582 Mr. C. A. B. Garrett on the 
aside by a force applied at a point on the bar distant a 
quarter of its length from the free end. 
In obtaining this result, he quotes expressions derived in a 
former chapter for the kinetic and potential energy of the 
vibrating bar. Moreover, he simply gives the result as an 
interesting fact, and gives no experimental justification of 
his assumptions. 
It occurred to me that it might be possible to put the 
solution ina somewhat simpler form, involving only very 
elementary differentiation and integration. This was done, 
and experiments were then conducted with a view to the 
justification or otherwise of the assumptions. These experi- 
ments show that a bar pulled aside by a force applied about 
one-fifth of its length from the free end, approximates very 
closely to the form assumed by the same bar when vibrating. 
Then the simple theory adapted to this state of things gives 
a value for the frequency within 1°4 per cent. of that obtained 
by the full theory. 
Simple Theory. 
Assume for a first approximation, as Rayleigh does, that 
the shape of the vibrating bar is the same as that of the same 
bar pulled aside by a force at the free end. Knowing the 
equation to the curve assumed by the bar, we can find the 
moment about the fixed end of all the forces supposed to be 
producing motion of the particles of the bar; then equating 
this to the bending moment at the fixed end, we get an 
expression enabling us to find the frequency of vibration. 
Fig. 1. 
Let OB be the vibrating var, fixed at O; OB, being the 
mean position ; 
L be the length of the bar ; 
« the radius of gyration of the cross-section about the ~ 
neutral axis ; 
U the velocity of longitudinal waves in the bar ; 
HE} Young’s modulus for the bar ; 
o the volume density of the bar ; 
A the area of cross-section of the bar ; 
x and y coordinates of P measured parallel and per- 
pendicular to OBo. 
py the radius of curvature of the bar at O. 
