238 Mr KELLAND, ON THE MOTION OF 



hence, '^ = - ^ .P<p{r + Sr) . (^x + Sa) - ^ 2 (/? + Aff) (Ax + A«); 



o- and 2 indicating the respective sums taken for all the particles which 

 are in motion. 



2. In order to reduce these expressions to an integrable form, it 

 is requisite to adopt some process of approximation. Suppose then we 

 omit Sa compared with Sx: this appears at the first glance a doubtful 

 process, for we cannot here suppose, as we did in the case of light, 

 that the particles have a very small motion; we know, in fact, that 

 this is not the case for sound; but all scruple will be removed when 

 we reflect that any particular ^a is the approach of two particles to each 

 other, whilst the corresponding Sx is their original distance; and from 

 the fact of the repulsive nature of the forces, we cannot, even in the 

 most unfavourable case, suppose the approach other than a small fraction 

 of the original distance. Or again, if Sa be the amount of recess of 

 the particles, since any recess must, in the case of vibrations, have 

 a corresponding approach, the same reasoning applies. 



3. Now (p{r + Ir) =■ <pr + FrAr nearly, where F{r) stands for 

 the differential coefficient of <p(r), taken with respect to r, 



{r + Srf =(Sx + Saf + If +h%'; 

 .-. 2rSr = 2SxSa 



Sr = — . da ; 

 r 



Fr 



.-. <p{r + Ir) . {Sx -f ^a) = ((^r + IxU) {Ix + la) 



Fr 



= (pr .Sx + ((pr H Sx^) Sa. 



Similarly, <p(R + Ai?).(Ax + Aa) = cpR.Ax + {<pR + ^Ax-)Aa; 



df ^ ^r J. I i 



-^^{cpR.Ax +((t>R + ^^Ax')Aa}. 



