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BELL SYSTEM TECHNICAL JOURNAL 



normal in four dimensions. If the variables be taken in the order given 

 above the moment matrix is, from equations (A2-2) of Appendix II, 



M 



(4.2) 



where the b's are defined by the integrals in equations (A2-1). The inverse 

 matrix is 



M-=' 



B 



(4.3) 



which may be readily verified by matrix multiplication, and the determinant 

 I If I is B-. The normal distribution may be written down at once when 

 use is made of the formulas given in Section 2.9 of Reference A. The sub- 

 stitutions 



Q, 



Ic = R'cos e - Rsin 6 6' 



Ic = i^cos 6 



I, = Rsin e, Is = R'sin 6 + Rcos 6 B' 



dicdisdicdis = RHRdR'dSdS' 

 enable us to write 



b,{il + i\) -^ b,{i7 + I?) 



-2bi(lcls - IJc) = b2{R' - 2QRC0S 6 + Q^) 

 + bo{R'^ + R^'^) 



-2hiR^' + 2hiQ{R'sm 6 + RB' cos B), 

 Consequently the probability density of R, R', B, B' is 



p{R, R\ B, 6') = ^ exp |-^ [h,{R' - 2QRcos B + Q') 



-f bo(R" + R'd'') - 2biR'e' + 2biQ(R'sm B + i^^'cos B)] 



(4.4) 



(4.5) 



In this expression R ranges from to oc , ^ from — tt to tt, and R' and B' 

 from — 00 to -|- oo . The probability density for R Siud R' is obtained by 



