J. G. Barnard on the Gyroscope. — 417 
Ant. XXXVI—On the effects of Initial Gyratory Velocities, and of Re- 
tarding Forces, on the Motions of the Gyroscope ; by Major J. G, Bar- 
Narp, A.M., Corps of Engineers, U.S.A.* 
Iv one of the concluding paragraphs of my first paper on the Gyro- 
scope (Am. Journal of Science, July, 1857) I stated that “an initial im- 
pulse may be applied to the rotating disk in such a way that the horizon- 
component of gravity g sin 0, would cause a horizontal motion without 
undulation.” 
The statement contained in the last sentence quoted, is not rigidly true; 
for besides the component of gravity, there is another force to be consid- 
ered, viz., the centrifugal force due to the gyratory velocity, which acts 
either in conjunction with, or in opposition to, the component of gravity, 
according as the axis of the disk is above or below a horizontal. = 
In this last position this force is null (as regards its effects in sustaining 
Statement quoted is true without qualification. The assum 
initial horizontal velocity requires only a new determination of constants 
for equations (a) and (c) (pp. 53, 54, July No.). 
If we make, in those equation 
a 6=a, p=90°, y=90°, a= —sin &, v, =m, vy =0, Vz = 2, 
(in which m is thé assumed initial velocity) and determine the constants 
A and / therefrom, the equations of motion will become 
‘ d y C n f 
sin ? 6 — —— (cos 6 — m sin & 
aca (cos 6— cos a) +- a 
; 2 2 
ONY hae sa 2A (00s 6—cos «) + m? 
dt? ' di2— 
and from them we get 
5 9 4a02 2 2C0mn. C2 n2 
sin? =| a csin’ ord ar Oi Az (cos 6— cos «) 
— mz? (cos + cos «)| (cos 8— cos a) (2) 
yy. 
From this we get u=0 when cos9— cosa=0; and as di 18 not zero 
for this initial elevation, it indicates, instead of a cusp, a tangency to the 
ae = h dulation, the other fac- 
€ curve described is horizontal without undulation, see 
tor of the member of eq. (2) hould ikewise become zero with 
$=a: an effect which may ensue from a suitable value given to m. 
ahi is i i igidly mathematical demonstration of 
— eects of ‘eeterding Soke ta oi ig . (January No.) of this Jour- 
ps and to give Sie. Uieky of the “motions” of the tyroscope & more gener: 
poke the introduction of “ Initial Gyratory ties. 
IND SERIES, Vor, XXV, No. 75.—MAY, 1858. 
53 
