208 The Rev. M. O'Brien on the Propagation 



which compose this equation, in order to determine what 

 terms may be neglected in approximating : let us take, for 



instance, the terms ( Smf{r) S x-j -j-^ and i^mf (r) Ix'^j 



J— J. Jf the motion we are investigating consist of a wave 



whose length is A, it is evident that a will be in some such 



form as a sin — [vt — x), and therefore -p^ will be of the 



, e. ., « ^ d^ ex. a. ^ . 



same order ot magnitude as — -^, and —j—^ as —^ ; also 6 or is 



d^ a. 

 of the same order of magnitude as r ; hence 2 mf[r) 8 oc^ , ^ 



and 2 mf (r) S x'^ -r—^ are respectively of the same order of 



,.2 ,.4 



magnitude as Sm/{r) -^ and Smf{r) — ^; if we take r to 



be the value of r, for which f (?) has its mean value, these 

 quantities are of somewhat the same order of magnitude as 



2 m f {r) -^ and 2 VI f {r) — , which quantities are in the 

 A A 



ratio —J. Hence it evidently appears, that if the distance at 



which the molecular force has its mean value be very small 

 compared with the length of the wave, all the terms in our 

 equation involving differential coefficients of the fourth and 

 higher orders will be very small compared with those involving 

 the second differential coefficients, and we may neglect them. 

 We shall assume this to be the case*, and therefore retain 

 only the second differential coefficients in our equation. It 

 is evident that the coefficients we have denoted by N and P 

 are not by any means to be neglected as compared with M, 

 for they are of exactly the same order of magnitude as M, as 

 appears immediately, if we observe that 8 x, 8 j/ are of the same 

 order as r, and rf (r) as/ (r). 



* The results arrived at in this paper are equally true, whether we retain 

 or reject these terms; if the hypothesis of finite intervals be true we must 

 retain them. But we know, from the fact of the dispersion of light being 

 small compared with the whole deviation, that these terms (if not quite 

 insensible) must be small compared with those retained. We therefore 

 reject these terms, as it is not our object to investigate that part of the 

 dispersion which may arise from them, but from other terms, namely, those 

 introduced into our equations, in consequence of the forces exercised by 

 the particles of matter on those of aether. 



