418 J. G. Barnard on the Gyroscope. 
The value of the deflecting force due to a given angular velocity m is 
(p. 64, July number) " n, and if we suppose this equal to the com- 
gr 
M 
ponent of gravity gsin «, we shall have m= Cin sin @, 
If we substitute this value of m in the second member of equation @) 
and assume « = 90° the factor in question becomes zero for 6=4, 
the maximum and minimum ag of @ are the same, indicating a - 
zontal motion without undulat 
For every other initial RS than 90° a different value of m is re- 
quired to produce this result, in consequence of the influence of the cen- 
trifugal force of gyration at other vena: 
With «= 90°, equation (2) bec 
2 
Gon the first factor of the second member equal to zero and solving 
with fash ee to cos 9 we get (recollecting the value given to 8 in our 
5a 
former art 
6=- J Pe pent 
cas Tay i “ata) Moy 
or m= . equation (3) expresses the cycloidal curve with cusps ‘; a’, 
8 , &e., as ‘ eas already shown in our former investigation. 
m > 0 but eth the minimum value of 6 derived from equation Sau is 
greater than i m is zero, while instead of a cusp (there is as bas 
lready been observed) a © cy at a, and the curve has the w wave 
ab, a’b’, (the points 6,5,'6,”, &e, being higher than 56'b'’).* 
When m= at the curve unites with the horizontal a@ 
there is no undulation; equation (4) giving cos@== 0, or 6= 90%. 
* In one the amplitudes, aa’, a’ a'’, of the reread a. inane oe 
same time that 
represented them 
the sagitte are diminished, but, for the of comparison, L have 
the same for each variety of curve. 
taal” and 
rep 
mee ee 
