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BELL SYSTEM TECHNICAL JOURNAL 



P{p I 0-34) and P{(T I p3i), defined respectively by (1.4) and by the correlative 

 of (1.4), can be expressed as 'single integrals,' as follows*: 



P(p I as,) = f * P(p,a) da, (1.6) P{a \ ps,) = H P{p,a) dp. (1.7) 



(?(P34 , (T34), defined by (1.5), can be expressed as a 'double integral,' funda- 

 mentally; but, for purposes of analysis and of evaluation, this will be replaced 

 by its two equivalent 'repeated integrals': 



Q(p3i , Cr 3i) 



f 



P{p,a) da 



dp 



= X^ I j ^(P.<^) dp\da, (1.8) 



the set of integration limits being the same in both repeated integrals 

 because these limits are constants, as indicated by Fig. 1.1. On account 

 of (1.6) and (1.7) respectively, (1.8) can evidently be written formally 

 as two single integrals: 



Q(P34, ^34) = / P(p 1 a34) dp = / P{a\ P34) da, (1.9) 



but implicitly these are repeated integrals unless the single integrations in 

 (1.6) and (1.7) can be executed, in which case the integrals in (1.9) will 

 actually be single integrals, and these will be quite unlike each other in 

 form, being integrals with respect to p and a respectively — though of course 

 yielding a com.m.on expression in case the indicated integrations can be 

 executed. 



The particular cases of (1.4) and (1.5) with which this paper will chiefly 

 deal are the following three: 



p{p < p' <p + dp, a, <a' < a^) = P{p | a^:) dp = P (p) dp, (1.10) 



Pipi <p' <p,a,<a' < a.) = Q{< p,a,o) ^ Q{p), (1.11) 



p{p <p' <p2,ai<a' < 0-,) - Q{> p,an) = (?*(p). (1.12) 



^ The single-integral formulation in (1.6) can be written down directly by mere inspec- 

 tion of the left side of (1.4). Alternatively, (1.6) can be obtained by representing the left 

 side of (1.4) by a repeated integral, as follows: 



Pip I (^34.) dp = 



pp-\dp P r'Ci 

 •' P L"'''3 



Pip, a)da 



dp = 



f Pip, <T)da 



dp, 



whence (1.6); the last equality in the above chain equation in this footnote evidently 



results from the fact that, in general 





fix)dx = f(x)dx, since each side of this equa- 



tion represents dA, the differential element of area under the graph of /(.v) from x to 

 X -f dx. 



