10 Mr. Tovey on the Relation between the Velocity and 
Now, let £ be a function of x and ¢, then for AE we may 
PE Az? dE Ag aE det 
dz ae) ae dB 23 “<d#° 93.4. 
If we substitute this value of A £ in equation (4.), and sup- 
pose the particles to be so arranged in their state of equili- 
brium that for every particle on one side of m, within the 
sphere of its influence, there is another at an equal distance 
on the opposite side, we may divide the sum © into two parts, 
one comprehending the positive values of Az, and the other 
the negative; and for every term in one part we shall have a 
term equal to it, and involving the same power of A z, in the 
other. But in the one part all the terms involving odd powers 
of A z will be positive, and in the other all negative; conse- 
quently all these terms will vanish: the other terms being 
all positive, the sum of one part will be equal to that of the 
other. Consequently equation (4.) will become 
d? ad? 4 2 
a2 = mE (r+ 0 ("Aa pa2(TE as ss - =a) 
the sum, in this equation, extending only to the positive 
values of Az. 
For m=. {$(r)+(r) Az*} Az*, write s*, and for m=. 
{> (r)+(r) Ax*} Azt write s’*, then the last equation will be 
2 2 4 
ae ae ey th (5.) 
t dz dz 
If we omit the last term, this equation becomes exactly of 
the same form as that for the transmission of sound, and gives 
then no relation between the length and velocity of the waves. 
But if we integrate the equation as it is, we shall obtain a re- 
lation of this sort; and this relation will afford a theory of | 
the dispersion of light. 
As § is a function of z and ¢, it may be expressed by a 
series of terms such as p sin t+ q cos nt, where p and g are 
functions of z, and ” a constant quantity*. Suppose then 
& = psinnt+qcosnt; substitute this value of £ in equa- 
tion (5.), and it will become 
Il 
© 
+(9 eee ea et aS od \ ee 
q * dz Sait, =e, 
* Poisson, T'raité de Mécanique, No. 514, edit. 2. 
