354? Mr. R. Moon on Polarization and Double Refraction, 



_ 4>?<.<|>(m + 2a))...4)(^ + 2(v-l)cu) . 



^''^~ <?^2«)...<f52(v-l)co • • ^^'' 



Expressing co in terms of K, K', we have vH =Z»'K — a'K'i, 

 ~ vH'* = iK— «K'i, and .-. vco = {b'f--bf)K-{a<f~af) K'«. 

 Let^,g' be any two integer numbers having no common factor, 

 which is also ft factor of v, we may always determine a, b, a'j i', 

 so that v«;=^'K— g'K'». This will be the case \fg=b'f—bf' 

 ^ = a'f—aj'. One of the quantities/,/' may be assumed 

 equal to 0. Suppose /'=0, then ^ = 6'/, g' = a'f; whence 

 ag — bg' = vf. Let k be the greatest common measure of ^, g'f 

 so that g = kgp g' = kg'i; then, since no factor of A^ divides v, 

 k must divide/, or f=kf,, hut g, = b' J, g'^ — a f, and a', b' are 

 integers, or /must divide g,^; whence / = 1, or/=/f. Also 

 ag^—bg', = v, where g, and g'l are prime to each other, so that 

 integer values may always be found for a and b ; so that in the 

 equation (I.), _gK — g'K'i 



g,g' being any integer numbers, such that no conimon factor 

 o^gig' also divides v. 



The above supposition,/' = 0, is, however, only a particular 

 one, omitting it, the conditions to be satisfied by a, b, a', b\ 

 may be written under the form 



aU — a'b = V, "I 



ag — bg' = 0[moA.v]i V (8.) 



a' g — b'g' ^0 [mod. v], J 

 to which we may join the equations before obtained, 

 vH = ^.'K-«'K',,-l 

 -vH'* = 6K-aK., /••••• (^') 

 which contain the theory of the modular equation. This, 

 however, involves some further investigations, which are not 

 sufficiently connected with the present subject to be attempted 

 here. 



LXir. On Polarization and Double Refraction. By R. 

 Moon, M.A.^ Fellow of Queen's College, Cambridge, and of 

 the Cambridge Philosophical Society*. 



TN a paper published in this Journal some months agofj I 

 *- endeavoured to give a popular explanation of the phjeno- 

 mena of diffraction, and I attempted to show that those phae- 

 nomena may be accounted for on the hypothesis, that the 

 waves composing common light consist of recurring cycles of 

 waves, the individuals of each cycle being related in a certain 

 manner to each other, and occurring at regular finite inter- 

 * Comrrmnicated by the Author. f S. 3. vol. xxiv. p. 81. 



