4 EIGHTH REPORT — 1838. 



tical law of dispersion, which was first discovered by Cauchy, namely, 

 the expression 



v' = Ao- A,X-2+ A.jX-*&c. ' (11.) 



in which 



' 1.2.3.4..... (2e + 2) Y ^ ■> ^ 2r ^ ^ '' J ^ ^ 



But, in order that this law may agree with the phenomena, it is es-- 

 sential that tlie series (11.) should be convergent, even in its earliest 

 terms ; and this consideration enables us to exclude the supposition 

 which has occurred to some mathematicians, that the particles of the 

 ether attract each other with forces which are inversely as the squares 

 of the distances between them. For if we suppose rf(r) — r'^, and 

 therefore/(r) = r-^,f' (r) = — 3 r-^ we shall have 



A.= i 



(2.y 



J — r-^+Sr-s Aa.'2J 



^ 1. 2.3 A.... (2 i + 2) 

 Aa;2' + 2; (13.) 



and by extending the summation to particles, distant by several times 

 the length of an undulation from the particle which they are supposed 

 to attract, these sums (13.) become extremely large, and the terms of 

 the series' (11.) diverge very rapidly at first, though they always finish 

 by convergino-. In fact, if we conceive a sphere, whose radius = « \ 

 = n times the length of an undulation (n being a large multiplier), 

 and whose centre is at the attracted particle ; and if we consider only 

 the combined effect of the actions of all the particles within this sphere, 

 we may, as a good approximation, convert each sum (13.) into a triple 

 definite' integral, and thus obtain, for the general term of the series 

 (11.), the expression 



(—l)'4!7rmn"-\'^ (2-n-nY* /, , -, 



(- ly A.-X— = ^— (,^ ^ ,) ,3 • ,,^,L (2i + 3 ) ' ^''-^ 



€ being the mean interval between any two adjacent particles of 

 the ether, so that the number of such particles contained in any 



sphere of radius r, is nearly = -^, if r be a large multiple of e. 



And hence we find, by taking the sum of all these terms (14.), the ex- 

 pression . 



o X''"* r 1 . cos.2 7r?^ sm. 2 tt yt \ _ . . 



^ =-^ |3 + 727^ - 12^^ I • ^'^•> 



so that, by taking the limit to which v'^ tends, when n is taken greater 

 and greater, we get at last as a near approximation 



t'^=^. (16.) 



3 rrc' 



and 



^ = a/— ""• (17.) 



t> V m 



