The curves generally employed in shaping the gear teeth are the cycloidal and the involute.

Cycloidal Curves. Cycloid. The cycloid is a curve generated

Fig. 90. Construction of Rectangular Hyperbola.

SECTION, AND REAR AND FRONT ELEVATIONS, OF TWO-STORY FLAT BUILDING FOR MR. J. WM. THORSON, CHICAGO, ILL.

W. Carbys Zimmerman, Architect, Chicago, 111, by a point on the circumference of a circle which rolls on a straight line tangent to the circle, Fig. 91.

Built on a 40-foot Lot. Building Faced with Red Paving Brick on All Four Sides. Roofs of Shingles. Inside Trim of Red Oak and Birch. Dining Rooms Have Beam Ceilings. Tile in Bathrooms. Cost of Building Complete, \$9,000. Side Elevation Shown on Opposite Page. For Plans, See Pages 26 and 42.

SIDE ELEVATION OF TWO-STORY FLAT BUILDING FOR MR. J. WM. THORSON, CHICAGO, ILL.

W. Carbys Zimmerman, Architect, Chicago, 111. Front and Rear Elevations Shown on Opposite Page. For Plans, See Pages 26 and 42.

The rolling circle is called the describing or generating circle, the point on the circle, the describing or generating point, and the tangent along which the circle rolls, the director. In order that the curve described by the point may be a true cycloid the circle must roll without any slipping.

Fig. 91. Geometrical Construction for a Cycloid.

Fig. 92. Geometrical Construction for a Hypocycloid.

Hypocycloid. In case the generating circle rolls upon the inside of an arc or circle, the curve thus generated is a hypocycloid, Fig. 92 If the generating circle has a diameter equal to the radius of the director circle the hypocycloid becomes a straight line.

Epicycloid. If the generating circle rolls upon the outside of an arc or circle, called the director circle, the curve thus generated is an epicycloid, Fig. 93.

Involute Curves If a thread or fine wire is wound around a cylinder or circle and then unwound, the end will describe an involute curve. The involute may be defined as a curve generated by a point in a tangent rolling on a circle, known as the base circle, Fig. 94.

Fig. 93. Geometrical Construction for an Epicycloid.

Fig. 94. Geometrical Construction for an Involute.

The details of the construction of the ellipse, parabola, hyperbola, cycloid, and involute will be taken up in connection with the plates.