CHARLES B. WARRING. ‘158 
3. Ifthe surface is inclined, the top crosses it at right 
angles to the inclination, and travels towards that side 
of the spiral which is up hill. 
4. To change its direction, change the inclination at 
right angles to the way you wish it to go. 
WHY THE FIRST CURVE OF THE SPIRAL IS THE LARGEST. 
Let figure 2 represent a top exaggerated for conveni- 
ence of representation, the actual ‘‘point” being less 
than one-twentieth of an inch across. If much larger 
than this, the top will rise so quickly, and describe a 
path with so large a radius, that it is difficult to study 
it. The face of the ‘‘point,’’ except where the con- 
trary is stated, is flat. 
Suppose the top at the moment represented in our 
diagram to be revolving rapidly on its axis in the direc- 
tion indicated by the ellipse around the shaft. The 
upper end, m, will be approaching the observer, and 
the lower end, p, will be receding from him. In other 
words, the instrument is gyrating around the instan- 
taneous axis, q a, both motions being with the hands of 
a watch. 
The rolling of the ‘‘point’’ would carry the top ina 
straight line from the observer, if its axis kept parallel 
to itself. But as m swings around the vertical q a, p 
correspondingly changes its direction, and the distance 
which it (p) will travel while m is swinging through, 
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