234 Royal Irish Academy, 



tioned, and, if not to account for them mechanically, at least to 

 advance a step higher in the inquiry. In that theory it was sup- 

 posed, (and the supposition holds good in all known crystals, ex- 

 cept quartz,) that the molecules of the aether vibrate in right lines, 

 the displacements remaining always parallel to each other as the 

 wave is propagated; and it was shown that the function v, by 

 which the motion is determined, then depends only on the relative 

 displacements of the molecules. But when this is not the case, — 

 when, as in quartz, each molecule is supposed 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 ad- 

 vanced in its vibration. This idea, analytically expressed, intro- 

 duces a new term v into the value of the function 2 v ; and, if the 

 plane of the wave be taken for the plane of x y, it is easy to show 

 that 



ydzdz"^ dzdz^J' 



Now if we integrate by parts the expression 

 fffdxdydz Zv, 



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

 duced form of the triple integral will be 



from which it appears that the quantities 



dz^' dz^' 



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

 tions of X and y respectively. These are the very terms in the addi- 

 tion of which the hj^othesis 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 Boe- 

 tian numerical Contractions*." 



The author refeiTed, at the commencement of this communication, 

 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. Tliis opinion is contained in the following extract : 

 — " It would be impossible, with the few materials yet brought 

 to light, to conjecture with any great probability, how far these 

 Boetian contractions may have influenced the introduction, or co- 

 operated with the Arabic system, to the formation of our present 

 numerical notation. It appears to me highly probable that the two 

 systems became united ; because the middle age forms of the figure 



* Papers by Mr. Halliwell on subjects immediately connected with the 

 above, will be found in the last and present vols, of Lond. and Edin. Phil. 

 Mag. 



