554 Mr. Moon on Fresnel's Tlieoi-y of Double Refraction. 



vibrations is due to Young only or not, 1 pledge myself to 

 prove that the researches of Fresnel have not advanced us 

 one step beyond it. 



Fresnel's first step in the mathematical part of his theory, is 

 to prove the existence of the axes of elasticity : " a proposi- 

 tion," says Mr. Smith (Cambridge MathematicalJournal,vol.i. 

 p. 1), "on which the whole theory of double refraction de- 

 pends, and which Fresnel has proved by a metliod which has 

 the advantage of geometrical distinctness, but which is long 

 and rather difficult to follow out on that account." On the 

 same account I shall prefer to give Mr. Smith's elegant ana- 

 lysis in preference to the cumbrous processes of Fresnel, trust- 

 ing my readers will take my word for it, that whatever it may 

 want in "geometrical distinctness," it gains in logical clear- 

 ness. 'jV'j fj _ ^i '^ ^ _ ^1.1^ 



" The proposition is thu§ state¥^-Irf any s'^tem of parti- 

 cles acting on each other with forces which are functions of 

 their mutual distances, there are three directions at right an- 

 gles to each other, along which if a particle be displaced, the 

 forces of restitution will act in the same direction, u-iuur: l>ii« 



"Let xy zhe the co-ordinates of the attracted point a?ij/]Z, ; 

 •^23/2 ^i '" be the co-ordinates of the attracting points ; r^ r^r^ 

 ... the distances between the attracted and attracting points; 

 ^\ (^i) <^2(^2) ^3 (^3) ••• ^^^ attractions; XYZ the total resolved 

 forces along the axes, then we sUall hjify^^gg ^^ ^j- ^^^^^ ^^^ 



.--(- 



\o 90idt X'== —- 4>i (^1) + -\ — ♦a (^2) + ^^iol« jisdJo 



1 '2 



ill zio& noituMizii't 



and similarly for Y and Z. Now let j ^?i jdb) sdj juS 



•i-iL 



in /^ _ __ 



J«^9em aoiiuihzo-t ')o ooioi 3di dbrna ^'f9'/ ad oi r- — '7 



rtnoo oi oldiMii^ lk^ m Zd^iTlbi([ I '.\n oJ lunoij-joqo'i* 

 ^^itinib^o-c■■J oiW ddirf/fful) ,!)np>t?/f9l)nu r»b I siffj iud j Inij-A^iq 

 moil ^^ov ,L»tobyrfK3 S"'-' '\#h^iPte'piif«iite^4«'»«q ^i*' ^'^ 



Mfbo odi loY = -7— >s?):P»,>t equilibrio. '^. f^S:!^^ oJ ': 



oit .a:8 ,5,5: feV , ]\ V. "no/ (u. ;;. -.^.. .ijom arli I0 eoL... ... , 



■; .3i8 ^$ + Z = ^ i*s C^- ^ V^ «i- S + I- «^ + ilfi'^^ +'r^ 



" Let the particle recelVe fe' stnall displacement, the projec- 

 tions of which on the co-ordinate axes are ScT, Sj/, '6z. Then 

 supposing the displacement to be very small, the force of re- 

 stitution may be taken as proportional to it, so that we have 



