PROBABILITY THEORY AND TELEPHONE ENGINEERING 61 



In particular, the formulas (3*4), (35), (36) for the last three of the 

 "leading distribution parameters" of the original given chance- variable 

 Z are immediately obtainable by merely letting the point a (Fig. 12) 

 coincide with the center c; for then equations (42), (43), (44) reduce to 



z = Z - Zc = Z -Z, (45) 



t2 



_2 



= Z' - Z , (46) 



\z\^ = \Z\^ - \Z\\ (47) 



2.1. Proofs of Formulas (23), {25), {26) 



With W = U -\- iF here denoting any complex quantity,^ formulas 

 (23), (25), (26) will be proved by starting with the three identities^" 



2UV=\mW\ (48) 



2C/2 = |PF|2 + ReTF2, (49) 



2F2 = |PF|2 - RqW\ (50) 



In order to apply these identities in proving formulas (23), (25), 

 (26), which relate to Fig. 11, we evidently must identify the IF appear- 

 ing in these identities with the W in Fig. 11, and also must introduce 

 the relation existing between W and Z in Fig. 11, namely 



I'F = Zexp( -i^A). (51) 



To prove (23) we substitute (51) into (48) and take the mean value 

 of the result, thus getting 



2£7F= |Z^| sin(agZ^- 2^^). (52) 



For the general case in which jZ^j is not zero, this equation shows 

 that the necessary and sufficient condition for UV to be zero is that 

 -^A shall have any of the special values ^^' satisfying the following 

 equation, in which n is real: 



agZ"2 - 2^/ = WTT, {\n\ = 0, 1,2,3, • • •)• (53) 



^° These are equivalent to the identities 



i4:UV = W" - iv^, 



4:U' = W^ + W' + 2WW, 

 -4F2 = H'2 + #2 - 2WW, 



which are immediately obtainable from the pair of simpler identities 2U = W -^ W 

 and i2V = W ~ W. However, formulas (48), (49), (50) can be readily verified by 

 merely substituting W = U -\- iV. 



