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BELL SYSTEM TECHNICAL JOURNAL 



where Do = \, Di = an , Dr,r = Dr-i , and Drs is the cofactor of a^r (or 

 of Ors because they are equal) in Dr : 



D. = 



ail ai2 • • • air 



air • • • Clrr 



hr = [Dr-iDrr'\ 



then, if none of the Dr's is zero, 



n 



2 arsXrXe = yl + yl + • • • + /„ 

 1 



From (3.5-2); the Jacobian d{xi , • • • x„)/d{yi , • • • >) is equal to D~^'^. 

 Applying our transformation to the exponent: 



xi = yi — aU^^'^yi 



a;2 = + D^^'^y^ 



Di= \ - a 



Since Xi runs from to oo so must y<i . The expression for X\ shows that yi 

 runs from a U^^''^y2 to oo . The intesrral is therefore 



J = DT'" [ dy2 [ 



dyi 



We now change to polar coordinates: 



yi = P cos £ 



yi = P sin 6 

 yz >0 gives < < tt 

 yi ^ aDJ^'^yi gives cot 6 > aD^^'^ 



dyi dyi = p dpdd 



and obtain 



/ 



JO 



r - 2 



dd j pe " dp 

 Jo 



= ^D7"' cot-^ iaD^'^') 



where the arc-cotangent lies between and tt. This may be written in the 

 simpler form 



T l/i 2\— 1/2 —1 1 



J = ^(1 — a ) cos a = ^ip CSC cp 

 where 



a = COS <p, 

 it being understood that < (p < r. 



