396 
Proceedings of the Royal Society of Edinburgh. [Sess. 
it with dt i ; { RD ] denotes a realisation by taking the difference of what 
follows it with ± i divided by 2 i. As an example of flexural waves in 
an infinite elastic rod, arising from a given initial displacement, we may 
take the solution 
y = { RS} 
l 
where 
+ H) 
1 / xH 
X 2 
e 4 Kb(z+it) 
-■'t“ (j A-2 L ' V “” 
T = J(z 2 + t 2 ), and r = tan 
-i (t 
■ ( 12 )- 
In what follows, z is taken as 1 and *6 as -r- in order to allow us to use 
4 
Lord Kelvin’s hydrodynamical results in our present problem. 
§ 7. Taking the origin of co-ordinates at the middle of the bar, the 
initial configuration is given by 
X- 
— c 4/c b 
y = e 
. (13). 
Almost immediately after the commencement of motion, an infinite number 
of waves are formed along the bar, with amplitudes diminishing according 
X 2 
to the law e 4 * &T2 and with the distances from zero to zero becoming shorter 
and shorter as we pass from the middle toward the ends of the bar. 
The zeros come into existence at the ends of the bar, and begin travelling 
inwards to the middle. The first zero formed comes almost instantaneously 
to the middle region, and the others follow it in their order of formation. 
The inward progress of the zeros soon ceases, the first zero never quite 
reaching the centre ; in a short time they begin to move in the opposite 
direction and continue to do so for ever, and the amplitudes at any point x 
ultimately fall off according to The middle point of the rod subsides 
to its undisturbed position nonvibrationally, while the distance from it to 
the first zero on either side continually increases after a certain time, 
being given by the equation x 2 = 3k&tt£. 
§ 8. The seven curves given by Lord Kelvin, as space curves for water- 
waves, on page 191 of his paper, Proc. Roy. Soc. Edin., vol. xxv., Feb. 1904, 
show the condition of things in our present problem at seven different points 
near the middle of the bar, as t increases from 0 to oo , provided the curves 
be continued to meet the axis of t asymptotically at infinity (see fig. 1 and 
§ 9 below). These curves show that points very near the origin never 
pass through their initial positions, but fall back nonvibrationally to them ; 
points farther from the middle rise slightly and then behave in the same 
