476 BELL SYSTEM TECHNICAL JOURNAL 



curves with the sun having been presented as if for its own sake. 

 Much work in the field does stop at this point, and is not without 

 value in spite of its stopping there. The theorist indeed may care 

 for an {h' ,f) curve only as material for deducing the {N, z) curve — 

 the distribution-in-height of the ions — to which he aspires. It is, 

 however, fortunate that this is not the only value of the {h' ,f) curves, 

 for as we now shall see, the derivation of the {N, z) curves from them 

 is full of difficulties. 



Perhaps the greatest of these difficulties springs from the dependence 

 of the signal-speed on N, which is to blame for the difference between 

 virtual height h' and true height h. Even if it is fully justifiable to 

 identify signal-speed with group-speed v, the difficulty is not banished. 

 It resides in the fact that Qi' — h) depends not on things already 

 known but on the very thing one is striving to find out, to wit, the 

 distribution of ion-density in the ionosphere. The value of h for any 

 particular h' depends indeed not on the value of N at that height 

 alone, but on the values of N at all inferior heights. The problem 

 is somewhat like having to solve for x an equation in which x appears 

 badly entangled on both sides of the equals sign. The mathematical 

 technique is difficult and approximative. 



Another major difficulty resides in the fact that when N varies by 

 an appreciable fraction over a distance equal to a wave-length of the 

 waves, the consequences of the theory become a good deal more 

 complex than those embodied in the simple equations (4) and (5). 

 For instance, partial reflection may occur at a level where N is rising 

 rapidly, though as yet far below the mirror-density Nc. One cannot 

 therefore say that whenever an echo is observed on a frequency /, 

 there must somewhere exist an electron-density related to / by (5). 

 The literature is full of allusions to mystifying echoes, some of which 

 are ascribed to partial reflection. Figure 2 shows an £-echo and an 

 F-echo received from the same signal; and many a {h',f) curve shows 

 the ^-branch running along for quite a distance underneath the 

 jp-branch, instead of stopping at just the abscissa where the F-branch 

 begins.^ Yet on the other hand, the £-layer may be denser at some 

 places than it is at others of equal altitude, and parts of a signal may 

 be reflected from the places of high density while other parts slip 

 between these and go on to the F-layer. The assumption that N 

 depends on z only, which hitherto has been taken for granted in this 

 paper as it is in most theory, is in fact very assailable; and people are 



^ This is so well-known a phenomenon that lengthy papers have been written 

 about it under the name of "abnormal £-ionization," though it seems too common 

 to deserve the adjective "abnormal." 



