156 Proceedings of the Poyal Society of Edinburgh. [Sess, 
27t/^. In the case of the top we have and the period is 
therefore 
A 27t 
C w 
(12) 
in agreement with the usual more elaborate theory. 
3. The deviating force Cnv, being proportional to and at right angles to 
the velocity, is easily resolved into components in any system of co- 
ordinates. Thus, to obtain the general equations of motion of a solid 
of revolution in terms of the usual spherical polar co-ordinates 6, \jr, we 
note that since the components of the velocity v in and perpendicular to 
the plane of (9 are 0 and sin 6 \jr respectively, those of deviating force will 
be — Cn sin 0 yjr and Cnd. Hence, assuming the known expressions for the 
accelerations of a point in spherical polars, we have at once 
A((9 - ij/^ sin 0 cos B)= — On\p sin 0 ® j 
I 
(13) 
where 0, 'P are the moments of the external forces tending to increase 0 
and respectively. 
The theory of the nearly vertical top, which is a little troublesome to 
deal with on the basis of equations such as (13), is easily treated directly 
If X, y be the projections of the unit vector OC on fixed horizontal rect- 
angular axes through O, the components of deviating force will be 
whence 
- Cny, Cnx, 
Kx= - Qny + Ai) = Gnx + lAghy . . . (14) 
If we put z = x-\-iy, these may be combined into the single equation 
Az - iCnz - M.ghz = 0 ..... (15) 
^ ^ . . . . . (16) 
oy^lOnjA, v = lJ{CV-iAMgh) . . . . (17) 
the solution of which is 
where 
and the (complex) constants H, K are arbitrary. This represents motion in 
an elliptic orbit which revolves about the origin with the angular velocity 
00 , the period in the ellipse being 27 t/i/. 
Before leaving the ordinary top we may recall the familiar experiment 
where a projecting material axis, or stem, is observed to follow the wind- 
ings of a metal arc or wire brought into contact with it. The friction of 
the wire causes the cylindrical stem to roll along the arc, and the deviating 
force called into play tends to maintain the contact (see fig. 2). 
