386 
is not the case,—when, as in quartz, each molecule is sup- 
posed to vibrate in a curve—then it is natural to conceive 
that the function v may depend, not only on the relative 
displacements, but also on the relative areas which each 
molecule describes about every other more or less advanced 
in its vibration. This idea, analytically expressed, intro- 
duces a new term v into the value of the function 2v; and, 
if the plane of the wave be taken for the plane of zy, it is 
easy to show that 
pa o(tn ve _ de ti) 
dz dz* dz dz?) 
Now if we integrate by parts the expression 
SS dadydzev, 
so as to get rid of the variations of differential coefficients, 
the reduced form of the triple integral will be 
a aE 
2c SiS dedydz (TI x — én) ; 
from which it appears that the quantities 
dn @E 
Cae? ° ae 
are to be added to the usual expressions for the force in the 
directions of 2 and y respectively. These are the very 
terms in the addition of which the hypothesis before alluded 
to consists. 
The Secretary read a paper*by James Orchard Halliwell, 
Esq. F. R.S., &c., entitled “ an Inquiry into the Period of 
the first Use of the Zero by those Writers who adopted the 
Notation of the Boetian numerical Contractions.” 
The author referred, at the commencement of this com- 
munication, to the opinion which he had formerly expressed 
on the nature of the change from the use of the abacus, to 
that of local position, and the cipher. This opinion is con- 
