AS EXPLAINED BY THE HYPOTHESIS OF FINITE INTERVALS. 183 



Let „ e, be any values of , and ^, then the position of the nu- 

 merator, omitting the known factor, for these particular values of the 

 V and ^ IS 



This sum being taken for all values of ? from to infinity. But in 

 the limits of a wave, , and ^ will have equal corresponding values, so 



that there will be a term ^' ~ *>i^) m^ .. „ y 1 • , 



(p+,,= +Knl' *^^ ^' ^^'"S here a particular 

 value of »/. ' ^' 



And for the next wave the expression becomes v ('?■'- ^1') i^-%Y 

 which gives a result ^of the same form. Hence the sunf of 't'hlsf two 



*^™' '' ^ (m^T^ *''''''" °"^^ °" """^ ''^^' "^'""''^ ^^ ^" essentially 

 positive quantity; and this is true for every particular value of , and r 

 and IS therefore true for the sum of all the values; whence the nume- 

 rator above is a positive quantity. Similar to the corresponding 

 term of the denominator is ^^Jll±fLlS^L^, ^hich is also essentially 

 positive. * 



This result is necessary to the reasoning I adduced above, in order 

 to shew that the forces which the particles exert on each other are 

 attractive. 



I wisli it were in my power to offer any considerations relative to 

 the phenomena of polarization by reflexion from the surface of jjlass 

 and so on. There appears to be little doubt of the truth of the results 

 vv-liich have been deduced by M. Fresnel relatively to the coefficients 

 of the intensity of reflected and transmitted light produced by the 

 different vibrations. I cannot however think that the hypothesis of 

 the aether within the glass being more dense than that without in the 

 ratio of M : 1 IS altogether satisfactory, but I forbear making anv re 

 marks on that subject further than to shew what is the corresponding 

 relation of the densities deducible from our hypothesis 



