412 NEUMANN ON A METHOD OF ESTIMATING THE 
+ (A — A’)sin¢ = + A”sin >) 
(A + A’) cos ¢ = A" cos ¢! f a a ay 
C+C = GC 
and on account of the equality of the components of the pres- 
sure on the refracting plane being perpendicular to the plane of 
incidence, 
(Cc — ©) sin? ¢ cos $ = C" sin? 2 cos 4! 
A Xr 
sing sind! 
dey me 
(C — C’)sin ¢cos ¢ = C" sin g’cosg. . * (10.) 
The solution of the equations (9.) and(10.) reproduces Fresnel’s 
expression for the reflected and refracted magnitudes, supposing 
that the plane of polarization be allowed to pass through the 
ray and the direction of its vibrations. However, when a ray 
enters a lower refractive medium from a higher, and has passed 
beyond the limits of total reflection, i. e. when sin ¢! >1, cos ¢/ 
is imaginary. According to M. Cauchy’s observation, we must in 
this case substitute exponential magnitudes for sin (258 +) and 
or as 
cos (2228 *). We can satisfy the differential equations (3. and 
the third in 1.) by 
uw! = sin ge /—1cos 9! 
nv 
Lewis —— = ‘ etic % 
ane sin rr T 27+ B" cos =, ap) 2 
=e a. tiny J 
y! = VT cos gee Vt ost 
; . (11.) 
ern (ysing  ¢ mre pont 2). | 
x{B sin ( 7 ip) 27 —Al cos (— ip 2B 
wl = ea 2 —1 cos 4! 
SF 
eo oe & q! “ye i ne ¢! t i 
ae sin a F 2x%+D"cos i T oan 
Moreover, we know that in reflected light the component rays 
suffer a certain retardation, consequently we must add a member 
asngd+ysing ¢ 
to the equations (6.) in cos ee saptee 4 » so that 
