Theory of the Dispersion of Light. 307 
which, merely by the assumption of the coefficients and trigo- 
nometrical operations, is easily put under the form 
2a 
7 = % +7, COS es (u2—1+ 0), (3.) 
t) being entirely arbitrary, and 7 4, being also arbitrary, but 
small ; 7) is introduced only for greater generality. 
Differentiating in respect of ¢, we shall have 
@ Qn\2 2: 
oy =— (=) 4, COS (= (e—t+4)). (4) 
Also, by the method of finite differences, we have 
2 2 
—2 xn, cos (= (ea—t+t) ) (sin res) 
Ayj= bide erure (5.) 
— sin (= (we—-t+t,) (sin tad hv *) 
a 
Now (on precisely the same grounds as those adverted to 
in the analysis of Cauchy for deducing the equations (22.),) 
it will be seen that this expression is of such a form that if it 
were introduced in a summation, since we may assume half 
the values of A x as positive and half as negative, the second 
member involving the first power of the sine of a function of 
A, and the first member the square, the sums of all the 
values in the second member will destroy each other, but not 
those in the first. 
Thus on substituting this value of A in the differential 
equation (12.), or that above, (1.), we shall only have to take 
into account the first member, multiplied by the function of 
(vr); and it will thus easily appear that that equation (1.) is 
satisfied by these values derived from the assumed equation 
of the wave (3.), provided we suppose 
(22) = 8 {2m LO + PASE (ain HAF) 1, (6, 
or, in other words, the equation (3.) coupled with this last 
condition (6.) is a particular solution of the differential equa- 
tion of the motion of a system of molecules (1.). 
But also, this equation (6.) involves the relation between r 
- A 1 2 
and », (or between A and p, since we have — =—,) which 
y- ee pia? 
is expressed by writing, for abridgement, 
