420 J. G. Barnard on the Gyroscope. 
time ¢, counting from the commencement of motion, may be expressed 
thus 
n—f (t)* 
in which n is the initial rotary velocity of the disk. 
If we substitnte this expression for v, in the last two equations (3) (p. 
53, July No.,) and fotlow a siinilir process to that by which equations (4) 
of that paper are deduced, we shall get, for the equations of motion 
ea gem a "F (t) ad 0088 
(5) 
d6? 2m aa 
72 — = (cos 6~ 
For the as of simplicity suppose the — position of the axis be hori- 
zontal, or « =90 and the above beco 
sin29" — as cosa) 
d 
sino = FP eost~ 5 f° S(t) d. cos6 ’ 
8) 
dy? | d02_— 2Mgy ( 
26 4. — 6 
sin ey ae z= O88 
faff'a’ cages the cycloidal curve, and aee’e’’g’ the curve in 
gua: it will be observed that the angular velocity of the axis given 
by the 2nd saute (6) is the same for both, for equal values of 9, while 
the value of the horizontal component of that velocity, sind, is less 
than for the cycloidal curve, by the term = of: f(t) d . cos. 
As 6 diminishes, d cos 4 is positive i. ‘his term is subtractive and 
dy 
hence for any point ¢ or ¢’ on the descending branch, >; is less than for 
the corresponding point f or f’ of the eycloid, and the et aee’e”’ will 
be behind the eel aff’, and will descend lower. 
Asin 0 ao f 104. cos 4, attains its maximum, for as the 
curve ascends, 6 increases, and the increments of cos 4 become negative. 
At e”’ the term 
* When the retarding force is independent of the velocity, as in the case of 
tion, {t) in the snes expression is m is linear ; when this force is dependent Te 
the velocity, as ier the resistance of the air, /(¢) will, in general, an - te 
po f t; whether the force is due to either, oF ae 
Sethe conina, the above expres: sion for the velocity of rota 
however be used for the prose nt purpose. - 
Tw SE, cab, ges, be ese 
