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 xy, it is 

 easy to show that 



e; - C f — — - — ^^ 



"" \dz dss^ d% c?W* 



Now if we integrate by parts the expression 



J$5 dxdydz^v, 



so as to get rid of the variations of differential coefficients, 

 the reduced form of the triple integral will be 



from which it appears that the quantities 



d\ __ d^ 

 "" dz^' "^ dz^' 



are to be added to the usual expressions for the force in the 

 directions of x 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 HalHwell, 

 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- 



