162 Mk KELLAND, on the DISPERSION OF LIGHT, 



+ 2 .^^ SxSy { - 2(i sin^ M^ + 6 sin {ct-kp) sin kSp] 



4. 2 :^^^ Sx^!/ { - 27 sin' —^ + c sin (c#- *p) sin kSp] ; 



now it is manifest that ^(p{r) sin^^f> = 0; and also because sin {ct-kp) 



Fir) 

 is independent of 2, and the term 2 —7-' SxSij sin k^p is of an odd order, 



wherever there is a positive term, there will be a corresponding ne- 

 gative one: the whole expression denoted by the symbol 2 in this 

 term will therefore be identically equal to zero. 



Precisely the same reasoning applies to all. similar expressions. 



F(r) . . . „ kSp , . . . J 

 Further as regards the term 2 —p^ SxSy snr -^, bearing m mnid 



the remark above made with respect to p, it is manifest that the part 



2 ■' sin^ — - Sy will have one and the same value, for two equal 

 r 2 



values of Sx with different signs: thus, to assist the conception refer- 

 ring to the cube as at the commencement of this paper, the point P 

 being at its centre, suppose a particle at each of the two upper cor- 

 ners of the face on which you are looking, and 1/ vertical, then the 



expression 2 — — sin' — ^ Si/ is the same for each of them, but Sx in 



one corresponds to —Sx in the other, and the sum of the above func- 

 tion for two such particles vanishes. It is clear, therefore, that this 



expression iSxSy — ^ sin' — ^, and all analogous ones are identically 



equal to zero. 



Our equations then become much reduced, and assume - tlie re- 

 markably simple form, 



^ =- a22{0. +^^Msin'*^ = -;ra, 

 df <^ r ' 2 



5|._/322{0. + ^a2^}sin'*^ = -.'^, 



