150 
Proceedings of the Royal Society 
4. On the Vibration of a Uniform Straight Spring. 
By Edward Sang, Esq. 
In this paper the solution of the well-known and hitherto refrac- 
tory problem, “ to investigate the law of motion of a uniform 
elastic plate,” is presented as an example of the application of the 
theory of primaries to physical research. This problem, on account 
of its importance in the theory of acoustics, has attracted the 
attention of the most celebrated mathematicians. 
If we suppose a physical elastic line, flexible only in one plane, 
to have a number of masses fixed at intervals along it — if we in- 
vestigate the law of the motions of these masses — and if, there- 
after, we imagine the masses to be indefinitely subdivided and 
their parts distributed along the intervals, we shall arrive at the 
law of vibration of an elastic plate. Now, when we examine the 
general features of the vibration of such a discrete series, we dis- 
cover that every motion of which it is susceptible may be regarded 
as the combination of simple movements, the number of which is 
equal to the number of the connecting ties ; these simple move- 
ments being such that, if one of them existed alone, all the- parts 
of the series would come simultaneously into their mean positions. 
The demonstration of this theorem is identic with that which 
was given in a paper on linear vibration read before the Society 
during the session 1856-57, and therefore omitted from the present 
paper. From this theorem it follows, that the trembling of a con- 
tinuous spring must be composed of an infinity of simple vibrations, 
each of which is performed in its own peculiar periodic time. 
Hence our attention must be given to the characters of a simple 
vibration. 
In order that a number of bodies may perform simultaneous 
oscillations, it is necessary that their tendencies to return to their 
mean positions be proportional to their distances from these posi- 
tions, and to their masses jointly. Hence, if a number of equal 
masses be arranged uniformly along a thin elastic uniform bar, 
the form which that bar assumes when making a simple vibration 
must be such that the pressures necessary to retain it at rest in 
that shape are proportional to the distances from the mean position. 
