462 Proceedings of Philosophical Societies. [Dec. 
mechanics, M. Poisson makes an interesting application of it to 
one of the most extensive and curious branches of acoustics ; that 
is to.say, to the vibrations of elastic plates, to the figures which 
they present, and to the sounds which they emit, during their 
movements. We may suppose that the plate to become sonorous 
separates very little from a fixed plane. ‘This consideration puts it 
in our power to neglect all the quantities of the second dimension 
with respect to one of the three co-ordinates. Abstracting, then, 
the weight of the plate, and supposing that each point of the plate 
remains during the movement in the same perpendicular to ‘the 
fixed plane, the author obtains a new formula which divides itself 
into two others, according as the one or the other of the two con- 
stant quantities which it contains are reduced to 0. One of these 
particular equations had been already found by Euler; the other 
occurs without sufficient proof, or even without any demonstration, 
in a piece sent for the prize proposed by the Class of the Sciences, 
on the Mathematical Theory of the Vibration of Sonorous Plates, 
verified by a Comparison with Experience. This prize is still open 
till the ist of October, 1815, the Class not having hitherto re- 
ceived any piece worthy of attention, except that to which it has 
given an honourable mention on account of this same formula, 
The author satisfies it by particular integrals composed of exponen- 
tials of the sine and cosine. In this he followed the example given 
by Euler, To each of these integrals corresponds a particular figure 
of the sonorous plate, and the sound which it emits depends in 
general on the number of nodal lines which form during these 
vibrations. The tones thus calculated agree in a satisfactory 
manner with the experiments of Chladny, and with other experi- 
ments made by the anonymous author. This conformity was the 
principal cause of the honourable mention made of that memoir. 
M. Poisson points out another kind of comparison, much more 
difficult, and which would be relative to the figure produced, after 
a given manner of .putting the plate in a state of vibration. He 
would wish, likewise, that the results of the calculus were deduced 
from the general integral, and not from some particular integrals. 
Unfortunately this equation cannot be integrated in a finite form 
except by definite integrals, which contain imaginary quantities 
under arbitrary functions ; and if we make them disappear, as M. 
Plana has done in the case of vibrating cords, we obtain an equation 
so complicated that it appears very difficult to make any use of it. 
We see, then, that the question of sonorous plates offers still to 
the analyst sufficient difficulties to surmount to account for the deci- 
sion of the Class, who have put off the term of deciding the prize 
till the Ist of October, 1815. But the double demonstration of 
the fundamental formula is a very important step. It may be here- 
after taken as ‘a datum of the problem ; so that the candidates will 
turn all their efforts towards the integration of the formula, and the 
different methods of comparing it with experience. Those who 
