Length of a Wave, in the Undulatory Theory of Light. 9 
df (r) Inge Sd Ar 
Now,f(r+Ar) = f(r) += Ar FERp eT Ro gm 
consequently, 2 
(r+Ar) mein). (2 Ces oO : 
Pg Ast Aes at (Ce 2D) ar} 
(Ax + Aé) 
=O nay LO) a gy (one i) 
(ArAEt+AyAn+Az 8) Az, 
by substituting for Ax its value previously found. 
If in this expression we write ¢ (7) for a, wv (r) for 
(4 ote 
3 dy an and substitute it in the first of the equa- 
tions (2.), we shall have by virtue of the first of the equa- 
tions (1.), 
X=ms.{o(r)AE+ (r) (ArAE+AyAn+AzA$) 2}. 
The second and third of the equations (2.) are similar to 
the first, consequently if we transform them in the same man- 
vats f @E dy 
ner, and (by the principles of dynamics) put WP? dP 
oe. 
qe for X, Y, Z, we shall have 
23 = m=.{o(r)AE+U(r).(ArAg + AyAy + AzAS)Ar}, 
d*y 
ape E.{o(r)Ant(r). (AvAg+ AyAn+ AzAg)Ay}, (3.) 
os = mE Lo(r)AL+ (7) . (APE + AyAn) +AzAd)As}. 
From these general equations, a number of integrals, 
adapted to particular cases, may be found. Let us suppose 
the vibrations of the particles to be performed in straight 
lines, all in one direction. This is a case of polarized light. 
Let x be taken in the direction of the vibrations; then y and ¢ 
will be zero, and the first of the equations (3.) will give 
iM 
26 = mz. {o(r) + (rr) Aa} AE. (4.) 
Third Series. Vol. 8. No. 43. Jan. 1836. C 
