J. G. Barnard on the Gyroscope. 419 
dG , 
~ When pee iets ; becomes still zero with 6=a= 90°: but this 
oa of a maximum is now a minimum value of 6, for the value of nf 
which satisfies equation (4) is genta than 90°, and the curve ab, a'b,', 
&e,, undulates above the ey 
Fi inally when m— sare , equation (4) will give cos6— we at and a 
snbetitution of this in the first equation (1) (making = 90°), wil give 
* 7 9: showing that the curve makes cusps at its superior culminations, 
and that the common cycloidal motion is resumed. In fact the value of . 
dy 1] wage 
— al? (p. 59, July number) at the lowest point 4 of the cycloid, is, 
(substituting the values of 8 and 4) exactly.equal to 
value of the sagitts u corresponding to eb is what we have just found for 
— r and the 
Cos 6, or eb,, Miis se: 
T 22 
tf now, retaining m constant at this value to which we have brought 
it, we increase the rotary velocity, m, or vice versa, a curve with loops, (fig. 
2,) may be described, as it can be shown that, for the maximum value 
of 6, “ry becomes negative.* 
Tn my su ae aper in the January number of this Journal I 
have oti to hoe tok the theoreti rcloida] motion of a sim- 
Solid of revolution is modified by the retarding forces of friction and 
the resistance of the air, and to show that the theory explains all the 
Phenomena observed in the ordi inary gyroscope. 
It may be objected re tie er that the nature of the curve given in 
Fig. 1, (p. 69,) is in some degree assumed, and I ay wish to show 
t it can be confirmed by mathematical demonstra 
The rotary velocity n of the disk is supposed to - gradually destroyed 
gh the retarding forces of friction at the extremities of the axle, 
and of the resistance of the air at the surface. 
ae attempting to give analytical expressions for the retarding 
forces, t is sufficient to say that the” rotary aise at the end of any 
Pej * The « ~_ corresponds, if Ta correctly i eu ringers 1 M os io 
ree, at the meeting of the American Association at Mo 
If backward initial sor a looped 
_m is made tive and small (i. ¢., a ‘a’, results. ee carne. (n 
8 always su »sed Vv +) are but the diffe h 
boven a fit ip A gre cloids; the common—a partiendar case of 
a8 prolate, c common curtate cycloix 
i the problem in which the initial 
respon the pakeder ease of | pro oMes 
Cn 
8yratory 
