70 J. G. Barnard on the Motion of the Gyroscope. 
done had it started /rom rest with its diminished value of n; 
and, for the same reason as before, will not be able to rise 
again as high as its starting point a, but to a somewhat lower 
oint @, and with an increased horizontal velocity. These im 
crements of horizontal velocity will constantly ensue as the cul- 
minations become lower and lower, while on the other hand, the — 
ha become less and less marked, as indicated by the — 
gure. 
I have stated in my former paper (p. 71) that a certain iniiial — 
horizontal angular velocity such as would “ make its correspond: 
ing deflecting force equal to the component of gravity, g sin 4 
would cause a horizontal motion without undulation.” This — 
horizontal velocity is rapidly attained through the agencies just 
described: or, at least, nearly approximated to, and the axis, a8 _ 
observation shows, soon acquires a continuous and uniform hort — 
zontal motion. Be 
On the other hand, this sustaining power being ere pro- 
portional to the rotary velocity of the disk, as well as to the ai 
gular velocity of the axis, diminishes with the former, and asi 
diminishes, the axis must descend, acquiring angular velocity 
due to the ea of fall: hence the rapid gyration and the de 
scending spiral motion which accompanies the loss of rotary 
velocity. 
the living force generated by gravity be directly el 
the height of fall, and involves as a corollary that t 
* The first of these equations (as I have remarked in a note to p. 59) is the 
pression of another fundamental principle—more usually called the “ principle | 
areas. : : 
