J. G. Barnard on the Gyroscope. 421 
_ But as the values of ¢ on this branch of the curve are nearly double those 
for equal values of 0 of the descending one, the integral [7 (t)d.cosé 
a 
will become zero at some point g’, before 6 has regained its initial value, 
: ee 
at which point — will be the same as for the corresponding point g of 
the cycloid. Above the point g’ the term asnig F(t) d.cos 6 be- 
comes negative and (with its negative sign) becomes additive and there- 
dy 5 
qr always greater than for corresponding 
fore, above g’ the values of 
eoKe 
points of the eycloid. Hence the angular velocity of the axis can never 
me zero and consequently the axis cannot rise to its initial elevation 
and form a cusp, but must make an inflexion and culminate at a, below 
me initial elevation. 
train of reasoning to that just gone through for the first undulation proves) 
as high as a,: and pari ratione, each succeeding wave will be more 
flattened and extended than the preceding, until they soon virtually dis- 
appear, and the curve becomes a descending helix. 
; velocity (a 
measured by f(t), it is evident that the future character e 
be determined by this function. 
with the horizontal velocity oo its square may be neglected in the 2nd 
d 
€quat., (6); and, equating the values of sind deduced from these 
two equations, we shall have 
5 [4.0088 co a cos 6—sin 6 Ae cos 6, 
By differentiating both members and making various reductions we get 
Moy 3sin?0—2 C0, 
“a Jaubeaad Ae” 
an equation which, after the disappearance of the undulations, gives the. 
Value of 6 in terms of t. does 
to the limit corre- 
As f(t) increases @ diminishes in the first member, 
sPonding to sin? 6—= 2 which makes the numerator o the fraction in the 
m 
imum ; showing, to that limit, 
ve, 
e . 
_ As the val f f(t) beyond f(t)=7 do not belong to the question, 
there can bs a6 ed Oecd fies that value of 9 which reduces the 
member to zero; or beyond sin?6= 4. 
