158 Mr KELLAND, ON THE DISPERSION OF LIGHT, 



the symbol 2 having reference to the sum of similar expressions, taken 

 for all the particles whose action on P is sensible. By expansion we 

 obtain 



<p{r + p)=cp{r)+Fir).p'^ 



F {r) being the differential coefficient of (p {r) taken with respect to r, 



but {r+p'f={hx^UY + {hj + hiiY + {U + hy; 

 .: r- + 2rp' = t^ + 2 {Sa;Sa + hj^fi + ^zSy), 

 omitting powers of p and Sa, ^/3, ^7; 



.-. p' =:-^{Sa;Sa + Si/Sli + ^zSy), 

 and by substituting this value in the above equation it gives 



4!f!= 2.5d)/- + —(M^» + ^!/^(^ + ^^h) + } {S.v + Sa) 



at- '^ r 



but 2^ (r) . ^x is manifestly the accelerating force, resolved parallel to 

 X, on the particle P in its state of rest, and consequently is equal to 

 zero; we have then 



which we will call equation (1). 



Previous to the solution of this equation in its general form, let 

 us examine what it becomes in that particular case where those par- 

 ticles only which are in the immediate vicinity of P sensibly affect its 

 motion, an hypothesis which is tantamount to supposing all the par- 

 ticles very near each other, as it is manifest that on the latter sup- 

 position the sum of the forces exerted by those particles nearly in con- 

 tact with P, is beyond all comparison greater than those of particles 

 at a finite interval from it. Proceeding on this supposition, Ave obtain 



. da » da ^ _ da . 



Sa = -r- ^x + -J- Sy + ^ Sz 

 dx dy "^ «s 



