66 BELL SYSTEM TECHNICAL JOURNAL 



where we have integrated by parts in getting ^^tii • Setting r = and using 

 ^0 = gives 



h'ni = hm ==0 



In order to obtain the matrix M of the second moments /Xrs in a form 

 fairly symmetrical about its center we choose the 1, 2, 3, 4 order of our 

 variables to be ^i , tji , 772 , ^2 . From equations (3.4-13) etc. it is seen that 

 this choice leads to the expression (3.4-2) for M. 



When we put ^1 and ^2 equal to zero, we obtain for the probability density 

 function in (3.4-12) the expression 



M 



1-1/2 



47r2 



exp - , , (M22171 + 2Mnriiy]2 + M33772) 



Because of the symmetry of M, M22 is equal to M33 . When, in the integral 

 (3.4-12) we make the change of variable 



r M22 Y' r 



Mo 



1/2 



we obtain 



dXl dX-l \ M f'- r" r" _j,2^y2j^2(M2zlM-22)o:y 



M22 



/ X dx I dy ye 

 Jo Jo 



The double integral may be evaluated by (3.5-4). Let 



^ = cos-M - ^M = cot-' i-H), H = M^slMh - MhV'" 

 \ M22/ 



where H is the same as that given in (3.4-2). Our expression now becomes 

 dxidx2 I M \'i' 



-4^ 7^ir=^3 ^' + ^ '°' ^-""^^ 



From a property of determinants 



M22M33 - M23 = I M I (l/'o - lAx) 



Using this to eUminate | M \ and dividing by 



27r L '/'o J 



which, from (3.3-10), is the probability of going through zero in .vi , ;vi + dxi 

 with positive slope, gives the probability of going through zero in dxo with 



