174 REPORT—1850. » 
tion to a parabola of which and e are co- ordinates, measured from a point in the 
curve itself, perpendicular and parallel to the axis respectively. 
Tables* are given in the original paper in which the hyperbolic and parabolic law 
are compared together, and with the results of experiments; and it is shown that 
the former has the advantage of having a mean error between one-third and one- 
fourth of the error of the parabolic formule. The same comparison is extended to 
the deflection of beams, for which formule of similar forms are adopted. 
Taking the ‘‘ hyperbolic ” formula for direct longitudinal compression (0) of a rod 
of a unit of length and a unit of sectional area by a direct force to be 
@ 
—=y—6 ἕ 
c if τ 
and the ‘‘ hyperbolic” formula for central deflection (f) of a rectangular bar by 
a transverse pressure (w) applied perpendicularly at its centre to be 
Ξξε-- ζω, 
the paper proceeds to show that the following equation is very nearly correct where 
the deflection is of small magnitude compared with the length of the beam :-— 
fates (udaty)— 2 27 pag th +18), 
a+y 
where P is half the deflecting pressure, d half i thickness of the beam in the di- 
rection of that pressure, » the breadth, and a half the length. 
The breaking weight of rectangular beams is then found in terms of the constants 
a, 8, y, δ; and the resulting expression is, as far as its form is concerned, found to 
be similar to that ordinarily used for determining the breaking weight of rectangular 
beams. 
From the formulz for direct tension and for deflection may be obtained (by sub- 
stituting in them the numerical values of «, 8, ε, ¢, deduced from experiment) the 
numerical values of y and ὃ by means of the connecting equations last given. 
This method seems to give more accurate results than the experiments on direct 
compression detailed in the Report, which are so irregular as evidently not to be 
trustworthy. They were made by compressing bars enclosed in tubes of which the 
sides resisted the flexure of the bars. This lateral resistance had of necessity great 
effect in sustaining the external force applied to compress the bars, and was probably 
the principal cause of the vitiation of the results. 
The numerical values of the constants for compression are obtained in the original 
paper from several independent experiments, and agree with each other in a very 
satisfactory manner. It is however to be observed, that a great desideratum exists 
for perfecting the “ hyperbolic ” or any other hypothetical law of elasticity, namely 
a knowledge of those variations of the strength and elasticity of cast metal which 
depend on the magnitude of the castings. It is greatly to be desired that this want 
of experimental data may not long remain unsupplied. 
On the Water Sirene. By Professor Joun Donatpson, Edinburgh. 
Professor Donaldson explained, that in regard to the vibratory movement of fluid 
masses, it had long been known that when solid bodies are brought into collision 
under water, the liquid is agitated directly in all the points where it touches. the 
solid vibrating bodies—acquiring thereby an undulatory movement, producing sound, 
heard of course at a greater or lesser distance, corresponding to the violence of the 
shock. It was also known, that by a direct shock, the nortnal vibrations of discs, 
and longitudinal vibrations of rods, threw water, mercury and other liquids, into an 
undnlatory or vibratory motion; and it had therefore been very generally supposed, 
that the shock of solid bodies is essential to the production of an undulatory or vi- 
bratory movement in fluid masses. But the Baron de la Tour found that sounds 
could be produced under water without the percussion of solid bodies, by throwing 
the water into rapid undulatory motion, by the play of the sirene, the instrument 
about to be described. 
* These tables and the analytical investigations were presented before the Cambridge 
Philosophical Society, Nov. 1850. 
