516 PROFESSOR KELLAND ON THE VIBRATIONS OF 



appears that the equations do not depend on the lave of force, but are equally 

 true in all cases. 



Before I proceed with any further discussion of these equations, I desire to 

 prove some important theorems relative to the values of the constants in symme- 

 trical media. 



THEOREM 1. Relation between the sums of powers of one co-ordinate, and pro- 

 ducts of powers of different ones. 



Let/ g, h be the co-ordinates of a particle; x-f, y-g,zh those of another 

 measured from it. Then 



Suppose the particle whose co-ordinates are/, g, h, to be moved to a point 



f+a, ff+/3, A + 7, then 2 ?</, becomes 2w0^(#-/ a) 2 + (# .? /3) 2 + (z-A 7) 2 . 

 Also ( a ._/_a)+(y-- j 8) s -(*-A-7) s =r s 



Let $ be the distance through which the particle is moved, and let 

 -g)P+(z-}i)y be denoted by e; then 



2 m $ r,=2 m </> Vr 2 -2e+8 2 



/2e-S 2 \ 2 

 = 2m0r + . . . +2mfrl - 5 \ + . . . 



Now this must be a function of $ : consequently 

 2&/,-e 2B must equal P 8 2n 



or 2 WZ / / r{a(^-/) 



P being some function of r. 

 By expanding each side we get 



Hence we obtain 



1.2 



