396 Royal Society. 
the disc indefinitely near to M, but more remote from A, the direc- 
tion in which the disc will presently be supposed to revolve being 
AMM'B. 
A given force F is applied at N perpendicular to the plane 
ANBS, so that the dise may describe an angle @ round AB in the 
time ¢ ; whereby the points Mand M! describe the two ares MP=y 
and M'P!=y! simultaneously. Suppose the circumference of the 
circle AMB to be divided into four quadrants, commencing at A, 
where y=O, and corresponding with the four ranges of value of 0 
through each of four night angles; suppose M and M' to be in the 
first quadrant, so that y! is greater than y ; then if the disc be sup- 
posed to revolve, a particle at M is c carried from the line MP to the 
line M’P’, so as to acquire an increase of velocity from the plane 
AMM’ independently of the force F, and consequently (by the first 
of the two verbal formule) all the momentum so acquired by the 
particle is lost to the disc, ring, &e., which are thus impelled as by 
a force in the direction PM or P’M!, so as to oppose the rotation 
imparted by F, but to impart another round O in the. direction 
ANB in the plane of the ring; ¢. ¢. in a plane perpendicular to that 
in which F acts. A force having the same tendency is found, by 
means of one or the other of the two verbal formule, in the er 
three quadrants, and thus every par ticle (dm) of the disc contributes 
to the one effect. This effect is due to the difference of the velocities 
oe) ene - Gi Pand P’, or to the momentum (3 ~ 5p) am lost or 
dt 
gained by the particle dm in the time dé. 
The value of = is obtained from the enaeen y=r@ sin 0, making 
ao ¢ and @ to vary ; but the value of = is obtained from that of 
= J yy making @ only to vary. It is es fa Sil that 
(4 = #) dm= (cos De oy .@ sin a) rwdtdm. 
It is thence shown, by taking the moments about AB, and ap- 
plying D’ Alembert’s principle, 
Fag 
atu fs? edd—w 2 find. cos 6dd= sei 
the integrals applying to 6@ only, S between the limits O and 27 ; 
i. e. to all the particles of the disc simultaneously and independently 
of g or ¢. From this is obtained the result 
_ 4Fag .. 5 fwt\. 
o Wat S 
W being the weight of the disc. 
This value being periodical, and ranging between the limits O and 
the maximum — ;» Shows that the disc makes oscillations which are 
wW 
of less extent and duration, as the spinning of the disc is more 
rapid ; 2. e. a3 w” is made greater compared with ; and thus if F 
