TRANSACTIONS OF SECTION A. 743 
3. On the Pulsations of a Rotating Bell. By G. H. Bryan. 
Tt is well known that if a vibrating elastic rod of circular section be rotated 
upon its axis, the plane of vibration remains fixed in space, instead of turning with 
the rod—an experiment frequently used to illustrate the corresponding property of 
polarised light. If, on the other hand, a tuning-fork be rotated, beats will be heard 
which indicate that the vibrations turn with the fork. 
The object of the present paper is to show that when a vibrating ring, 
cylinder, bell, or other elastic shell in the form of a surface of revolution is 
_ rotating about its axis of figure, the nodes and points of maximum radial motion 
will not remain fixed either in space or in the body, but will turn about the 
axis with angular velocity less than that of the body. The author first proves 
this mathematically for a ring or cylinder. As in the investigations of Hoppe and 
Lord Rayleigh, this is supposed inextensible, and in order to show more fully the 
difference between the actual effects of rotation and the purely statical effects of 
centrifugal force, the author supposes the ring to be acted on by an attraction to 
the centre varying directly as the distance, which may be so chosen as to counteract 
the latter force. Taking the type of vibration, which has 2n nodes, it is shown 
that these nodes rotate about the axis with angular velocity— 
n* — Li. 
n+l’ 
where is the angular velocity of the ring. Instead, therefore, of hearing 2n 
beats per revolution, as we should if the nodes remained fixed relatively to the 
ring, we actually only hear 
on n? —1 
n> +1 
beats per revolution. Putting n=2, 3, &c., we find the corresponding numbers to: 
be 2°400, 4:800, 7:059, 9:231, 11:351, &c., approximately. 
The author finds that the results of experiment agree pretty closely with 
theory. A champagne-glass was clipped on a microscopist’s turn-table, which 
was set in motion by a string twisted once round its axle. One end of this string 
was held in the hand and the other passed over a smooth peg, and was attached to 
a weight. The glass having been struck, the number of beats was counted while 
the weight was drawn up from the floor to the peg, the number of revolutions 
being counted separately. Two glasses were used, and rotated with various 
angular velocities; the results for the fundamental tone being respectively 2°6 
beats per revolution (11 observations), and 2°2 beats per revolution (26 observa- 
tions). Considering the vast difference between the champagne-glasses used in 
these rough experiments and the ring or cylinder of Hoppe, the agreement of 
observation with theory is remarkable ; and more especially so as the mean of the 
observed results is exactly the number found by theory. 
4. On the History of Pfaff’s Problem. By A. R. Forsytu, F.R.S. 
The paper was an abstract of Chapter III. of the author’s ‘Theory of 
Differential Equations,’ Part I., Exact Equations and Pfaff’s Problem, subse- 
quently published. 
5. On some Geometrical Theorems relating to the Powers of Circles and 
Spheres. By Professor WILLIAM WooLSsEY JOHNSON. 
The determinant of the powers of three circles relatively to three other circles 
was shown to be sixteen times the product of the areas of the triangles, whose 
vertices are the centres of the circles of each group into the relative power of the 
circles orthogonal to the groups. It vanishes only when these circles cut at right: 
angles. 
