TROPOSPHERIC PROPAGATION AND RADIO METEOROLOGY 165 
¢ CENTER .OF EARTH 
Ficure 39. Application of Snell’s law of refraction. 
Figure 39A. If there are several boundaries it is 
readily seen that Snell’s law generalizes (Figure 
39B) to 
No COS ao = M1 COS a1 = M2 COSaz = ---, 
and for a continuously variable layer it becomes 
nm COS a = No COS ao, 
where n and @ are continuous variables which are 
functions of the height and the index 0 designates an 
arbitrary reference level. 
Snell’s law for a curved earth may be derived from 
Figure 39C. For successive boundaries it is found: 
M Sin Bo = sin B’y, 
m sin By = nz sin B’,, etc. 
Multiply the first equation by 7, the second by 7, 
ete. Then 
Noro SIN Bo = Maro sin B'o , 
Mr) Sin By = ner; sin B's, etc. 
But from the triangle OAB 
sin B’) _ sin By 
See) SE aie,» 
viet To 
so that: 
Noro SID Bo = NM Sin By = Nore sin Bp = --- 
Again introducing the angle a with the horizontal 
and making the transition to a continuously variable 
refractive index gives 
mr COS a = Noo COS ao , 
which is the generalization of Snell’s law for a curved 
earth. rp may be chosen as any convenient height, 
say a for the surface of the earth or a + hi; for the 
height of the transmitter, and 7» is the corresponding 
value of n. 
